Embed Size (px)
Citation preview
1
FRACTIONAL ORDER CONTROLLERS AND
APPLICATIONS TO REAL LIFE SYSTEMS
By
SHANTANU DAS Reactor Control Division BARC
ENGG01200704021
CI: Prof. Dr M.S. Bhatia LASER & PLASMA TECHNOLOGY DIVISION
BARC
A thesis submitted to Board of Studies in Engineering Science In partial fulfillment of the requirements
for the degree of DOCTOR OF PHILOSOPHY
of HOMI BHABHA NATIONAL INSTITUTE
August 2007-2013
2
Recommendations of viva voce board
As members of Viva Voce board, we certify that we have read the dissertation prepared by Sri Shatanu Das ENGG01200704021 entitled “FRACTIONAL ORDER CONTROLLERS AND APPLICATIONS TO REAL LIFE SYSTEMS” and recommend that it may be accepted as fulfilling the dissertation requirement for the Degree of Doctor of Philosophy. -----------------------------------------------------------------------------------------Date: Chairman -----------------------------------------------------------------------------------------Date: Guide/Convener -----------------------------------------------------------------------------------------Date: Co-Guide-1 -----------------------------------------------------------------------------------------Date: Co-Guide-2 -----------------------------------------------------------------------------------------Date: Co-Guide-3 -----------------------------------------------------------------------------------------Date: Co-Guide-4 -----------------------------------------------------------------------------------------Date: Member (1) -----------------------------------------------------------------------------------------Date: Member (2) -----------------------------------------------------------------------------------------Date: Member (3)
Final approval and acceptance of this dissertation is contingent upon the candidate’s submission of final copies of dissertation to HBNI
I hereby certify that I have read this dissertation prepared under my direction and recommend that it may be accepted as fulfilling the dissertation requirement.
Date: Place:
3
STATEMENT BY AUTHOR This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at Homi Bhabha National Institute (HBNI) and is deposited in the Library to be made available to borrowers under rules of HBNI. Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgement of source is made. Request for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the Competent Authority of HBNI when in his or her judgment the proposed use of the material is in the interest of scholarship. In all other instances, however, permission must be obtained from the author.
SHANTANU DAS
4
DECLARATION I, hereby declare that the investigation presented in the thesis has been carried out by me. The work is original and not has been submitted earlier as a whole or in part for a degree/diploma at this or any other Institution/University.
SHANTANU DAS
5
ACKNOWLEDGEMENT I take this opportunity to thank all people who have supported and helped me in pursuing my PhD programme. I would like to thank my guide Dr. M. S. Bhatia, Professor HBNI, Laser and Plasma Technology Division BARC, and my Departmental coordinator Mr. B. B. Biswas Head of Reactor Control Division who has been a constant source of inspiration, guidance and support during the programme tenure. I would like to express my gratitude to Dr Srikumar Banerjee Chairman AEC and Dr Ratan. Kumar Sinha Director BARC, for their guidance, and encouragement to write a book, on new topic, published, by Springer Verlag Germany namely Functional Fractional Calculus for System Identification and Controls, (280 pages) published on 16-10-2007, and II-edition namely Functional Fractional Calculus (612 pages), published on 1-06-2011; the same are used worldwide. I acknowledge contribution of, the students Mr. Suman Saha, Mr Saptarishi Das, Mr. Abhishekh Choudhury, Sri Indranil Pan Sri Basudeb Mazumder and Sri Sumit Mukherjee of Department of Power Engineering University of Jadavpur; Sri Subrata Chandra, Ms Moutushi Dutta Chaudhury and Ms Soma Nag of Department of Physics University of Jadavpur,, Ms Rituja Dive of VNIT-Nagpur, Sri Tridip Sardar of Heritage Institute of Technology Calcutta, Sri Jitesh Khanna and Sri Vamsi of IIT Kharagpur who have contributed for development of on Fractional Calculus along with me. The professors, I humbly acknowledge are Prof. Mohan Aware (VNIT), Prof. Ashwin Dhabale (VNIT), Prof. Sujata Tarafdar, Prof. Amitava. Gupta (Univ. of Jadavpur), Prof. S Sarkar, Prof U. Basu (Univ. of Calcutta). Prof. S Sen , Prof. K. Biswas (IIT Kharagpur), Prof S Saha Ray (NIT Rourkella) and my inspiration Prof. Rasajit Bera (Heritage Institute) pioneer in ADM method; to have given me patience hearing to my ‘absurd’ ideas be it on engineering aspect, be it on mathematics aspects, be it on physics aspects of Fractional calculus. I take this opportunity to thank Professors of Calcutta University, and Jadavpur University for instituting this subject Fractional Calculus as formal course for M. Phill, Ph. D and Masters Students; and have given me opportunity to teach this subject at University class rooms in detail. These are first Universities to try to induct this subject formally. . Place: Mumbai Shantanu Das, October-2013
PhDNo.ENGG01200704021
http://scholar.google.co.uk/citations?user=9ix9YS8AAAAJ&hl=en www.shantanudaslecture.com
6
CONTENTS Abstract
List of Figures
Chapter-1
Figure-1: Dividing the function interval into small slices of
Figure-2: Plot of half-derivative of xe and xe .
Figure-3: Number line & Interpolation of the same to differintegrals of fractional calculus.
Figure-4: Fractional differentiation Left Hand Definition (LHD) block diagram.
Figure-5: Fractional differentiation of 2.3 times in LHD.
Figure-6: Block diagram representation of RHD Caputo
Figure-7: Differentiation of 2.3 times by RHD.
Figure 8: Step response of the system for different values of n using a=b=1 and y (0) =0.
Figure 9: Step response for different values depicting increased damping for greater values of a.
Figure 10: Step response for different values of parameter b.
Figure 11: Effect of initial conditions on a system with n =1, 1.75 for the step response
Chapter-2
Figure-1 Circuit for constant current discharge method
Figure-2 Voltage characteristic between capacitor terminals.
Figure-3 Constant current (50 mA) charge-discharge pattern of 10F, 20 F and 25 F aerogel
supercapacitors, studied by using Super Capacitor Test System.(Courtesy CMET Thrissur),
Figure-4 Voltage profile of charge discharge of super capacitor considering fractional order
impedance in super-capacitor.
Figure-5 The super-capacitor construction.
Figure-6 The SEM image of super-capacitor electrode showing roughness & porous nature
(Courtesy CMET Govt. of India Thrissur, Kerala).
7
Figure-7 Distribution ( r ) of aggregate pores of several sizes, on the electrode surface.
Figure-8 Charge distribution at cleavage of electrode crystal and formation of double layer
capacity.
Figure-9 a) Showing distribution of pores size, b) Corresponding distribution of capacity.
Figure-10 Depicting circuit picture of a rough disordered electrode.
Figure-11 Impedance Spectroscopy showing Warburg Region of Super-Capacitor.
Figure- 12 The constant voltage charging of super-capacitor.
Figure-13 Constant voltage charging and discharging voltage profile at super-capacitor
Chapter-3
Figure-1a A snapshot of the film (inner blob) superposed on the photograph of the film taken about 2 s
earlier (outline visible along the periphery) shows the shrinking of the film. The colors have been
adjusted for clarity. Courtesy Dept. of Phys; University of Jadavpur Kolkata.
Figure- 1b An area-time plot (castor oil on perspex). Courtesy Dept. of Phys; University of Jadavpur
Kolkata.
Figure-2 The non-Newtonian area-time plot. Courtesy Dept. of Phys; University of Jadavpur Kolkata.
Figure-3 Plot show modulus of response function high passes characteristics when the order
distribution function is 0( ) ( )A z z z for 0z as fractional order of 0.2, 0.4, 0.6, and 0.8.
Figure-4 Plot show modulus of response function high passes characteristics when the order
distribution function is ( )A z h and with lower and upper limits of integration on the z .
Figure-5 Plot show modulus of response function low passes characteristics when the order
distribution function is 0( ) ( )A z z z for 0z as fractional order of 0.1, 0.3, 0.5, 0.7 and 0.9.
Figure-6 Plot show modulus of response function low passes characteristics when the order
distribution function is ( )A z h and with lower and upper limits of integration on the z .
Figure-7a Time domain presentation of the network induced stochastic delay.
Figure-7b Power Spectral Density of Network Delay.
Figure-8 Picturing the randon network delay via shot noise driving the fractional Langevin equation.
Figure-9 Diverging run-time variance of the network delay data (of figure-7).
8
Chapter-4
Figure-1 Block diagram showing decomposition and solution of second order differential equation,
Figure-2 Block showing solution of first order differential equation by decomposition
Figure-3 The RC circuit (a first order differential equation), with semi-infinite cable as fractional half
order element.
Figure-4 Block showing solution of first order differential equation by decomposition in presence of
fractional half order term.
Figure-5 Block diagram showing solution of by decomposition of a second order differential equation
in presence of fractional order term.
Figure-6 The oscillator circuit (a second order differential equation), with semi-infinite cable CRO-
probe acting as half order element.
Chapter-5
Figure-1 Bode plot of an FPP with slope of -20mdB/dec and its approximation as zigzag straight lines
with individual slopes of -20dB/dec and 0dB/dec.
Figure-2 Choosing the singularities for approximation by assuming a constant error between the -20
dB/dec line and the zigzag lines.
Figure-3 Showing expanded view of shaping of fractional pole by series of poles and zeros, with in
y dB error.
Figure-4 Bode plots of the transfer function C*(s).
Figure-5 Bode plots of the transfer function C(s).
Figure-6 Compensated Gain plots.
Figure-7 Compensated Phase plots.
Figure-8 Bode response for rational approximation of the fractional Lead Compensator.
Figure-9 Bode Response for error tolerance = 1dB for fractional order PID.
Figure-10 Asymptotic and exact phase plots illustrating the basic idea is drawn for req=-45o,
l=1rad/sec and h=1000rad/sec.
Figure-11 Fractional Order Impedance.
Figure-12 Practical circuit for semi-integrator.
9
Figure-13 Error decrease with increasing μ for various α.
Figure-14 Bode Plot for α=+0.5.
Figure-15 Phase plot for req=45o on (100, 10000) rad/sec, with ε 1o
Figure-16 Circuit diagram for implementing series FO Impedance.
Figure-17 Input and output waveform for α=+0.5.
Figure 18 Single port first order network with one zero.
Figure-19 Two port network and its transfer function.
Figure 20 CRO testing results of implemented Fractional Integrator Circuits.
Figure-21 Ideal Bode’s loop.
Figure-22 Gain and Phase plots for open loop ideal TF ( )L s .
Figure-23 Magnitude and Phase plot of PID controller ( )cG s with 1.0p i dk k k .
Figure 24 Gain and phase plot of PID controller ( )cG s with 1pk , 0.5ik and 1dk .
Figure-25 Magnitude and phase of Fractional Order PID controller with 1p i dk k k and
0.5 .
Figure-26 Comparison of PID and Fractional Order PID for degrees of freedom (a) Integer Order PID
and (b) Fractional Order PID.
Figure-27 Structure of Fractional Order PID connected to a plant.
Figure-28 Bode plot showing Iso-damping.
Figure-29 Nyquist Plot showing Isodamping.
Figure-30 Control system representation for sensitivity functions definitions.
Figure-31 Plant control system with fractional order integrator (phase shaper) as extra compensator
apart from conventional PID.
Figure-32 Isodamping lines in complex plane, with gain variation.
Figure-33 Bode plot showing Isodamping possible for few gains spreads in actuality.
List of Tables Chapter-1 Table-1 Cumulative sum showing 1st, 2nd , and 3rd integration of function for six points
10
Chapter-2 Table-1 Discharge conditions.
Chapter-4 Table-1 Decomposing the action reaction of second order mass spring system
Table-2 Modal force and displacements for second order system with fractional order damping.
Chapter-5 Table 1 Error tolerance and Order of Rational Approximation.
Table 2- Calculated values of R-C components
11
Contents in the Chapters Chapter 1: Introduction
Birth of Fractional Calculus
Let us try to find L’Hopital’s query
Paradox stems from unification of differentiation and integration which makes the differentiation a
non-local operation.
To get formulation for generalized derivative from first principle.
Defining factorial and binomial coefficients for non-integer and getting series formulation for
fractional derivative.
Try to answer L’Hopital’s question now!
Half derivative of a constant and other functions-and paradoxical case.
Fractional Derivatives with lower limit to minus infinity.
Repeated integration approach to get fractional derivative.
The Laplace Transform and Fourier Transform of Fractional derivative.
So what is fractional Calculus?
Fractional Derivatives Riemann-Liouville (RL) Left Hand Definition (LHD)
Fractional Derivatives Caputo Right Hand Definition (RHD).
Grunwald-Letnikov definition
Significance of non-integer order systems.
Applications of fractional calculus.
Fractional Differential Equations.
Conclusions
Chapter-2 Fractional Calculus approach to view anomalous charge discharge in
super-capacitor.
Introduction.
IEC-62931 Standard to test super capacitor-2007.
12
Actual Observed Voltage Profile for constant current charging/discharging and anomaly with IEC -
62931 standard.
Impedance Representation of super-capacitor.
Constant current charging-discharging current excitation and its voltage profile for determining super
capacitor parameter.
Revised Test Procedure.
Calculation of time at which power output of super-capacitor goes to zero.
Input Output Energy and Efficiency of Energy transfer.
Introducing Anomalous Transport Mechanism inside super-capacitor.
The disorder and its ordering for the porous electrode of super-capacitor by power law distribution.
The charge distribution & formation of electrochemical double layer capacity (EDLC). Calculation of
Capacity.
Distribution in capacity as power-law.
Debye and Non Debye Relaxation.
Impulse response function and impulse response for super-capacitor with disorder in porous electrode.
Appearance of Fractional derivative-in disordered electrode of super-capacitor.
Implication of Fractional Impedance.
Constant Voltage charging & discharging for determining the super capacitor parameters. Conclusion.
Chapter-3 Application to Real Life Physical Systems
Introduction
Spreading of viscous fluid and fractional calculus.
The anomalous behavior and fractional calculus.
Extension of fractional Calculus to continuous order differential equation systems.
Solving the continuous order differential equation.
Mechanism of random delay in networks of computer.
Random Delay a Stochastic Behavior.
About Levy distribution.
Fractional Stochastic Dynamic Model. Fractional Delay Dynamics.
The Random Dynamics of computer control system.
13
Conclusions.
Chapter 4 Solution of Generalized Differential Equation Systems
Introduction
Generalized Dynamic System & Evolution of its solution by principle of Action-Reaction
Generalization of Fractional Order Leading terms in differential equations formulated with Riemann-
Liouvelli and Caputo definitions and use of integer order initial/boundary conditions with
decomposition method.
Proposition.
Physical Reasoning to Solve First Order System and its Mode Decomposition.
Physical Reasoning to Solve Second Order System & its Mode Decomposition.
Adomian Decomposition Fundamentals and Adomian Polynomials.
Generalization of Physical Law of Nature vis-à-vis ADM.
ADM Applied to First Order Linear Differential Equation and Mode-Decomposition Solution
ADM Applied to Second Order Linear Differential Equation System and Mode-Decomposition.
ADM for First Order Linear Differential Equation System with Half (Fractional) Order Element and
Mode-Decomposition.
ADM for Second Order System with Half Order Element and its Physics.
Application of Decomposition Method in RL Formulated Partial Fractional Differential Equations
Linear Diffusion-Wave Equation and Solution to Impulse Forcing Function. Application of
Decomposition Method in RL formulated Fractional Differential Equation (Non-Linear) and its
solution.
Conclusions.
Chapter 5. Realization of Fractional Order Circuits and Fractional Order Control
System.
Introduction.
Singularity Structure for a single Fractional Power Pole (FPP).
14
Geometrical Derivation of recurring relationship of Fractional Power Pole for fractional integration.
Singularity Structure for a Single Fractional Power Zero (FPZ).
Fractional Order Integrator and its Rational Approximation.
Fractional Order Differentiator and its Rational Approximation.
The Phase Shaper of fractional order.
Illustrative Example.
Fractional Order Lead Compensators and its rational approximation.
Fractional PIλDµ Controller and its Rational Approximation.
Illustrative
Example of Rational approximation of Fractional Order PID.
Hardwire Circuit Technique to realize Fractional Order Elements.
Getting the Rational Approximation of s .
Pole zero calculation.
Fractional Order (FO) Impedance.
Algorithm for Calculation for Pole-Zero position of FO-impedance.
Design and Performance of FO.
Implementation of FO impedance.
Realization of impedance function by Analogue Network.
Impedance function of Single Port Network.
Impedance Function for a Two Port Network.
Improved Two Port Network.
Bode’s Ideal Loop.
Integer Order PID controllers.
Fractional Order PID Controllers.
Parameters for Tuning of Controllers.
Tuning of Fractional Order PID.
Isodamping a plant having integer order PID tuned system by topping with ‘fractional’ phase shaper.
Conclusions
References
15
Abstract
The topic selected for this PhD study is based on ‘Fractional Calculus’. Though it was first
discussed three hundred years ago by L’Hopital and Leibniz in 1695; yet is only finding
application recently for description of dynamic systems and controls. Still many are skeptical
of this subject of fractional calculus, perhaps due to no other reason but unfamiliarity. So a lot
of effort and space is devoted to make this subject accessible and make its interpretation more
lucid and clear so that the inevitable applications should follow. Keeping application in mind
the choice of this subject as Ph D study was chosen as ‘Fractional Order Controllers and
Application to Real life Systems’. The trigger point to have this study was, a real life control
of Nuclear Power Plant (PHWR 500MW) by logarithmic logic that gave a better governance
than observed in earlier plants, where the error was simply the difference of two linear
powers (the actual linear power minus demand power). This linear power error based control
though working in all the PHWR 235 MW Nuclear Power Plants may be a cause of spurious
actuations of control rod drive mechanisms-observed regularly; compared to the logarithmic
logic used for effective power error in PHWR 500MW.
When asked why?-the reason perhaps was that the way to govern a natural exponential system
by logarithmic power error matches the two domains and thus may be increasing the
efficiency of the control action. Perhaps, in the logarithmic case, ‘we are talking to the plant’
which is naturally exponential-in the language of the process; this gives efficient way of better
control. The point which is emphasized here is if we communicate in the language of the
dynamic system then we will be communicating better: naturally communicate in French with
persons in France!! This conjecture was presented at ICONE-13 Beijing.
Therefore, this leads us to a question the control action is generally proportional error, integral
error, and derivative error – if the system dynamics behave that way? We have been
habituated to think about dynamic equations that follow integer order derivative and integer
order equations-but in reality it is an approximation of what the system behavior actually is.
We consider as an approximate on Markovian nature of dynamics; but in reality the system
dynamics are with ‘memory, history, non-local’ and this memory based system is better be
16
described by fractional calculus.
So what is fractional calculus? Can we have circuits and systems which can do ½, ¼, ¾,
differentiation and integration? If we have them, we can thus have controllers that
communicate with the natural dynamics of the plant. We developed those circuits and
systems. Therefore, first we see how fractional calculus is? Why memory, history and non-
local characters are there in fractional derivatives? Why fractional derivative is not slope at a
point?
In a letter to L’Hopital in 1695 September 30, Leibniz raised the possibility of generalizing
the operation of differentiation to non-integer orders, and L’Hopital asked what would be the
result of half-differentiating x ; Leibniz replied “It leads to a paradox, from which one day
useful consequences will be drawn”. The paradoxical aspects are due to the fact that there are
several different ways of generalizing the differentiation operator to non-integer powers,
leading to non-equivalent results. We can say this query dated 30th September, 1695 gave
birth to “Fractional Calculus”; therefore this subject of fractional calculus with half derivative
and integrals etc, are as old as conventional Newtonian or Leibniz’s calculus. However, this
subject was dormant till the beginning of the century, and only now have started finding the
applications. We will answer L’Hopital’s query in the thesis, along with other definitions of
fractional differentiation and integration, we will show how memory is included in the
fractional derivative operation. Thus we will develop the concept of fractional calculus while
trying to answer L’Hopital.
Thesis is put into five chapters covering topic “Fractional Order Controllers and Application
to Real Life Systems”.
In the chapter-1, Introduction, we have introduced idea as what is fractional calculus, the
fractional order differentiation and fractional order integration, and addressed very basic
issues regarding memory. We discussed various integral representations of fractional
derivative, in convolution integral form, and also derived computational expression in terms
of conjugation to classical integer order calculus. In this chapter, we have also demonstrated
17
the Fractional Differential equation and its solution briefly. In this chapter we also
demonstrated via computation method the solution to differential equation and response to a
step-input excitation, for varying fractional order. This gives a feeling of dynamic system
described by non-integer order differential equation where the order gives the characteristic
response. As an example we applied concept of fractional divergence to get reactor flux
profile, in earlier publication. The details of these anomalous non-Fickian diffusions are
detailed in earlier publications.
In the chapter-2, Fractional Calculus approach to view anomalous charge discharge in
super-capacitor we take a very important aspect of charging discharging of super-capacitor,
which is entirely different from normal capacitor charging discharging voltage and current
patterns. We point out that IEC-62931 method describing the voltage profile measured to a
constant current charge and discharge is not correct, as we show there is fractional order loss
component present in the super-capacitor cell; give a different voltage profile (backed by
experimental determination too). Therefore parameter extraction by following the IEC-62931
standard does not reveal the correctness of super-capacitor parameters. We propose here a
new scheme to extract parameters of super-capacitors, by actual charge discharge profiles
observed in our super-capacitor experiments.
In order to complete the study we derive efficiency in energy transfer while charging and then
discharging the super-capacitor, in constant current mode and constant voltage mode. We
infer that the efficiency is independent of the discharge time and charge time, and excitation
(current or voltage), but only is depending on fraction i.e. ratio of discharge to charge
excitation (current/voltage). Also we evaluate maximum efficiency of energy transfer, and
that is function of the fractional index of loss impedance. For a no-loss case of ideal capacitor
this fractional index is unity and efficiency too is unity. With the inclusion of loss component
of fractional order we show the charge discharge curves of super-capacitors are different to
what IEC-62931 standard says, and thereby we propose to use this new method to extract
parameters of super-capacitor in future.
The loss component in super-capacitor is due to the fact, the electrodes are rough. The
18
observation of micro-structural roughness of electrode material of super-capacitor; returns
time fractional derivative in the transfer function; this is discussed in this chapter. Here we try
and relate index of heterogeneity that is the exponent of power law distribution of the rough
porous electrode to the order of fractional differ-integration. This treatment is not being
carried out before in detail earlier. Many researches pertaining to impedance spectroscopy,
report this type of phenomena; perhaps treatment of this type will be beneficial to the
mathematical physics aspect of those researches to relate microscopic disorder with fractional
calculus; with this new mathematical process developed and described here.
We have tried to evolve fractional differ-integrations as constituent of transfer characteristics
for super-capacitors-which are fractional loss element, and also in this chapter tried to
evaluate loss tangent and stated that loss tangent is frequency independent. Whereas classical
loss tangent is frequency dependent when classically expressed via lumped resistor and
lumped capacitor. The frequency independence of loss tangent (as called di-electric loss) is a
feature of several di-electrics used in insulators and conventional di-electric capacitors. The
reason that fractional differential equations appear is due to rough disordered electrode of
super-capacitors which are purposely made to enhance the effective electrode surface area to
get capacity of Farad ranges in small volume. The reason of disorder as power law in packing
of pores in electrode is identified as cause of several modes of electrical relaxation to external
impulse to super capacitors; this manifests as fractional differential equation as constituent
expression for super-capacitor, with fractional order related to exponent of power law of
distribution relaxation rate. Further practical research is required to relate and quantify the
exponent of power law of disordered electrode structure vis-à-vis exponents of fractional
order differential equations.
In the Chapter 3, Application to real life Physical systems we have put some real life
problems where we can invoke fractional calculus, to explain anomalous behaviors observed
in observations and experiments. The examples we have taken from experiments on visco-
elasticity, and applying concept of fractional calculus to see physics of delays in computer
network systems. Here we also give generalized treatment (extending this research further to
have if possible) for a ‘continuous order system identification approach’. Wherever possible
19
the distinction of this approach is highlighted.
A non-Newtonian fluid reveals anomalous visco-elastic properties as compared to Newtonian
fluid. The experiment of spreading of viscous sample is conducted where arrowroot solution
is kept between two glass plates and steady load is applied. A camera is kept below to capture
snap shots regularly to record the spreading pattern. The area is calculated graphically later,
and its plot with respect to time for various loads reveals interestingly the observation of
oscillatory nature of the spreading (especially when load/stress is high). We relate area to
strain and following analysis show that the fractional differential equation gives suitable
explanation of this anomalous behavior, of non-Newtonian relaxation (with memory).
The visco-elastic equations of classical integer order equation is generalized, via fractional
derivative of order ‘q’, to representation of the stress-strain in distributed spring and dashpot
system for a non-Newtonian fluid. When, the order 1q , then normal integer order equation
is recovered. The strain built up for any relaxation process may be treated as convolution
integral of a strain variable with suitable integral kernel. In this chapter we have used the
experimental data to get the value of fractional order ‘q’ for various fluids under stress; and
discussed the nature of kernel of the above convolution for several types of relaxation with
and without memory.
Dynamics of delay in any systems demonstrate the stochastic behavior. The delay of random
nature has wide spikes and if a statistics be taken, it is like a power law, with pronounced tail.
This we demonstrated on a delay data of the network. We have developed a new extension of
fractality concept for dynamics of random delay. We have proposed a possible fractional
calculus approach to model the evolution of stochastic dynamics of random delay. We
consider the fractional form of Langevin type stochastic differential equation, and replace
standard ‘white noise’ Gaussian stochastic driving excitation force, by ‘shot-noise’ whose
each pulse has randomized amplitude. The proposed fractional dynamic stochastic approach
allows obtaining the probability distribution function (pdf) of the modeled random delay. As
an application of the developed general approach we thus derive the equation of pdf of
increments of random delay as a function of increment of time.
20
We estimated from a gathered data of delay of packets in a heavily loaded network, that its
graph is irregular, and the irregularity index of Hurst exponent, exponent of its power spectral
density and the graph’s fractal dimension points towards a fractional Brownian Motion (fBM)
like system.
Effect of network delay in control system is very widely researched topic, and has practical
relevance to modern computer control industry. A Brownian motion to model the stochastic
process of ‘random delay dynamics’ is proposed in this chapter, through fractional equivalent
of Langevin equation driven by ‘shot-noise’. A shot noise results when a ‘memory-less filter’
is excited by train of impulses derived from a homogeneous Poisons Point Process (PPP). We
consider the fractional form of Langevin type stochastic differential equation, and replace
standard ‘white noise’ Gaussian stochastic driving excitation force, by ‘shot-noise’ whose
each pulse has randomized amplitude. The force is acting on a delay generating block where
the Fractional equivalent of Langevin equation is dynamic representation of the system The
driving force is train of pulses, will give a delay function of time, which also may be called
fractional stochastic variable, from this above dynamic system as. The fluctuation dynamics
of this variable is studied in this chapter.
At the end of the day one has to solve Fractional Differential equations (FDE). We treat a
new method to solve FDE in chapter-4; Solution of Generalized Differential Equations
Mathematical modeling of many engineering and physics problem leads to extraordinary
differential equations (Non-linear, Delayed, Fractional Order). We call them Generalized
Dynamic System. An effective method is required to analyze the mathematical model which
provides solutions conforming to physical reality. For instant a Fractional Differential
Equation (FDE), where the leading differential operator is Riemann-Liouville (RL) type
requires fractional order initial states which are sometimes hard to physically relate.
Therefore, we must be able to solve these dynamic systems, in space, time, frequency, area,
volume, with physical reality conserved.
The usual procedures, like Runga-Kutta, Grunwarld-Letnikov Discretization with short
21
memory principle etc, necessarily change the actual problems in essential ways in order to
make it mathematically tractable by conventional methods. Unfortunately, these changes
necessarily change the solution; therefore, they can deviate, sometimes seriously, from the
actual physical behavior. The avoidance of these limitations so that physically correct
solutions can be obtained would add in an important way to our insight into natural behavior
of physical systems and would offer a potential for advances in science and technology.
Adomian Decomposition Method (ADM) is applied here in this chapter by physical process
description; where a process reacts to external forcing function. This reactions-chain
generates internal modes from zero mode reaction to first mode second mode and to infinite
modes; instantaneously in parallel time or space-scales; at the origin and the sum of all these
modes gives entire system reaction. By this approach formulation of Fractional Differential
Equation (FDE) by RL method it is found that there is no need to worry about the fractional
initial states; instead one can use integer order initial states (the conventional ones) to arrive at
solution of FDE.
This new finding is highlighted in this chapter, which eases out solving for the generalized
dynamic systems. We have placed a ‘new’ method of solving Fractional Differential
Equations taking into consideration only integer order initial states. This method is close to
nature where system reacts to external stimulus.
In chapter-5 Realization of Fractional Order Circuits and Fractional Order Control
System we have given how we can practically realize the fractional order differentiator,
integrator and PID. The development of the theory we have done with relevant proofs, thus
these give computational algorithms to do the realizations of these ‘fractional’ order
components-we must say realization of fractional Laplace operator. When we extended this
method we realized that getting circuit components to realize fractional order elements are
impossible. We worked on improved schemes and thus generated methods to realize circuit
hardwire to do this job.
A Fractional slope on the log-log Bode plot has been observed in characterizing a certain type
22
of physical phenomena and is called the fractal system or the fractional power pole (or zero).
In order to represent and study its dynamical behavior, a method of singularity function is
discussed in this chapter, which consists of cascaded branches of a number of poles-zero
(negative real) pairs. Moreover, the distribution spectrum of the system can also be easily
calculated and its accuracy depends on a prescribed error specified in the beginning. This
method would thereafter be used widely in approximating fractional order transfer functions
for the discussed Lead Compensators as well as the PIλDµ controllers. This chapter presents
an effective method for the approximation by a rational function, for a given frequency band,
of the fractional-order differentiator sm and integrator s-m (m is a real positive number), and the
fractional PIλDµ controller) .First, the fractional-order integrator s-m (0 < m <1) was modeled
by a fractional power pole (FPP) in a given frequency band of practical interest. Next, this
FPP is approximated by a rational function, using the method of singularity function
approximations). The above idea was used to model the fractional-order differentiator sm (0 <
m <1) by a fractional power zero (FPZ). Then, the approximation method of the FPP was
extended to the FPZ to obtain its rational function approximation. Therefore, with this
method, one can achieve any desired accuracy over any frequency band, a rational function
approximation of the fractional-order differentiator and integrator. The rational function
approximation of the fractional PIλDµ controller is just an application of the above method.
These building blocks make fractional order PID at ease. The performance results are shown
in this chapter regarding this circuit development, along with illustrative examples. We also
write the possible theories regarding Fractional Order Controllers the Fractional Order Phase
Shapers and its usage in ‘efficient control’ of making plant reaction overshoot independent of
gain changes (rather plant parametric uncertainties). This phenomenon is called iso-damping.
The idea is to have a constant phase for a seemingly wide range of frequency-which enables
system gain to vary so that the feedback quantity remains same-thereby giving same
overshoot or undershoot. Here we also show experimental results of controller effort and
compare the conventional controller with fractional order controller and state that fractional
order controller takes lesser control effort compared to conventional integer order controller.
Well this was idea dream of H W Bode who stated in 1950 that, “wish I could have some
circuit what will be doing ‘partial’ integration!” Well, we have realizable circuits with us to
23
do this Fractional Calculus.
The reference section contains list of various work on this subject, from which the ideas were
drawn, also lists several publications by the author.
Shantanu Das
PhDNo.ENGG01200704021
October, 2013
http://scholar.google.co.uk/citations?user=9ix9YS8AAAAJ&hl=en www.shantanudaslecture.com
24
Chapter 1
Introduction Birth of Fractional Calculus In a letter to L’Hopital in 1695 September 30, Leibniz raised the possibility of generalizing the
operation of differentiation to non-integer orders, and L’Hopital asked what would be the result of
half-differentiating x ; that is: 1 2 1
21 2
d ( ) ? or [ ] ?d xx D x
x
Leibniz replied “It leads to a paradox, from which one day useful consequences will be drawn”. The
paradoxical aspects are due to the fact that there are several different ways of generalizing the
differentiation operator to non-integer powers, leading to in equivalent results.
We can say this query dated 30th September, 1695 gave birth to “Fractional Calculus”; therefore this
subject of fractional calculus with half derivative and integrals etc, are as old as conventional
Newtonian or Leibniz’s calculus. However, this subject was dormant till the beginning of the century,
and only now have started finding the applications. Since posing the question to Leibniz, L’Hopital
never worked on this subject though.
Let us try to find L’Hopital’s query Let us try and find out if we can differentiate the function ( )f x by ½ to get
1 2
1 2
d ( ) ? for ( )d
f x f x xx
In other words we try to find answers to L’Hospital’s querry. Differentiation and integration are
usually regarded as discrete operations, in the sense that we differentiate or integrate a function once,
25
twice, or any whole number of times. However, in some circumstances it’s useful to evaluate a
fractional derivative. Half derivative Quarter derivative semi-integration etc.
In some ways the most natural and appealing generalization is based on the exponential
function ( ) axf x e whose nth derivative is simply n axa e
dd
nax n ax
n e a e nx
This is n -fold repeated differentiation or integer order derivative of ( ) axf x e . This
immediately suggests defining the derivative of order (not necessarily an integer) as
dd
ax axe a ex
Negative values of represent integrations (anti-derivative) and we can even extend this to allow
complex values of , or even to a continuous distribution of this order in some interval.
( )d
( )d
d d d d( ) ( ) ( ) ( )d d d d
b
a
b
a
k q qp iq
p iq k q qf x f x f x f x
x x x x
Any function expressible as a sum of exponential functions can then be differentiated in the same
way. For example, the generalized derivative of the cosine function according to this approach is
given by
/ 2 / 2 2 2
d d ( ) ( )cos( )d d 2 2
( ) ( )2 2
cos2
ix ix ix ix
i x i xi ix i ix
e e i e i exx x
e e e e e e
x
Since / 2 / 2( ) ( )i ii e e , we have the nice result
d cos( ) cosd 2
x xx
26
Thus the generalized differential operator simply shifts the phase of the cosine function (and likewise
the sine function by 0(90) ), that is in proportion to the order of the differentiation. For
differentiation the process advances the phase, needless to say the integration makes the phase lagged.
After all for 1 and 1 , we have 1
1
d dcos( ) cos cos( ) cosd 2 d 2
x x x xx x
Needless to say, this approach can be applied to the exponential, i.e. ½ derivative of axe should be 1/ 2
1 21/ 2
d ( )d
ax axe a ex
This approach is correct in some cases as we will see in subsequent section. This approach is also
called Liouvelli’s approach.
The exponential approach seems to give a very satisfactory way of defining fractional derivatives but
we have yet to answer L’Hopital’s question, which was to determine the half-derivative of ( )f x x .
Paradox stems from unification of differentiation and integration which
makes the differentiation a non-local operation There is no Fourier representation of this open-ended function ( )f x x , so it has no well-defined
‘spectral decomposition’. Of course, we can find the Fourier representation of x over some finite
interval, but what interval should we choose? This ambiguity gives a hint of why Leibniz considered
the subject to be paradoxical. Leibniz was well aware that the result of integrating a function is neither
unique nor local, because it depends on how the function behaves over the range for which the
integration is performed, not just at a single point. But he was used to thinking of differentiation as
both unique and local, because whole derivatives 2 2 3 3d/ d , d /d , d /d ....d / dn nx x x x happen to possess
both of those attributes. These conventional derivatives are local operator gives a slope at a point, i.e.
depend on local point, whereas integration 1 1 2 2d / d , d / d ........d / dn nx x x depends on the entire
interval, hence non-local in character; so does ½ , ¼, ….derivatives are. This we shall deal while
formulating the fractional differentiation.
27
The apparent paradoxes of fractional derivatives arise from the fact that, in general, differentiation is
non-unique and non-local, just as is integration. This shouldn’t be surprising, since the generalization
essentially unifies integrals and derivatives into a single operator. If anything, we ought to be surprised
at how this operator takes on uniqueness and locality for positive integer arguments.
To get formulation for generalized derivative from first principle To get a clearer idea of the ambiguity in the concept of a generalized derivative, it’s useful to examine
a few other approaches, and compare them with the exponential approach described above. The most
fundamental approach may be to begin with the basic definition of the whole derivative of a function
( )f x
0
d ( ) ( )( ) limd
f x f xf xx
Repeating n -times of this operation leads to a binomial series of following type
0 0
d 1( ) lim ( 1) ( )d
n nj
n nj
nf x f x j
jx
(1)
for any positive integer n . To illustrate, this formula gives the second derivative ( 2n ) of the function 4( )f x x as
2 24 4
2 20 0
4 4 420
2 2 2
0
d 1( ) lim ( 1) ( )d
1lim 2( ) ( 2 )
lim 12 24 14 12
j
jx x j
x
x x xh
x x x
We can generalize equation (1) for non-integer orders, but to do this we must not only generalize
the binomial coefficients, we also need to determine the appropriate generalization of the upper
summation limit, which we wrote as n in equation (1).
To clarify the situation, let us go back and derive “from scratch” the operations of differentiation
and integration in a unified context. Consider an arbitrary smooth function ( )f x as shown in the
figure below.
28
xx 2x 3x 4x 5x
( )f x
Figure-1: Dividing the function interval into small slices of
In addition to the point at x , we’ve also marked six other equally-spaced values on the interval
from 0 to x , each a distance from its neighbors. The number k of these points is related to
the values of x and by k kx . For convenience, we define a (backward)
shift operator E such that E ( ) ( )f x f x .
With this back ward shift operator we get differentiation as
0 0
0
( ) E [ ( )]d ( ) ( )D ( ) ( ) lim limd
1 Elim ( )
f x f xf x f xf x f xx
f x
With the help of the series expansion as
2 31 1 ....1
x x xx
and the shift operator as NE indicating N backward shift till (0)f the start point of the
29
function, that is:
E ( ) 0,1,2,3,....N f x N N
We obtain the following
11
0 0
11
0
2
0 0
1 E ( ) ( )D ( ) lim ( ) lim
1 ED ( ) lim ( )
lim 1 E E .. ( ) lim ( ) ( ) ( 2 ) ... (0)
f x f xf x f x
f x f x
f x f x f x f x f
As in limit goes to zero the operator 1D is a simply a differentiation operator and 1D is
a simple integration operator. We can do n times the above and write the following:
0
1 E( ) lim ( )n
nD f x f x
This reproduces the ordinary whole multiple derivatives. For example, the second derivative
of ( )f x
2
22
1 E ( ) 2 ( ) ( 2 )D ( ) ( ) f x f x f xf x f x
in the limit as goes to zero, which illustrated how we recover the binomial equation (1) for
any whole number of differentiations. However, strictly speaking, this context makes it clear
that we should actually write the second derivative as
2
22
1 E (1) ( ) (2) ( ) (1) ( 2 ) (0) ( 3 ) ...(0) (0)D ( ) ( ) f x f x f x f x ff x f x
It just so happens that, if n is a positive integer, all the binomial coefficients after the first
1n are identically zero
0nj
nC j n
j
so we can truncate the series, but for any negative or fractional positive values of n , the
30
binomial coefficients are non-terminating, so we must include the entire summation over the
specified range.
Consequently, the upper summation limit in (1) should actually be 0( ) /x x , where 0x is
the lower bound on the range of evaluation. We often choose 0 0x by convention, but it is
actually arbitrary, and we will see below some circumstances in which the lower bound is not
zero.
In any case, we can re-write equation (1) in the more correct form that does not rely on n
being a positive integer 0
0 0
d 1( ) lim ( 1) ( )d
x xn
jn n
j
nf x f x j
jx
(1a)
The bracket ... is floor operator makes the value to nearest lower integer.
Defining factorial and binomial coefficients for non-integer and
getting series formulation for fractional derivative To define the binomial coefficient for non-integer values of n , recall that for integer
arguments these coefficients are defined as
!!( )!
n nj j n j
so we need a way of evaluating the factorial function for non-integer arguments n .
Notice that for any positive integer n we have the definite integral 1 2 1 2
2
1
2 ( !)(1 ) d(2 1)!
nn nx x
n
From above we get
12
2 11
(2 1)!! d (1 )2
nn
nn x x
31
The argument of the factorial on the right side is 2 1n , so the right hand expression is well-
defined for half-integer value of n such that 2 1n is non-negative. Hence this is a well-
defined expression for the factorial of any such half-integer argument. For example, setting
1/ 2n and using 1 2 1/ 2
1d (1 )x x
in above expression we get
1 !2
Furthermore, now that the factorial of all (positive) half-integers is defined, the above formula
allows us to compute the factorial of any quarter-integer, and then every sixteenth, and so on.
Hence, using the binary representation of real numbers, and using the
identity ( 1)! ( 1)( !)x x x , we now have a well-defined factorial function for any real
number.
This is traditionally called the gamma function, with the argument offset by 1 relative to the
factorial notation, so we have
( ) ( 1)!n n
Therefore
1 1 1! 1 !2 2 2
for any positive integer n . The gamma function has several formulations and its integral
representation is
1
0
( ) duu e u
The fundamental recurrence formula for the gamma function is therefore ( 1) ( ) Note the reflection relation is
( ) (1 )sin( )
x xx
We have the following values for positive half-integer arguments
3 4 1 1 3 1 5 3, 2 , , ,2 3 2 2 2 2 2 4
32
Now that we have a general way of expressing “factorials” for non-integers, we can re-write
equation (1a) in generalized form, replacing each appearance of the integer n with the real
number . This gives
0
0 0
d 1 ( 1)( ) lim ( 1) ( )d ! ( 1 )
x x
j
jf x f x j
x j j
(2)
If is an integer n , the vanishing of the binomial coefficients for all j greater than n implies
that we don’t really need to carry the summation beyond j n , and in the limit as goes to
zero the n values of ( )f x j with non-zero coefficients all converge on x so the derivative
is local.
Try to answer L’Hopital’s question now! Choosing 0 0x as the low end of our differentiation interval, the formula (2) for the general
derivative becomes /
0 0
d 1 ( 1)( ) lim ( 1) ( )d ! ( 1 )
xj
jf x f x j
x j j
(3)
With this, we are finally equipped to attempt to answer L’Hospital’s question. Taking the
function ( )f x x with 1/ 2 , signifying the half-derivative, this formula gives
1 2
1 2 0
1 1 11( ) ( ) ( 2 ) ( 3 )...2 8 16d 1( ) lim 1/ 2 1/ 25d ( 4 )... (2 ) ( )
/ 2 / 1128
x x x xx
x xx x
In this equation the binomial coefficients symbol is understood to denote the generalized
function, with the factorials expressed in terms of the gamma function. As explained
previously, the coefficients in the above expression are just the coefficients in the binomial
expansion of 1/2(1 E ) . Evaluating this expression (numerically) in the limit as goes to
zero, we find that the half-derivative of x is (almost)
33
1/ 2
1/ 2
d ( ) 2d
xxx
(4)
We can thus write the above as 1/20 ( ) 2 /xD x x indicating that differentiation is starting at
start point 0x . This concept of defining the fractional derivative with backward shift
operator gives [ ( )]a xD f x , the forward derivative of function defined in the interval ,a b . We
can have backward derivative too as [ ( )]x bD f x , by using forward shift operator; in this case
future points of the function is to be known a priory. In a sense therefore forward derivative
constructed by back-shift operator is causal.
This is exactly what we would expect based on a straightforward interpolation of the
derivatives of a power of x . Recalling that the first few (whole) derivatives of mx are
2 3
1 2 32 3
d d d( ) , ( ) ( 1) , ( ) ( 1)( 2)d d d
m m m m m mx mx x m m x x m m m xx x x
Thus we expect to find that the general form of the nth derivative of mx is
d !( )d ( )!
nm m n
n
mx xx m n
Replacing the integer n with the general value , and using the gamma function to express
the factorial, this suggests that the a fractional derivative of nx is simply
d ! ( 1)( )d ( 1) ( 1)
m m mm mx x xx m m
(5)
1/ 21 (1/ 2)
1/ 2
d 1! 1( ) 2d (1 [1/ 2] 1) (3/ 2)
xx xx
which is exactly the same answer to L’Hopital’s question as we got previously, i.e., the half-
derivative of x is given by (4). The (5) is Euler formula holds for 1m ; that is requires
function to be better behaved than 1x .
Now, since analytic functions can be expanded into power series ( ) kkk
f x a x we can use
equation (5), applying it term by term to determine the fractional derivatives of all such
34
functions. Furthermore, applying this formula with negative values of gives a plausible
expression for the fractional- integral of a power of x . For example, to find the whole integral
of 3x we set 3m and 1 and then compute 1
3 3 ( 1) 4 4 41
d 3! 3! 6 1( )d (3 [ 1] 1) (5) 24 4
x x x x xx
Note that the above integration is valid only if the initial point be zero, else initial value is
subtracted.
So, in a sense, equation (5) is an algebraic expression of the fundamental theorem of calculus,
i.e., the inverse relationship between the operations of differentiation and integration, since
the n th derivative of the n th derivative (integration) is the identity, (provided the initial
values at the start point of the function is zero). The unification of these two operations makes
it even less surprising that generalized differentiation is non-local, just as is integration.
Half derivative of a constant and other functions-and paradoxical
case Incidentally, the generalized derivative as developed so far gives some slightly surprising
results. For example, the half-derivative of any constant function 0Cx is 1/ 2
1/ 2
d 0! C(C) Cd (1/ 2)x x
(6)
Thus, not only is the half-derivative of a constant with (respect to x ) non-zero, it is infinite
at 0x , and decays to zero at x . Nevertheless, equations (3) and (5) are agreeably
consistent with each other, giving some confidence in the significance of this generalization
of the derivative. Given this equivalence, one might wonder about the value of the elaborate
derivation of equation (3) when it seems to be so much easier and more direct to arrive at
equation (5). In answer to this there are two points to consider. First, equation (3) applies to
fairly arbitrary functions, whereas equation (5) applies only to functions expressible as power
series. Still, a very large class of functions can be expressed as power series, so this in itself is
not an overriding factor. More important is the fact that equation (5) gives no hint of the non-
locality of the generalized derivative, i.e., the dependence on the function over a finite range
35
rather than just at a single point, and the need to specify (implicitly or explicitly) the chosen
range. The importance of this can be seen in several different ways. Perhaps the most
significant reason for taking care of the derivative interval is brought to light when we try to
apply equation (3) or (5) to the simple exponential function. We previously proposed that the
general n th derivative of axe is simply ( )n axa e , and yet if we expand the exponential
function xe into a power series 2 3
2 31 ...1! 2! 3!
ax a a ae x x x
and apply equation (5) to determine the half-derivative, term by term, we get 1/ 2
2 3 41/ 2
d ( ) 1 4 8 161 2 ...d 3 15 105
xe x x x xx x
A plot of this function, along with xe , is shown in the figure below
0.5 1 .0 1.5 2.00
2
4
6
8
½ Derivative of xe
xe
x
Figure-2: Plot of half-derivative of xe and xe
Here we see one of the paradoxes that might have intrigued Leibniz. According to a very
reasonable general definition we expect any derivative (including fractional derivatives) of the
exponential function to equal itself, and the exponential goes to 1 as x goes to zero, and yet
36
our carefully-derived formulas for the half-derivative of the exponential function goes to
infinity at 0x . Clearly something is wrong. Must we abandon the elegant exponential
approach, along with its beautiful explanation of the trigonometric derivatives as simple phase
shifts, etc? No, we can reconcile our results, provided we recognize that the derivative is non-
local, and therefore depends on the chosen range of differentiation; that is like integration it
has lower and upper limits.
Fractional Derivatives with lower limit to minus infinity The lower terminal to minus infinity is Liouville formulation, and this refers also to steady
state systems. In these cases our simplified results of phase shifting of trigonometric functions
do hold. Consider the two anti-differentiations (integrations) shown below
0
0 0
443 0d d
4 4
x xxu x
x x
xxu u e u e e
The first integral shows that when we say 3x is the derivative of 4 / 4x we are implicitly
assuming 0 0x , which is consistent with our derivation of equation (3). However, the
lower integral shows that, by saying xe is the derivative of xe , we are implicitly
assuming 0x .
Thus ranges of integration/differentiation we have tacitly assumed for these two
definitions are different. To get agreement between the interpolated binomial expansion
method and the definition based on exponential functions we must return to equation (2),
and replace the condition 0 0x with the condition 0x . This is easy to do, because it
simply amounts to setting the upper summation limit to infinity, i.e., we take the
following formula for our generalized derivative
0 0
d 1 ( 1)( ) lim ( 1) ( )d ! ( 1 )
j
jf x f x j
x j j
(7)
With this, we do indeed find that the th derivative of axe is simply ( ) axa e , consistent
with the purely exponential approach.
37
Repeated integration approach to get fractional derivative Still another approach to fractional calculus is to begin with a generalization of the
formula for repeated integration. Suppose the function f ( )x has the specified values
0 1 2 3 4 5f , f , f , f , f , f at equally spaced intervals of width . The integral of this
function from 0x to 5 can be approximated by times the cumulative sum of these
values, and the integral of this new function is times the cumulative sum of those
values, and so on. This is illustrated in the table below.
0
1
2
3
4
5
ffffff
0
0 1
0 1 2
0 1 2 3
0 1 2 3 4
0 1 2 3 4 5
ff ff f ff f f ff f f f ff f f f f f
1 0 1 2 3 4 5f f f f f fS
0
1
2
3
4
5
ffffff
0
0 1
0 1 2
0 1 2 3
0 1 2 3 4
0 1 2 3 4 5
f2f f3 f 2f f4f 3f 2f f5 f 4f 3f 2f f6f 5f 4f 3f 2f f
2 0 1 2 3 4 5(6 0)f (6 1)f (6 2)f (6 3)f (6 4)f (6 5)fS
3 0 1 2 3
4 5
(6 1)(6 0) (6 0)(6 1) (6 1)(6 2) (6 2)(6 3)f f f f2 2 2 2
(6 3)(6 4) (6 4)(6 5)f f2 2
S
0
0 1
0 1 2
0 1 2 3
0 1 2 3 4
0 1 2 3 4 5
f3 f f6f 3f f10f 6f 3f f15f 10f 6f 3f f21f 15f 10f 6f 3f f
0
1
2
3
4
5
ffffff
1 f ( ) d d 1S x x x
2 f ( ) d d 1S x x x
3 f ( ) d d 1S x x x
Table-1: Cumulative sum showing 1st, 2nd , and 3rd integration of function for six
points
Thus the 1st, 2nd, and 3rd “integrations” yield the values (the is common term not shown)
1 0 1 2 3 4 5
2 0 1 2 3 4 5
3 0 1 2 3
4 5
f f f f f f(6 0)f (6 1)f (6 2)f (6 3)f (6 4)f (6 5)f(6 1)(6 0) (6 0)(6 1) (6 1)(6 2) (6 2)(6 3)f f f f
2 2 2 2(6 3)(6 4) (6 4)(6 5)f f
2 2
SS
S
38
As we divide the overall interval into more and more segments, becomes arbitrarily small,
and so do the differences between the factors in any given term, so the successive integrations
give 1
1 2
2
1 2 120 0 0
32
1 3 2 130 0 0 0
d f ( ) f ( )d d ( )f ( )dd
d 1f ( ) f ( )d d d ( ) f ( )dd 2!
ux x
u ux x
x u u u x u u ux
x u u u u x u u ux
and so on like following. 11 2
11 1 1
0 0 0 0 0
1
0
d 1f ( ) ............. f ( )d d .....d ( ) f ( )dd ( 1)!
1 ( ) f ( )d( )
nuu ux xnn
n nn
nx
n
x u u u u x u u ux n
x u u un
Thus we have Cauchy’s expression for repeated integrals, which is 11 2
11 1 1
0 0 0 0 0
1............. f ( )d d .....d ( ) f ( )d( 1)!
nuu ux xn
n n
n
u u u u x u u un
which we can express using the gamma function instead of factorials, for fractional order as
1
0
d 1f ( ) ( ) f ( )dd ( )
x
x x u u ux
(8)
The (8) is Riemann fractional integral formula. The convergence properties of this formula
are best when has a value between 0 and 1.
Let us evaluate double integral of x take then 2 and apply (8) 2
20 2
0
3 3 5
00
5
d 1 ( ) dd (2)
2 2( )d3 5
415
x
x
u xx
u
xD x x u u ux
x u u u x u u
x
Now let us do semi integration of x , take then 1/ 2 and apply (8)
39
1/21/2
0 11/2 10 02
2 2 20 0 2
20
d 1 1 ddd (1/ 2) (1/ 2) ( )( )
1 1 dd(1/ 2) (1/ 2)
24 4 2
1 d(1/ 2)
4 2sinPut d (cos )d
2 20; / 2 ; / 2
x x
x
x x
x
2
u u uD x ux u x ux u
u u uuxu u x x uu x
u u
x xu
x x xu u
u u x
With these substitutions we proceed
/21/2
0 2 2/2 2
/2
2/2
/2/2
/2 /2
sin(cos )1 2 2[ ] d
(1/ 2)sin
4 4sin(cos )
1 2 2d(1/ 2) 1 sin
2
1 sin 1 cosd(1/ 2) 2 (1/ 2) 2 2
1(1/ 2) 2
x
x x x
D xx x
x x x
x
x x x x
x
2x
The same we will get via Euler formula as shown below
0
1 11/2 1/2 2 2
0
( 1)[ ]( 1)
1 1for se mi i n tegration of2 2
1 1(3 / 2) 32 ( )
1 1 (2) 2 212 2
m mx
x
mD x xm
x m
D x x x x x
40
There are two different ways in which this formula might be applied. For example, if we wish
to find the (7/3)-rd ( 7 / 3 )derivative of a function ( 7 / 3 7 / 3d ( ) / df x x ), we could begin by
differentiating the function three whole times (taking nearest integer say m just greater than
; that is 3m ), and then apply the above formula with to “deduct” two thirds i.e.
( ) (3 [7 / 3]) 2 / 3m of a differentiation
Alternatively we could begin by applying the above formula with and then
differentiate the resulting function three whole times ( 3m ).
These two alternatives for fractional derivatives are called the Right Hand Definition (Caputo)
and the Left Hand Definitions (Riemann-Liouville) respectively. Although these two
definitions give the same result in many circumstances especially when the start point of the
process is at , they are not entirely equivalent, because (for example) the half-derivative of
a constant is zero by the Right Hand Definition-Caputo, whereas the Left Hand Definition
gives for the half-derivative of a constant the result given previously as equation (6). In
general, the Left Hand Definition is more uniformly consistent with the previous methods, but
the Right Hand Definition has also found some applications.
Equation (8) highlights (again) the non-local character of fractional operations, because it
explicitly involves an integral, which we have stipulated to range from 0 to x . For any whole
number of differentiations we don’t need to invoke this integral, but for a non-integer number
of differentiations we must include the effect of this integral, which implies that the result
depends not just on the values of function at x , but over the stipulated range from 0 to x .
To illustrate the use of equation (8), we will (again) determine the half-derivative of ( )f x x ,
as L’Hopital requested. Using the Left Hand Definition, we first apply half of an integration
to this function using equation (8) with 1/ 2 , giving
41
1/ 21/ 2
1/ 20
0 0 01/ 2 1/ 2
3/ 23/ 2 3/ 2
3/ 2
d 1( ) ( ) ( )d put , d dd (1/ 2)
1 ( ) 1( d ) d d(1/ 2) (1/ 2)
1 2 1 42(1/ 2) 3 (1/ 2) 3
43
x
x x x
x x u u u x u z u zx
x zz z z xz zz
xx x
x
Then we apply one whole differentiation to give the net result of a half-derivative 1/ 2 3/ 2
1/ 2
d d 4( ) 2d d 3
x xxx x
in agreement with equation (6).
In operator sense we have for Riemann-Liouvelli fractional derivative as
111/2 1 12
0 0
( )0 0
d[ ( )] ( ) here 1d
d( ) ( ) 0 ( 1) ,d
x x
mm m m
x x m
D f x D D f x m Dx
D f x D D f x m m m Dx
In this case the Right Hand Definition (Caputo) gives the same result. Choose 1m then
differentiate the function ( )f x x once to have (1) ( ) 1f x , do the semi-integration of
this (1) ( ) 1f x that is 1/2 1/20 [1] (1) / (3 / 2) 2 /xD x x . In this process we first
differentiate the function m times, then follow up by remainder fraction and integrate it by
that fraction. In operator sense the Caputo derivative is as follows (we write C here to
distinguish from Riemann-Liouville fractional derivative)
C 1/2 1/2 1 10 0
C ( )0 0
d[ ( )] ( ) here 1d
d( ) D ( ) 0 ( 1) ,d
-x x
mm m m
x x m
D f x D D f x m Dx
D f x D f x m m m Dx
Now suppose we apply this method to the exponential function. Since our definition has been
based on the range from 0 to x , whereas we’ve seen that the “exponential approach” to
fractional derivatives is essentially based on the range from -∞ to x , we expect to find
disagreement, and indeed for the half-derivative of xe we get (by applying (8) with 1/ 2
42
and then differentiating one whole time) 1/ 2
1/ 2
d ( ) erfd
xx x ee e x
x x
This is identical to the half-derivative of xe given by equation (5), shown in red in the plot
presented previously (when we only had the series expansion of this function). Again, we can
reconcile this approach with the “exponential approach” by changing the lower limit on the
integration from 0 to -∞. When we make this change, equation (8) gives 1/ 2
1/ 21/ 2
d 1( ) ( ) ( )dd (1/ 2)
xx u xe x u e u e
x
and of course the whole derivative of this is also xe , so the half-derivative of xe by this
method is indeed xe , provided we use a suitable range of differentiation.
We note that Riemann-Liouville fractional derivative does not require function to be
differentiable (needs only to be continuous), whereas in Caputo case differentiability is
essential.
The Laplace Transform and Fourier Transform of Fractional
derivative The Laplace Transform of fractional derivative-integral of order operation is
1
10 0 at 0
0( ) ( ) ( )
nk k
x x xk
D f x s f x s D f x
(9)
Where Laplace Transform defined as
def
0
( ) d { ( )}sxf x x e f x
In Laplace definition above the order of differ-integration ; and the integer
n such that ( 1)n n . In this expression (9) when 0 , that is operation is
fractional integration, the term involving summation becomes zero for any function,
( )f x with available Laplace Transform. Also one can have similar to Laplace Transform
of fractional differ-integrals of ( )f x ; a Fourier Transform of fractional differ-integral
43
operation. A function ( )f x , which is “well-behaved” at x , we can have
( ) (i ) ( )xD f x f x (10)
and therefore we have fractional derivative/integral operation as inverse Fourier
transformed one
1( ) (i ) { ( )}xD f x f x
Where the Fourier and Inverse Fourier Transform is depicted as following
def def
i 1 i1( ) ( ) d { ( )} ( ) ( ) d { ( )}2
x xf x F x e f x f x F e F
In some cases (especially for steady state systems with lower terminal of differ-
integration a ) the Fourier Transformation method is another way to find fractional
derivative/fractional integration of function ( )f x . That is
(i) Obtain the Fourier Transform of ( )f x as ( )F .
(ii) Then this transformed ( )F in frequency domain we multiply by (i ) ,
where .
(iii) The resulting function (i ) ( )F we inverse Fourier transform, to get ( )xD f x .
So what is fractional Calculus? Well we must simply say that it is generalization of normal calculus to real or complex ‘order’
differentiation and integration. Like a continuum between two integers we have real numbers so is the
case that between two integer order integration and differentiation (figure-3) we have wonderful world
of mathematics the universe of fractional calculus. The two definitions of the fractional derivative are
shown below in figure 4 to figure-7.
Let us consider n an integer and when we say xn we can quickly visualize x multiply n times will give
the result. Now we still get a result if n is not an integer but fail to visualize how. Like to visualize 2
is hard to visualize, but it exists. Similarly the fractional derivative we may say now as
d ( ) / df x x though hard to visualize does exist. As real numbers exists between the integers so does
44
fractional differintegrals do exist between conventional integer order derivatives and n fold
integrations.
See the following generalization from integer to real number on number line as
n
n xxxxxx ............ n is integer
xnn ex ln n is real number
nnn )1....(3.2.1! n is integer
)1(! nn n is real
and Gamma Functional is
0
1)( dttex xt
Therefore the above generalization from integer to non-integer is what is making number line general
(i.e. not restricting to only integers). Figure 3 demonstrates the number line and the extension of this to
map any fractional differintegrals. The negative side extends to say integration and positive side to
differentiation,. 2 3
2 3
d d d, , , ,...d d df f fft t t
..., d d d , d d , d ,t t f t t f t f t f
Writing the same in differintegral notation as represented in number line we have figure-3
45
3 2 1 2 3
3 2 1 2 3
d d d d d d... , , , , , ,...d d d d d d
f f f f f fft t t t t t
-3 -2 -1 0 1 2 3 )3(f )2(f )1(f )0(f )1(f )2(f )3(f INTEGRATION DIFFERENTIATION
Figure-3: Number line & Interpolation of the same to differintegrals of fractional calculus
Fractional Derivatives Riemann-Liouville (RL) Left Hand Definition
(LHD) The formulation of this definition is:
Select an integer m greater than fractional number
(i) Integrate the function )( m folds by RL integration method.
(ii) Differentiate the above result by m.
Expression is given as:
0 10
d 1 ( )( ) dd ( ) ( )
tm
t m m
fD f tt m t
The Figure-4 gives the process block diagram & Figure-5 gives the process of differentiation 2.3 times
for a function.
)(tf ( )d m
d m d ( )t f t
Figure-4: Fractional differentiation Left Hand Definition (LHD) block diagram
46
INTEGRATION DIFFERENTIATION
)3(f )2(f )1(f )0(f )1(f )2(f )3(f
0.7 2.3
(i) 7.0)( m
3m (ii)
Figure-5: Fractional differentiation of 2.3 times in LHD
In this LHD the limit of integration is from 0 to t . We thus denote the derivative by notation 0 ( )tD f t .
In fractional calculus we find limit of derivative-i.e. derivatives are taken in interval. We call this as
‘forward derivative’. Now if the limits of integration are changed to ( t to 0 ) the derivative is denoted
as 0 ( )t D f t the ‘backward derivative’. The backward derivative is related to forward derivative by
0 0d( ) ( 1) ( )d
mm m
t tmD f t I f tt
Therefore in order to obtain fractional derivative of a function at a point (say 0) we should have the
values of these two derivatives same: forward derivative should equal the backward derivative. This
implies not only we should know the function from past to the point of interest (say 0) but also the
function should be known into the future-in order to have point fractional derivative at a point!
47
Fractional derivative of purely imaginary order i.e. i ,( 0 ) with real part as ‘zero’ is expressed
in Riemann-Liouvelli notation (with 1m ):
ii
1 d ( )( ) d(1 i ) d ( )
x
a xa
f uD f x ux x u
and associated integral of purely imaginary order in Riemann-Liouvelli definition is:
i i 1 i id 1 d( ) ( ) ( ) ( ) ( )dd (1 i ) d
x
a x a x a xa
D f x I f x I f x x u f u ux x
Fractional Derivatives Caputo Right Hand Definition (RHD) The formulation is exactly opposite to LHD.
Select an integer m greater than fractional number
(i) Differentiate the function m times.
(ii) Integrate the above result )( m fold by RL integration method.
In LHD and RHD the integer selection is made such that mm )1( . For example differentiation
of the function by order will select 4m . The formulation of RHD Caputo is as follows:
( )
0 1 10 0
d ( )1 1 ( )d( ) d d
( ) ( ) ( ) ( )
m
t t mmCt m m
ffD f t
m t m t
Figure-6 gives the block diagram representation of the RHD process and figure-7 represents
graphically RHD used for fractionally differentiating function 2.3 times.
)(tf d m
( )d m d ( )f t
Figure-6: Block diagram representation of RHD Caputo
48
INTEGRATION DIFFERENTIATION )3(f )2(f )1(f )0(f )1(f )2(f )3(f 2.3 3m (i) (ii) 7.0)( m
Figure-7: Differentiation of 2.3 times by RHD
The definitions of Reimann-Liouville of fractional differentiation played an important role in
development of fractional calculus. However the demands of modern science and engineering require a
certain revision of the well established pure mathematical approaches. Applied problems require
definitions of fractional derivatives allowing the utilization of physically interpretable “initial
conditions” which contain )(),(),( )2()1( afafaf and not fractional quantities (presently unthinkable!).
The RL definitions require 1 2
1 2lim ( ) lim ( )a t a tt a t aD f t b D f t b
In spite of the fact that initial value problems with such initial conditions can be successfully solved
mathematically, their solutions are practically useless, because there is no known physical
interpretation for such initial conditions, presently. It is hard to interpret. RHD is more restrictive than
LHD. For RL )(tf is causal. For LHD as long as initial function of t satisfies 0)0( f . For RHD
because )(tf is first made to m-th derivative i.e. )()( tf m , the condition
0.....&0)0( 21 mffff is required. In mathematical world this is vulnerable for RHD may be
deliberating. For LHD C 0 C / (1 )D t , the derivative of constant C is not zero. This fact led
to using the RL or LHD approach with lower limit of differentiation a in physical world this
posses problem. The physical meaning of this lower limit extending towards minus infinity is starting
of physical process at time immemorial! In such cases transient effects cannot be then studied.
49
However making a is necessary abstraction for consideration of steady state process, for
example for study of sinusoidal analysis for steady state fractional order system.
Grunwald-Letnikov definition The derivation is done above section. This is used for computational purposes. For an integer n, this is
formulation repeated differentiation and integration
( )
00
d ( ) 1 !( ) lim ( 1) ( )d ! !
n nn j
n nh j
f t nf t f t jht h j n j
Now for a non-integer, ( 1)n n
0 0
1 (( ) lim ( 1) ( )( (
nj
nh j
D f t f t jhh j j
Significance of non-integer order systems The mathematical concept and formalism of real or complex non integer differentiation stems from the
work of Liouville and Riemann at the beginning of the 19th century. Its synthesis and applications to
physics and engineering appeared in the last decade of the 20th century, after the introduction of new
linear capacitor model proposed by Westerlund in 1993, the significance of fractional order systems in
electrical systems.
Dynamics of many systems known can be explained by integer order systems but they do not and
cannot explain the behavior of all the systems, viz. the systems characterized by long memory
transients and semi-infinite dimensional and distributed structures. That is where the application of
Fractional Calculus comes as better alternative to classical Integer Order Calculus. According to
Westerlund ,” It is an engineering type of theory that does not in any way 'explain' the nature of the
internal processes of the system, but can reproduce and predict its behavior much better than any other
theory that we know of.” Therefore fractional calculus describes the nature better than integer order
calculus.
Applications of fractional calculus Fractional order systems are already introduced in various fields such as electrochemistry
electromagnetism etc. Practically, the modeling of fractional systems turns to out to be useful in
50
achieving better control over the system. Understanding complex processes involving anomalous
diffusion is possible only through fractional order calculus. Transmission lines electrical and electronic
noises dielectric polarization and anomalous diffusion are some of the fields having non integer
physical laws. In Electronics, advantages due to fractional order filters are realized and design of
Fractional order Differentiator and Integrators are studied. Fractional order digital IIR filter is designed
keeping in view the advantages offered by a Fractional order controller.
In addition to the fields specified, there is a large number of Electro-Mechanical, Visco-Elastic systems
where only fractional order system modeling is done to completely study the system properties. Some
other fields which are modeled using fractional order calculus are as follows:
New mathematical models of real materials, Viscoelasticity.
Modeling of hereditary processes, ageing of materials and systems.
Dynamical processes in fractals, Diffusion and other transfer processes in fractals.
Fractional-order controllers. PIλDµ-controllers, CRONE controllers. Active noise
control.
Mathematical models in economy, Econophysics.
New types of electrical circuits, Fractances.
Physiology, Models of live tissues.
Chemistry, Electrochemistry, Voltammetry, Polarography
Fractional Differential Equations Here we demonstrate a simple way to solve Fractional Differential equation and state some
counterintuitive properties relating to initial conditions and system order. However, our contribution in
solving generalized differential equation systems we have summarized in chapter-3; the solution via
principle of action and reaction. In this small demonstration we use Grunwald-Letnikov discretization
procedure to solve ‘tracking filter’ and its step input solution.
In this algorithm, the fractional differential term is directly replaced by the numerical approximation
definition given by Grunwald. But in order to do that, we have to discretize the time with a sampling
period satisfying the Nyquist criterion. Thus we can have t = h*n and f (t) is denoted by f (n) where f
51
(n) is the discretized function. Using Grunwald’s approximation, as mentioned in previous section we
write the following.
0 0
1 (( ) lim ( 1) ( )1) 1)
nj
th j
D Kh f Kh jhh j j
Where K is the number of data points available.
Consider a system bounded by the fractional order differential equation given by:
d ( ) ( ) ( )d
n
n
y ta by t u tt
, (0 1)n
Using approximation in above, we can discretize the differential equation
0
1 (( 1) ( ) ( ) ( )( (
nj
nj
na y Kh jh by Kh u Khh n j j
System output is given by
0
1 (( ) ( 1) ( )( (
( )
nj
nj
n
nu Kh y Kh jhh n j j
y Kh abh
Above equation directly gives the solution for a simple n-th order FO system. Step response of the
system represented by equation is studied for various values of n.
d ( ) ( ) ( )d
n
n
y ta by t u tt
, (0 1)n
For all the plots, the total number of points taken for evaluation is 1000. Following are plots.
The step response of different values of n is plotted. It can be seen that the time constant of the system
increases as the value of n reaches from 0 to1. The step response for values of n greater than one is
interesting. Since the order of the system is more, the response is more of an oscillatory kind. Thus to
have oscillatory under damped response we need not have second order system alone. A first order
system can have oscillatory under damped response (rather can even have sustained oscillations). Thus
order definition in classical sense is perhaps not fit in fractional derivative cases!
52
Figure 8: Step response of the system for different values of n using a=b=1 and y (0) =0
Now the values of the coefficients a & b are changed for n = 1.2. We observe that these parameters
control the amount of damping experienced by the system as well as the steady state value of the
solution.
53
Figure 9: Step response for different values depicting increased damping for greater values of a.
Figure 10: Step response for different values of parameter b.
54
Unlike any integral differentiator operator, fractional differential operator is not a local operator. For
integer derivative at a point P, only the knowledge of function values in the neighborhood of P is
sufficient. When n is not an integer, it is not sufficient to just know the functional values of
neighborhood of P in order to evaluate Dny (t), rather we need to have information about the entire
history of the function from initial instant t0 to t .
Ordinary and fractional derivatives differ in several ways. The signal to which a fractional derivative
is applied starts at t = 0. Before that, the signal is always assumed to be zero. If that is not the case, t =
0 must be moved to a time before which the signal does not differ from zero. The fractional derivative
of a constant is not zero because the signal changes from zero to a finite value at t = 0 gives a
contribution to the fractional derivative at all later times. A derivative of integer order also assumes a
value different from zero at the step at t = 0, which we call a delta functional, but there is no
contribution at later times since the delta functional is very short. The fractional derivative is not zero
even if the signal is zero because the signal might have differed from zero at an earlier time.
The plot given below compares the memory for initial values for an integer order system with n=1 and
a fractional order system with n=1.75.
55
Figure 11: Effect of initial conditions on a system with n =1, 1.75 for the step response
From the above plot, it can be concluded that the effect of initial conditions in case of integral systems
dies out very fast when compared to that of fractional order systems. For n=1.75, the effect of initial
condition is present feebly even after 25 sec while this effect dies down in 3 sec for the integer order
system
Conclusions In this chapter we introduced the concept of fractional calculus-a generalization of the classical integer
order calculus. We showed the difference between classical integer order calculus and fractional order
calculus. We also pointed out that the fractional derivative is not a local property but requires entire
history and memory; therefore the systems which are non-Markovian in nature are best suited to be
governed by fractional differential equation.
56
Chapter-2
Fractional Calculus approach to view anomalous charge discharge in super-capacitor
_______________________________________________________ Introduction
We take a very important aspect of charging discharging of super-capacitor, which is entirely different
from normal capacitor charging discharging voltage and current patterns. We point out that IEC-62931
method describing the voltage profile measured to a constant current charge and discharge is not
correct, as we show there is fractional order loss component present in the super-capacitor cell; that
gives a different voltage profile (backed by experimental determination too). Therefore parameter
extraction by following the IEC-62931 standard does not reveal the correct super-capacitor parameters.
We propose here a new scheme to extract parameters of super-capacitors, by actual charge discharge
profiles observed in our super-capacitor experiments.
In order to complete the study we derive efficiency in energy transfer while charging and then
discharging the super-capacitor, in constant current mode and constant voltage mode. We infer that the
efficiency is independent of the discharge time and charge time, and excitation (current or voltage), but
only is depending on fraction i.e. ratio of discharge to charge excitation (current/voltage). Also we
evaluate maximum efficiency of energy transfer, and that is function of the fractional index of loss
impedance. For a no-loss case of ideal capacitor this fractional index is unity and efficiency too is
unity. With the inclusion of loss component of fractional order we show the charge discharge curves of
super-capacitors are different to what IEC-62931 standard says, and thereby we propose to use this
new method to extract parameters of super-capacitor.
The loss component in super-capacitor is due to the fact, the electrodes are rough. The observation of
micro-structural roughness of electrode material of super-capacitor; returns time fractional derivative
in the transfer function; this is discussed in this chapter. Here we try and relate index of heterogeneity
that is the exponent of power law distribution of the rough porous electrode to the order of fractional
57
differ-integration. This treatment is not being carried out before in detail earlier. Many researches
pertaining to impedance spectroscopy, report this type of phenomena; perhaps treatment of this type
will be beneficial to the mathematical physics aspect of those researches to relate microscopic disorder
with fractional calculus; with this new mathematical process developed and described here.
We have tried to evolve fractional differ-integrations as constituent of transfer characteristics for super-
capacitors-which are fractional loss element, and also in this chapter tried to evaluate loss tangent and
stated that loss tangent is frequency independent. Whereas classical loss tangent is frequency
dependent when classically expressed via lumped resistor and lumped capacitor. The frequency
independence of loss tangent (as called di-electric loss) is a feature of several di-electrics used in
insulators and conventional di-electric capacitors. The reason that fractional differential equations
appear is due to rough disordered electrode of super-capacitors which are purposely made to enhance
the effective electrode surface area to get capacity of Farad ranges in small volume. The reason of
disorder as power law in packing of pores in electrode is identified as cause of several modes of
electrical relaxation to external impulse to super capacitors; this manifests as fractional differential
equation as constituent expression for super-capacitor, with fractional order related to exponent of
power law of distribution relaxation rate. Further practical research is required to relate and quantify
the exponent of power law of disordered electrode structure vis-à-vis exponents of fractional order
differential equations.
IEC-62931 Standard to test super capacitor-2007 We briefly describe what is given in the standard. The measuring circuit is shown in the figure-1,
where a constant current charging is used and then super-capacitor is charged to a rated voltage (in our
case each cell is rated to 2.5 to 2.7V). The switch in the figure-1 is thrown to a discharge position,
where the constant current discharge is applied. The voltage profile of the charge discharge is recorded
and that is in figure-2
58
S
XC
A
V
co n s tan t cu rren t /co n s tan t vo ltag ep o w er s u p p ly
C C C V
co n s tan t c u rren td isc h arg e r
Figure 1 – Circuit for constant current discharge method
Key
A d.c. ammeter
V d.c. voltmeter
S changeover switch
Cx capacitor under test
The measuring method is the following
a) If the d.c. voltage of the constant current/constant voltage power supply is not specified in the
individual standards, set at the rated voltage (UR).
b) Set the constant current value of the constant current discharger to the discharge current
specified in Table 2.
c) Turn the switch S to the d.c. power supply, and unless otherwise specified in the individual
standards, apply voltage and charge for 30 min after the constant current/constant voltage
power supply has achieved the rated voltage.
d) After a charge for 30 min has finished, change over the switch S to the constant current
discharger, and discharge with a constant current.
59
e) Unless otherwise specified in the individual standards, measure the time t1 and t2 where the
voltage between capacitor terminals at the time of discharge reduces from U1 to U2 as shown in
Figure 2, and calculate the capacitance value by the following formula:
3 0 m in
2U
2t
1U
RU
1t
3U
V o ltag e ( V )
T im e (s )
3 : d ro pU IR
Figure 2 – Voltage characteristic between capacitor terminals
C = I x (t2-t1)
U1-U2
Where
C is the capacitance (F);
I is the discharge current (A);
U1 is the measurement starting voltage (V);
U2 is the measurement end voltage (V);
t1 is the time from discharge start to reach U1 (s);
t2 is the time from discharge start to reach U2 (s).
60
f) The discharge current I and the voltages U1 and U2 at the time of discharge voltage drop shall
be as per Table 1. The method classification shall be in accordance with the individual
standards.
Table 1 – Discharge conditions
Classification Class 1 Class 2 Class 3 Class 4
Application Memory
backup
Energy
storage
Power Instantaneous
power
Charge time 30 min 30 min 30 min 30 min
I(mA) 1 x C 0.4 x CUR 4 x CUR 40 x CUR
U1 The value to be 80% of the charging voltage (0.8 x UR)
U2 The value to be 40% of the charging voltage (0.4 x UR)
NOTE C is the rated capacitance in F (Farad), and UR is the rated voltage in V (Volt)
NOTE The discharge current I shall be set in accordance with the following:
a) If ΔU3 exceeds 5% (0.05 x UR) of the charging voltage in the initial characteristics, the current
value may be reduced by one half, one fifth or one tenth.
b) This ΔU3 is IR drop at start of ‘constant current’ discharge. The R is equivalent series resistance
ESR.
c) The number of significant figures for the discharge current value of 10 A or less shall be one
digit; the second digit of the calculated value should be rounded down.
d) The number of significant figures for the discharge current value exceeding 10 A shall be two
digits; the third digit of the calculated value should be rounded down.
Actual Observed Voltage Profile for constant current
charging/discharging and anomaly with IEC -62931 standard The actual test discharge voltage profile recorded is depicted in figure-3
61
The anomaly between the figure-2 and figure-3 is that the voltage profile while charging and while
discharging is not a linear ramp up or linear ramp down. Instead it appears as if there is exponential
charge discharge like / /charging discharging( ) (1 ) , ( )t RC t RC
R Rv t U e v t U e . But is it actually
exponential? We will analyze this aspect. The first anomaly is that we consider that impedance of the
super-capacitor as standard text book formula, that is 1( )Z ssC
; (in Laplace domain), with C in
Farads. In actual measurements of impedance spectroscopy it is found that super-capacitors are having
impedance as 1 1( ) ,2n
n
Z s ns C
. Here the capacitor behavior is like fractional order impedance. In
real life the fractional order impedance (rather exactly half order impedance) is for lossy distributed R
C transmission line. In super capacitor therefore the new fractional unit of capacity is 1Farad / sec n .
Presently research is directed to give interpretations to these new types of fractional order units arising
out of fractional differential equations. If the impedance of super-capacitor is fractional order then the
volt current time domain relation is d 1( ) ( );d 2
n
n ni t C v t nt
; as against the classical volt current
expression (what we all are used to) that is fractional order d( ) ( )d
i t C v tt
. With this changed law of
Constant Cur.(50 mA) CDC pattern
0.00
0.50
1.00
1.50
2.00
2.50
0 2000 4000 6000 8000 10000 12000
Test time (s)
Vol
tage
(V)
AG-25F AG-20F AG-10F
Figure 3- Constant current (50 mA) charge-discharge pattern of 10F, 20 F and
25 F aerogel supercapacitors, studied by using Super Capacitor Test System.
(Courtesy CMET Thrissur)
62
capacitor theory, we now try to explain the anomalous observed charge discharge voltage profile, to a
constant current charging and constant current discharging excitation to a super-capacitor.
Impedance Representation of super-capacitor We represent a super-capacitor via (i) a pure (loss-less) capacity C in Farads; (impedance1/ sC ), this
contribution is due to basic Electric Double layer (EDLC), or Helmholtz layer near the electrodes
(separation is order of Angstroms). The (ii) part is lossy fractional order impedance call it / nF s , in
Ohm per (fractional order) 1sec n .This part is due to fractional capacity nC , with impedance1/ nns C ,
arising due to rough electrode. Obviously, 1/ nF C in our impedance assumption. The (iii) part is pure
resistance call it ESR (Equivalent Series resistance) R in Ohms. Therefore we say that our s domain
impedance, for a super-capacitor is 1( ) nFZ s Rs Cs
.
Constant current charging-discharging current excitation and its voltage
profile for determining super capacitor parameter The constant charging and discharging is given by summation of step currents as
1( ) ( ) ( ) ( ) ( ) ( )
0c d
t Ti t Iu t I If u t T Ifu t T u t T
t T
The function ( )u t T is Heaviside step input. Where c
d
IfI
, cI is the charging current in our case it
is I , which flows into the capacitor from time 0 cT , and dI is the discharge current (in our case it is
fraction f of the charging current, that is If ), flowing out of the super-capacitor (refer figure-1) from
time cT to dT . In our current excitation case we are not considering the rest period of 30 min, as
depicted in the figure-2.
We use 2
1 1 1( ) , ( ) ,sTu t u t T e ts s s
, 1
1!
n
ntn s
and we generalize the
factorial by Gamma function for non-integer n , by ! ( 1)n n , in the following derivations. From the
current excitation function ( )i t , we write the Laplace transformed expression for current excitation as
63
(1 )( ) c dsT sTI f I IfI s e es s s
The voltage profile to the above current excitation is as follows (always we assume the initial charge in
super-capacitor is zero, therefore initial voltage (0) 0v )
1 1 1 2 2 2
1 (1 ) 1( ) ( ) ( )
(1 ) (1 ) (1 )
c d
c d c d c d
sT sTn
nsT sT sT sT sT sT
n n n
F I I f IfV s Z s I s R e e Fs Cs s s s C
IR IR f e IRfe IF IF f e IFfe I I f e Ifes s s s s s Cs Cs Cs
Taking Laplace inverse of the above expression we get the following voltage profile
( ) ( ) (1 ) ( ) ( )(1 )( ) ( )( ) ( ) ( )
( 1) ( 1) ( 1)(1 )( ) ( )( ) ( ) ( )
c dn nn
c dc d
c dc d
v t IRu t IR f u t T IRfu t TIF f t T IFf t TIFt u t u t T u t T
n n nI f t T If t TIt u t u t T u t T
C C C
Rearranging the above we obtain
(1 ) (1 )( ) ( ) (1 ) ( ) ( ) ( )( 1) ( 1)
( ) ( ) ( )( 1)
n nc c c
nd d d
IF I IF f I fv t IR t t u t IR f t T t T u t Tn C n C
IFf IfIRf t T t T u t Tn C
For 0 ct T , the voltage profile during charging is
( )( 1)
nc
IF Iv t IR t tn C
It has got a constant offset term IR , power law term (proportional to nt ), and linear ramp term
(proportional to t ). At end of charging time ct T , we should have the charged voltage less than or
equal to maximum rated voltage; i.e. ( )( 1)
nc c
c c m RIFT ITv T IR IR K V Un C
,
where( 1)
nc cIFT ITK
n C
.
Now for voltage profile for discharging, ( )v t for time c dT t T , we get
64
(1 )( ) (1 )( )( )( 1) ( 1)
( )(1 )( )( 1)
nnc c
d
n n c cc
IF f t T I f t TIFt Itv t IR IR IRfn C n C
IT If t TIFIRf t f t Tn C C
At time ct T , i.e. start of discharge time, we get ( )( 1)
nc c
d cIFT ITv T IRf IRf Kn C
,
where( 1)
nc cIFT ITK
n C
. The voltage profile is discontinuous at ct T , the point in time when current
changes direction from I to negative If . Therefore at ct T , we have ( )c cv T IR K , and for ct T ,
we have ( )d cv T IRf K .
I
I f
tdT
cT0
I R
R
I R KU
(1 )IR f
cT dT
( )i t
( )v t
t
( ) ( ) ( ) ( ) ( )
( ) ( )( 1)
(1 ) (1 )(1 ) ( ) ( ) ( )( 1)
( ) ( ) ( )( 1)
c d
n
nc c c
nd d d
i t Iu t I If u t T Ifu t T
IF Iv t IR t t u tn C
IF f I fIR f t T t T u t Tn C
IFf IfIRf t T t T u t Tn C
Figure-4 Voltage profile of charge discharge of super capacitor considering fractional order
impedance in super-capacitor
The charge discharge voltage profile is depicted in figure-4, which is what is obtained in our actual
experiments (figure-3). Therefore considering the fractional impedance of super capacitor, / nF s we
get the explanation of anomalous result, contrary to what IEC standard says.
65
Revised Test Procedure We thus state that instead of IEC method, we should follow a revised method, so that fractional
impedance of super-capacitor may be extracted. Therefore we recommend to charge the devise
(initially completely discharged super-capacitor) by a constant current I , till time ct T , such
that ( )c R mv T U V . In our case the cell has 2.5VRU (maximum 2.7V ). After that we immediately
discharge with constant current fI , till time dt T ; at this point of time ( ) 0dv T , that is all the energy
is extracted. At this time we switch off the current to zero, i.e. ( ) 0di T . Use the current excitation and
voltage profile equations to fit the curve of voltage profile to extract the
parameters , , (or ) ,nR C F C n , from following set of expressions
( ) ( ) ( ) ( ) ( )
(1 ) (1 )( ) ( ) (1 ) ( ) ( ) ( )( 1) ( 1)
( ) ( ) ( )( 1)
c d
n nc c c
nd d d
i t Iu t I If u t T Ifu t T
IF I IF f I fv t IR t t u t IR f t T t T u t Tn C n C
IFf IfIRf t T t T u t Tn C
Calculation of time at which power output of super-capacitor goes to
zero Power at time dt T is to be made zero, or we should make ( ) 0dv T , meaning that in the expression
for ( )v t , we place dt T , and equating to zero yields,
(1 )( ) (1 )( )(1 )( 1) ( 1)
n nd d d c d cFT T F f T T f T TR R f
n C n C
This is the case when power output of super-capacitor is zero at time dt T . In simple cases we may
point out the following observations
Power output is always zero for a pure R , as we cannot extract power from a resistor, there is no
energy store.
For a pure capacitor case (without fractional impedance i.e. 0F and with 0R ), we have
(1 )( ) 11d c dd c
T f T T T TC C f
66
For 1f , we have d cI I and 2d cT T , or discharging time is d c cT T T , equal to charging time
period. For 0.5f , the discharge current is half the charging current 0.5d cI I , gives 3d cT T , i.e.
the discharge time period is 2d c cT T T twice the charging time period. For 2f , the discharge
current is twice the charging current 2d cI I , and we get 1.5d cT T , meaning discharge time period is
½ the charging time period ( 0.5d c cT T T ).
For a pure lossy impedance that is ( ) (1/ )Z s R Cs , with, 0F , we have
11d cT T RCf
For a pure fractional impedance that is ( ) / nZ s F s , with 0R and C , we get
1
1
(1 )( ) (1 )( 1) ( 1) (1 ) 1
n n nd d c
d cn
FT F f T T fT Tn n f
For 0.5n , we have2
2
(1 )(1 ) 1d c
fT Tf
. For 1f , we have 43d cT T , meaning that for power out to be
zero, we have to have discharge time period as 1/3rd of charging time period. For 0.5f , we
have 95d cT T , the discharge time period is thus 4/5th of charging time period. For 2f , we
have 98d cT T , that is discharge time period is 1/8th of charging time period. For 0.414f that
is 2 1 , we have charge time period equal to discharge time period ( 2d cT T ).
Observation in this analysis for obtaining dT time at which power (or voltage) is zero for a pure loss
less capacitor and pure lossy capacitor implies that dT and cT are related by only the fraction f , that is
ratio of discharge current to charge current, and the fractional order of the impedance n ; and does not
depend on values of F (or nC ), or C .
Input Output Energy and Efficiency of Energy transfer For the charging period 0 ct T , the power input to the system is (using the expression for
( )v t obtained from excitation ( )i t ).
67
2 22( ) ( ) ( ) ( )
( 1)
n
inI Ft I tp t v t i t u t I R
n C
The energy is 2 1 2 2
2
0
( ) ( )d( 2) 2
cT nc c
in c in cI FT I TE T p t t I RT
n C
Current reverses at ct T , and is equal to If in the discharge period c dT t T . The power output is
thus
2 2
22
22
( ) ( ) ( ) ( )( ) (1 ) ( )
(1 )( )( ) ( )( 1) ( 1)
(1 )( )( )
out c
c cnn
cc c
cc
p t v t i t u t TI Rfu t T I Rf f u t T
I Ff f t TI Fft u t T u t Tn n
I f f t TI ft u t TC C
The energy out is
2
2 1 1 1
2 2 2 2
( ) ( )d
( )
(1 )( )( 2)
(1 )( )2
d
c
T
out d outT
d c
n n nd c d c
d c d c
E T p t t
I Rf T T
I F T T f T Tn
I f T T f T TC
The efficiency of energy transfer is
( )( )
out d
in c
E TE T
We calculate the energy transfer efficiency for pure fractional impedance as follows 2
2 1
0
( ) ( )( 1)
( ) ( )d( 1)
c
n
in
T nc
in c in
I Ftp t u tn
I FTE T p t tn
68
22
2 1 1 1
(1 )( )( ) ( ) ( )( 1) ( 1)
(1 )( )( ) ( )d
( 2)
d
c
nnc
out c c
n n nTd c d c
out d outT
I fF f t TI Fftp t u t T u t Tn n
I fF T T f T TE T p t t
n
Putting1
1(1 )
(1 ) 1
n
d cn
fT Tf
, as obtained in earlier section that time at which power out is zero, we get
1 11 121
1 1
(1 ) (1 )( ) (1 )( 2) (1 ) 1 (1 ) 1
n nn n
nc cout d c c
n n
f T f TI fFE T T f Tn f f
The efficiency is
1 11 1
11
(1 ) (1 ) 1 (1 )( ) ( )( ) (1 ) 1
n nn n
out dn
in c n
f f fE T fE T f
Doing d 0df gives value of 2 1nf , putting this value in
1
1(1 )
(1 ) 1
n
d cn
fT Tf
, we get
2d cT T ,i.e. optimum value of the discharge time to have power output of super-capacitor going to
zero. The maximum efficiency we obtain as 2max (2 1)n .
The derivation of maximum efficiency is as follows:
We optimize 1/ 1 1/ 1
1/ 1
[(1 ) ] [(1 ) 1] (1 )( )[(1 ) 1]
n n n n
n nx x xy x
x
Put (1 ) ; 1n nx z x z we get y as a function of z i.e.
1 1 2 1 1 1 2
1 1
2 1 2
1
( 1) ( 1)( 1)( 1)( 1) ( 1)
( 1) ( 1) ( 1)( 1) 1( 1) ( 1)
n n n n n n n n nn
n n
n n n n n nn
n n
z z z z z z z z zy zz z
z z z z z z z z zz z
69
2 2 1
2 1 1 2 ( 1) 1
22 1 1 1 2 2 1 1 1
1
d 1 d d( ) ( ) ( 1)d ( 1) d d
1 2 ( ) ( 1)( 1)
2 2( 1) ( 1) ( 1)
n n n n n nn
n n n n n nn
n nn n n n n n n
n n
y z z z z z nzz z z z
nz nz z z n z nzz
n z z nz z nz z z z nzz z z
The d 0dyz gives us
2 2 1 1 1 1 12 ( 1) ( 2) ( 1) 1n n n n n n nz z z z z z z z
The LHS is equal to RHS of the above, if 2z . Thus the optimum value is for 2z or 2 1nx . Also
it is verified that for 2z we get d 0dyz and 2z we have d 0
dyz . Thus 2; 2 1nz x gives
maximum value of the efficiency.
Observation is that for a constant current excitation for charging and (complete) discharging, the
efficiency is independent of cT that is charging period and the current amplitude I . As expected
0 for 0n (implying resistive element). The efficiency is 1 for 1n a pure capacitive element.
It is rather low value 17% for a half order fractional capacitor ( 0.5n ). Therefore energy transfer of
super capacitor cell can have maximum 17% energy efficiency in constant current charge discharge
mode as order of these cells are near half.
Introducing Anomalous Transport Mechanism inside super-capacitor In this section we study a electronic practical device that is super-capacitor, where the disordered
electrode structure (figure-5, figure-6, figure-10) manifests as fractional differ-integrals in the transfer
characteristic of the device. Nevertheless the experimental evidence exists where the impedance
spectroscopy tells that this super-capacitor is unlike normal capacitors; having an in-between
behaviour (neither true capacitor nor pure resistance); something in-between such as lossy capacitor.
This anomalous observation is not put to any theoretical framework vis-a-vis character of micro
structural disorder, in earlier researches.
70
A pure ideal capacitor with an Equivalent Series Resistance (ESR) is represented as
impedance 1ESR( ) i( )Z R C , where is angular frequency in radians/sec; and 2 (
frequency in Hertz Hz). At very high angular frequency we get the real part of the impedance;
and at very low the imaginary part dominates. Ideal plot is a straight parallel line to the negative
imaginary axis, cutting real axis at point ESRR ; as the frequency is varied from zero (very low) to
infinity (very high); that is making an angle 090 with the real impedance axis. Practically the
impedance plot of ideal capacitor is not straight line parallel to imaginary axis, but inclined at an
angle (close to 090 ); thereby writing the impedance as 1ESR( ) (i ) ; 0.85Z R C (the
symbol is not efficiency of energy transfer but a fractional number). The inclination is thus 0(90 ) in the case of practical capacitors. These practical capacitors have uniform electrode
devoid of any roughness and are having smooth structure (figure-7).
Collector
Collector
Figure-5: The super-capacitor construction
However the impedance spectroscopy, for super-capacitor, gives rather different conclusion; where we
see 0.5 , in the range of 5 Hz to 5 K Hz of frequency, and 0.9 , at the lower frequency (< 5 Hz).
Therefore the plot of real impedance with imaginary impedance is not a straight inclined line, as we
71
described above, but a curve having two distinct zones. Important one being > 5 Hz, zone where the
incline line is about 045 to real impedance axis-is caller Warburg region (figure-11).
The structure of super-capacitor is depicted in figure-5-7, shows that the electrode structure is rough or
disordered. This disorder or porous electrode enables to make a very high capacity of order of tens of
Farads, as this roughness of electrode increases the effective area to a very high value. The capacity
is /C A d , where the charge separation distance (in order of few Angstroms) is d ; A is effective
electrode area, and is the dielectric constant of the electrolyte. Referring to figure-5, the disordered
electrode is helping to increase the effective area to a very high value thus enhancing the capacity to
tens of Farads in small sized volume. This disordering is done in several ways, like using graphine,
carbon nano tube (CNT), gold foam, carbon foam etc.
The disorder and its ordering for the porous electrode of super-
capacitor by power law distribution The electrode of super-capacitor is ‘rough’, and not smooth. The carbon aero-gel (CAG) tape used as
electrode material its magnified picture is depicted in the figure-6, showing roughness, porous nature
rather we call it ‘disorder’, in the electrode material.
If we draw the packing of pores in 2-D, that will be looking like as in figure-7. This figure-7 show
good packing property, of different size distributions; also used to prepare concrete and is called Fuller
mix. The Fuller mix can be expressed in terms of the grain size ‘distribution function’ as follows 2.5
minmin max3.5( ) 2.5 rr r r r
r (1)
Where we assumed the aggregates to be spherical with diameter r ; and minr , maxr represent the minimum
and maximum pore diameters. The CAG pore diameter is in the nanometer range. The ( )r in (1)
represents the probability distribution function, i.e. ( )dr r is the fraction of grains with diameter in the
interval , dr r r , and max
min
( )d 1r
rr r . The equation (1) and the figure-7 show clearly that number of
small diameter nano-particles is higher than the larger diameter nano-particles.
72
Figure-6: The SEM image of super-capacitor electrode showing roughness & porous nature
(Courtesy CMET Govt. of India Thrissur, Kerala)
Figure-7: Distribution ( r ) of aggregate pores of several sizes, on the electrode surface
73
The disorder of figure-7 can be thus ordered via a ‘power law’ distribution as r ; . This is
depicted in figure 9 (a); the histogram of pore size approximated as power law.
The charge distribution & formation of electrochemical double layer
capacity (EDLC) For the electrode material we consider as simple case made of positive nuclei on fixed ‘regular’ grid
points inside a sea of homogeneous distribution of negative charge. This is depicted in Figure-8 (a).
By cleaving the electrode one obtains two halves, one of which can be considered as electrode
interface. The cleaving is at bx location, as depicted in the figure-8(a). Let us assume that cleavage has
made interface of metal (electrode) and organic (electrolyte); and the immediate picture of negative
charge sea is figure-8(b). This (immediate) new configuration figure-8(b) is energetically unfavorable,
and therefore relaxation of charge distribution takes place and we get the charge distribution as figure-
8(c). The positive nuclei remains fixed, the Q outside ( bx x ) the electrode (metal), leaves
Q deficient inside the metal ( bx x ). With an Electric field applied perpendicular to the electrode the
charge separation at the interface becomes as per figure-8(d). This spatial charge separation forms
“capacity”; the metal-electrolyte capacity mC .
The distance between the centers, as per the charge distribution finally plotted in figure-8 (e), are ( )mx
and ( )mx are given as (using moments), with respect to one dimensional charge distribution ( )Q x as
function of x . The ( )mx and ( )
mx are the x coordinate positions of the ‘peak’ density of negative and
positive ( )Q x in figure-8(e).
This argument states that planer charge density (figure-8 c, d, e) extends beyond the boundary cut of
electrode bx ; giving formation of EDLC. This phenomenon is base of formation of capacity mC at the
interface of electrode-electrolyte cut.
74
dbx
x
bx
bx
bx
x
( )a
( )b
( )c
( )d
( )e
Q
Q
Q
Q
Q Q
Q Q ( )mx ( )
mx
0
bx
x
x
x
Figure-8: Charge distribution at cleavage of electrode crystal and formation of double layer
capacity
Calculation of Capacity A very simple calculation we do based on Poisson’s equation; that is
2 ( ) 4 ( )V x x (2)
In (2) ( )V x is the potential function, and ( )x is the charge density function. Inverting the (2) twice we
get the following integral equation for potential at any point x as
( ) 4 ( )( )dx
V x x x x x
(3)
From (3) we can write plus potential ( )V x for region ( bx x ) and minus potential ( )V x for region
( bx x ) due to positive & negative charge densities as in figure-4e; as follows
( ) 4 ( )( )d ( ) 4 ( )( )db
b
x
b b b bx
V x x x x x V x x x x x
(4)
We can write the total charge at bx as following i.e. integrating the charge density function.
75
( )d ( ) ( )d ( )b
b
x
b bx
x x Q x x x Q x
(5)
Also note that total charge is zero that is also evident from figure-8(e)
( )d ( )d ( ) ( ) ( )d 0b
b
x
b bx
x x x x Q x Q x x x
(6)
Balancing the moments for the region bx x , we write the following
( ) ( )
( )( )d[ ][ ( )] ( )( )d
( )
b
b
x
x b
m b b mb
x x x xx Q x x x x x x
Q x
(7)
Similarly as above procedure, doing moment balance for the region bx x , we get ( )mx as
( )
( )( )d
( )bx
mb
x x x xx
Q x
(8)
The (7) and (8) show that there is charge separation. This separation of charges is the cause of EDLC,
and its capacity is defined as /mC Q V . Here Q represents separated charge that
is ( ) ( )b bQ x Q x ; with V as difference of potential. From (4) we write the potential difference
( V ) as
( ) ( ) 4 ( )( )d 4 ( )( )db
b
x
b b b bx
V V x V x x x x x x x x x
(9)
Dividing the (9) by ( ) ( )b bQ x Q x Q , we get the following
76
( )( )d ( )( )d4
( )( )d ( )( )d4
( ) ( )
( )( )d ( )( )d4
( ) ( )
b
b
b
b
b
b
x
b bx
x
b bx
b b
x
b bx
b b
x x x x x x x xVQ Q Q
x x x x x x x x
Q x Q x
x x x x x x x x
Q x Q x
(10)
Using (7) and (8) we write (10) as
( ) ( )4 ( )m mV x xQ
(11)
We obtain Electrochemical Double Layer Capacity (EDLC) as
( ) ( )
14 ( )m
m m
Cx x
(12)
The (12) can be approximated as 1/(8 )m mC x where ( ) ( )m m mx x x ; for symmetrical cases.
Distribution in capacity as power-law In the earlier section, we saw the formation of EDLC; if the electrodes were smooth, that is without
any disorder, we would be having same charge separation (say as in figure-8e) for the entire electrode.
In other words the capacity will be having same average value call it mC C . A small spread we are
assuming in distribution of mC in case of smooth electrode; as practically the perfect smoothness is not
attained. Also a small spread around the average mC C is due to distribution of relaxation rates due
to spread in activation energy and ambient temperature, by thermodynamics. We can assign the mean
value mC C to the inertia of the relaxing capacity, and its spread (standard deviation) to the degree
of interaction/disorder between the relaxing modes. This standard deviation is the cause of non-ideal
capacitor line with 0.85 , the slight inclined line (figure-12).
77
Due to spatial disorder in microstructures of the electrodes of CAG this charge distribution is spatially
spread and is not unique. Figure-8 (e) the distribution function is different for different cleavage. Some
cleavage may be symmetrical, as ideal as shown in figure-8 (a); some may have different nuclei
distribution near cleavage, with different numbers as per crystal face cleavage of electrode. However,
due to rough nature the charge distribution function figure-8(e) at each of the cleavage is different- the
distribution does not and need not be a normal, Gaussian type; tends towards asymmetric distribution.
This ‘fractal charge distribution’ can lead to a distribution of capacity of ‘rough’ electrode other than
normal or Gaussian distribution-leading to power law distribution ( c ), shown in figure-9 (b).
In previous section we have seen that the pore-size is having power law distribution, depicted in
figure-9(a). The larger sized aggregates in figure-7 have lower capacity than that of smaller sized
aggregates, depicted in figure-9(b). This is due to the fact that larger sized aggregates (or pores) have
number of charges more than that of the smaller sized ones. This fact gives rise to larger (height) of sea
of negative charge at cleavage (figure-8(a)); and the spillage (outside bx of figure 8 (c) (d) (e)) of this
larger sized charges will longer. Thus the negative charge for larger pore will be spilling to a greater
length thus giving ( )mx and ( )
mx greater than for the small sized aggregates (where spillage is to a
smaller distance). From this argument and expression (12) of the capacity we see that capacity is lower
for larger sized aggregates. Therefore we will get inverse of the power law of size statistics, as depicted
in figure-5 (b), following c law, for the distribution of capacity (EDLC) spread spatially in rough
electrode.
78
Frequency
minrmaxr
r
Pore - Size
Frequency
CapacitymC
( )a
( )b
Figure-9: a) Showing distribution of pores size, b) Corresponding distribution of capacity
However, the picture of rough electrode is different; could be electrically represented as shown in
figure-10. We can infer from figure-10 that on small scale the surface is rough; this leads to a charge
distribution on the electrode surface which will be distributed depending on crystal face which forms
the electrode surface, with size of aggregates. The capacities, mC as described above, are shown as
'Dl sc in figure-10.
79
R 1Ez 2Ez3Ez E kz E Nz
1D lc
2D lc
3D lc
D lkc
D lNc ( 1)D l Nc
1sz 2sz 3sz skz sNz
1lr 2lr 3lrlkr
lNr ( 1 )l Nr
Figure-10: Depicting circuit picture of a rough disordered electrode
Im Z
R e Z0 .1 0 .2 0 .3
0.2
0.4
0.6
5 H z
5 K H z
EDLCSuper-capacitor
Ideal-capacitor
Impedance SpectroscopyNyquist Diagram
Practical-capacitor 0
0
1E S R
0
( ) i( )
90
Z R C
1E S R0
( ) (i )
(90 )
Z R C
Region of constant phaseWarburg Region 0.5
2: rad ian / sec: H z
Figure-11: Impedance Spectroscopy showing Warburg Region of Super-Capacitor
80
The difference between the conventional capacitors and EDLC are not only in their structure but also
in the electrical characteristics as observed in to some extend we discussed this with impedance
arguments in the earlier section. In EDLC, the diffusion phenomena in the electrolyte and the size of
the electrode pores (figure-5) are very important. Thus when frequency is risen > 5 Hz, the number of
active porous layers accessible are reduced, diminishing the resistance and capacitance; gives
fractional behavior in this region having constant phase angle.
We will be concentrating on this particular region of frequency of operation region i.e. > 5 Hz, the
Warburg Region (figure-11). In figure-11 ideally when we increase the frequency to a very large value
from the Warburg region we will enter into positive imaginary axis of the Nyquist plot. This is due to
the fact a series ‘inductance’ will be prominent at very high frequency HFL will get added to overall
impedance function ( )Z . Therefore we are taking a cut-off of the practical frequency range for our
study.
Debye and Non Debye Relaxation Let a capacitor C be connected to a battery of BBV Volts, at time 0t ; obviously this capacitor will get
charged to the battery voltage. Let this capacitor is uncharged at 0t , thus there is no charge held by
it, therefore the voltage across the capacitor is zero at 0t , and (0 ) 0i . The voltage balance equation
assuming R be the total resistance of the circuit at 0t is the following
0
1 ( )d ( )t
BBi t t Ri t VC
Where ( )i t is the charging current flowing into the capacitor. The above equation is summing up the
voltage across a pure capacitor; got by charge accumulated at time t ; ( )Q t Coulombs divided by
capacitance value i.e. 0
( ) 1( ) ( )dt
cQ tV t i t t
C C with voltage across resistance ( ) ( )RV t Ri t and
equating it to the battery voltage BBV . The above integral equation may be differentiated and is put as
following, for 0t
81
1d ( ) ( ) ( ) ( )d
BBVi t i t t RCt R
The solution to the above system is ( ) ( / ) tBBi t V R e , that is the ‘impulse response’ of the circuit
equation. The relaxation current of the above system follows Debye’s relaxation, with one relaxation
rate (also termed as Maxwell-Debye law). From indicial polynomial point of view, we have indicial
polynomial for above differential equation as s . Thus the homogeneous solution of above is from
inverse Laplace of inverse of indicial polynomial, that is 1 1( ) ( )ts e i t . Physically, it
means that if we excite the above system with current input of unit delta function, the system will have
relaxation as ( ) ti t e . With this as Green’s function we find if the input is say step current, call
it 0 ( )I u t , where ( ) 1u t for 0t and ( ) 0u t for 0t ; then we get relaxation function for current as
convolution integral, i.e. ( ) ( )0 00 0
( ) ( ) d dt tt ti t I u e I e . From here we get the current
as 0( ) ( / )(1 )ti t I e . This is also obtained from Laplace inverse of the indicial polynomial
multiplied by Laplace of the Step input, i.e. 1 10( ) [ ( )]i t I s s . Doing partial fraction and
inverse Laplace we again get 0( ) ( / )(1 )ti t I e . In the above indicial polynomial the parameter s is
Laplace (complex) frequency; s ; with Re s , in time domain response signifies transient
behavior, and Im is ; with representing angular frequency (Radians/sec), states steady state (at
large time t )behavior in time domain. Thus is ; i 1 . Consider a partial differential
equation (PDE)
1
( , ) ( , ) ( )i t i t tt
(13)
With ( , ) 0i t , for 0t , and 0; . The above PDE (13) is having a free parameter . Now
if the free parameter 1 , then we have single time constant system ( 1 1( )RC ) with solution
as ( , ) ti t e . This is similar to relaxation current when a capacitor is connected to a battery of
1VoltBBV .Capacitor charging current with circuit resistance /( ) t RCi t e , with 1( ) ;RC RC ,
as unique time constant (rate), remaining, and its constitutive equation is
d ( ) ( ) ( )d
v t v t tt
(14)
82
The R in RC for (14) is ohmic resistance of the electrode plus collector metal, (plus any
resistance if put with the capacitor) and mC C , the average capacity, as for ‘smooth’ electrode
without any disorder. The standard Debye relaxation equation is (14); and thus valid for system
without disorder or roughness. This Debye system relaxes via (14) with only one mode that with only
one relaxation constant. We can thus name (13) as Non- Debye relaxation equation, where there are
variety of ways the system relaxes. Refer figure-6 there are several capacities of various values are
formed due to disordered electrode; hence this disordered system has varieties of ways of relaxation.
The distribution of capacity as discussed earlier follows a power law c .
Impulse response function and impulse response for super-capacitor
with disorder in porous electrode The figure-9 and figure-10, gives a picture of spatial disorder, and distribution of disorder. We may
assume that this disorder, manifests as several relaxation rates i 1,2,....i (ideally). Due to this
nature a particular charge/discharge or relaxation rate as representing by unique rate may not be
possible. These several time constants (relaxation rate) is taken as power law distribution
as (1/ )( )q , with 1/ 0 1q .The strong-relaxation or exponential charge/discharge
with one time constant follow a normal distribution with well defined average that represents average
time constant or relaxation rate, and that normal distribution has well defined standard deviation.
Unlike the normal distribution the ‘power-law’ distribution has no defined average or moments or
(standard deviation); and is representation of system which has variety of ways to relax. The
heterogeneity or the disordered system thus has varieties of ways by which dissipation mechanism
takes place. We can thus write solution to (13) as 1
( , ) ( , ) expi t h t t
(15)
( , )h t denotes the ‘impulse response function’. On integrating this ‘impulse response function’ (15) on
the free variable ( ) from 0 to , we get the function of time and that is called ‘impulse response’ or
the Green’s function (16)
1
0 0
(1 )( 0 , ) ( ) ( , )d exp di t g t h t tt
(16)
83
To get above substitute 1/ 1 (1/ ); ( / ) ; d ( / )dx x t t x , and by using definition of
Gamma function 1
0
( ) dy kk e y y
, and its property ( ) (1 )k k k the steps are as follows to get.
(1/ )
0
1
0
11
10 0
( ) d
d
( ) (1 )d d
x
x
x x
g t e xt
x xe xt t
xe x e x xt t t t t
(16)
Note that in case of Debye relaxation, the decay is te , that is the impulse response of the (14),
where as the impulse response of the disordered relaxation is a power law t ; with 0 1 , with a
long tail lingering in late times. With brevity we may also state that the infinite
series1
(1 )ktk
e t
; where probability distribution of is (1/ )( ) , 0 1 . Physically
meaning each Debye relaxation adds to give final power law decay in time.
Appearance of Fractional derivative-in disordered electrode of super-
capacitor Instead of delta function excitation let the Non-Debye system (13) be excited by ( )f t , a derivative of
function ( )f t ; so we write this as (17)
1
( , ) ( , ) ( )i t i t f tt
(17)
Then the response to this new excitation call it ( )i t is convolution of Green’s function (16) obtained
above with the forcing function, ( )f t that is as follows
0 0
( )( ) ( )* ( ) ( ) ( )d (1 ) d 0 1t t f ti t g t f t g f t
(18)
Multiplying and dividing the above expression with (1 ) and using the definition of fractional
integral that is 10 0
0
1( ) ( ) ( ) ( )d( )
t
t tI f t D f t t f
, we have (19)
84
( )
0
(1 )0
0
( )( ) (1 ) (1 ) ( )d(1 )
(1 ) (1 ) ( )
(1 ) (1 ) ( )
t
t
t
ti t f
D f t
D f t
(19)
Implying the appearance of fractional derivative for cases where several time-constants (ideally infinite
of them) define a relaxation process. Therefore a disordered relaxation (response) may well be
formulated by fractional differential equation, the order giving the ‘intermittency’ of relaxation
disordered process. The relation between relaxation index and the electrode roughness index , is a
topic of research.
Let this system with disorder as represented in (17) be excited by a current source which is a delta
function say 0 ( )I t . With this excitation the relaxation current would be fractional integral of the input
excitation that is from (19) (1 )0 0( ) (1 ) (1 ) [ ( )]ti t I I t . We have fractional integration of delta
function as 10 ( ) / ( )p p
xI x x p ; and using this formula we get 0( ) (1 )i t I t . This was what was
derived in (16) as impulse response. If the excitation current source is a step function as 0 ( )I u t ; then
the relaxation current is fractional integration (19) of order (1 ) ; that
is (1 )0 0( ) (1 ) (1 ) [ ( )]ti t I I u t . Using the formula for fractional integration of a constant
i.e. 0 C C / (1 )p pxI x p , we have; the relaxation current
1 1( ) [ (1 ) (1 )( )] / (2 ) [ (1 )( )]/(1 )i t t t
We have just identified that that the micro-structural disorder with pore size distributed as power law
r , with ; 1 gives a capacity distribution as power law c ; in turn gives a relaxation
rate distributed as power law, i.e. (1/ ) 0 1 ; which further gives a t relaxation (a long
tailed) as impulse response (instead of te ). Here we may point out that several relaxation rates
in , in the disordered system with power law index 1/ does not appear in the relaxation function
that is impulse response or other temporal responses. Now let us consider an impulse response
fractional differential equation as
d ( ) ( ) ( ) 0 1di t i t tt
(20)
85
The solution of this (20) is by one parameter Mittag-Leffler function, that is ( ) ( )i t E t ; where
we have 0
( ) ( ) / ( 1)nn
E t t n
.For 1 , we have 1( ) tE t e , (20) reduces to (14),
the system without disorder. The early and late times approximation of the one parameter Mittag-
Leffler function that is
11 ;(1 )
( )1 1;
(1 )
tt
E tt
t
(21)
A ‘two-parameter Mittag-Leffler’ function is , 0( ) ( ) / ( )k
kE z z k
, from here we
have ,1( ) ( )E z E z ; the one parameter Mittag-Leffler function. For large z , the expansion is
1,
1( )
( )
kpp
k
zE z zk
.For (21) only 1p is used.
It is interesting to observe that late time impulse response solution of (20) is t ; similar to one
obtained as t , for the case (13). Interestingly for the (20) impulse solution returns the index of
fractional order of the differential equation ( ) in the late time response, whereas the time response of
impulse solution of the (13) has the index of disorder ( ).
The (20) is generalized fractional relaxation equation let us write a general relaxation process for
( )i t as a convolution process from time beginning of relaxation process to present time, as (22)
0
d ( ) ( ) ( )d ( ) ( )d
t
i t k t i k t i tt
(22)
In (22) we have put a temporal kernel in the RHS convolution. This kernel we call Memory kernel.
With the choice of the memory kernel as delta function as ( ) ( ) ( )k t t , where is constant, then we
have (22) modified as
00
d d( ) ( ) ( ) ( )d ( ) ( ) ( ) ( ) ( )d d
tti t t i i t i t i t i t I e
t t (23)
In (23) we have initial condition (0 ) 0i ; and at 0t , we are exciting the system with 0 ( )I t . This
system (23) with memory kernel as delta function is called memory less system, with only one
86
relaxation mode given by unique relaxation (average) rate . This is a memory less system and
corresponds to disorder less electrode i.e. ideal normal capacitor. This (23) is same as (14) with
relaxation current to impulse excitation of value 0I is noted too. Now we change the kernel to a
‘power-law’ kernel as 20( )k t K t , and write the (22) as follows
20
0
d ( ) ( ) ( )dd
t
i t K t it
(24)
0K a constant. Manipulating (24) and using definition of fractional integration we get
2 100 0
0
( 1)d ( ) d ( )( ) ( )d ( 1)d ( 1) d
t
tKi t i tt i I K
t t
(25)
In (25) we apply 10 tI i.e. one-integer order integration to the both sides (or integrating once) to get the
following integral equation of fractional order
0( ) (0 ) ( )ti t i I i t
(26)
Taking Riemann-Liouville (RL) fractional derivative of order for both sides of (26) we obtain
0 ( ) (0 ) ( )(1 )ttD i t i i t
(27)
In doing (27) we assumed a constant initial value i.e. (0 )i and used formula of RL fractional
derivative of a constant, i.e. { }c as 0 { } ( ) / (1 )p pxD c cx p . With initial state as zero we write (27) as
d ( ) ( ) 0d
i t i tt
(28)
describing the homogeneous system of (20).
Here we have as appearing as index of fractional order derivative, having relation to time-decaying
memory kernel function ( )k t ; does this have some relation to power law index of (20), the
exponent of power law distribution of the relaxation rate, and the exponent power law for disorder;
is a topic of practical investigation.
87
Implication of Fractional Impedance In experimental observations we find that super-capacitor has fractional order impedance, and we tried
to relate the same with fractional differential equation, by stating the roughness of electrodes. The
impedance ( ) , 0 1nZ n has implication of dissipation theory. Practically on applying a step
input voltage 0U at 0t to a super-capacitor, we get a power-law decay of current.
0( ) 0nn
Ui t tk t
This is as per Curie relaxation law. The parameter nk is constant depends on capacity of super-
capacitor. This is from observation and the evaluation of order of power-law function is 12
n . The
Laplace of step input is 0 /U s . Taking Laplace of above power-law decay current, we get
0 01
(1 ) (1 )( ) n nn n
n U UnI sk s k s s
For 0 1VU , we get Transfer function of super-capacitor as
( ) (1 )( )( ) (1 )
n n nn n
n
kI s nH s s C s CU s k n
This is also admittance of super-capacitor or impedance equaling 1( ) , 0 1nn
Z s nC s
, this is
fractional impedance.
Let the super-capacitor be excited by a sinusoidal voltage (i )U . Then the current is also sinusoidal as
i / 2
(i ) (i ) (i )
[ (i )](i ) (i )
(i ) cos isin2 2
i sin (i ) 1 tan(1 ) i sin (i ) 1 tan2 2 2
nn
n n n nn n
nn
n nn n
I C U
C U C U e
n nC U
n nC U n C U
88
Where (1 )2 2 2 2
n n , this loss tangent is independent of . For pure ideal capacitor
with 1n , we have the ( )U i lags current ( )I i by2 , so tan delta is zero, and there is no loss in
ideal capacitor. We elaborate this below
From the above derivation we write,
1
1( ) i ( ) sin cos2 i 2
(i ) sin ( ) cos ( )2 2
nn
n nn n
n nI C U
n nC U C U
Recognizing the Fourier operator pair didt
, we write expression for ( )i t the current through super-
capacitor when ( )u t voltage is applied as
1 d ( )( ) sin cos ( )2 d 2
d ( ) 1 ( )d
n nn n
pp
n u t ni t C C u tt
u tC u tt R
Note that ( )u t in this is not necessarily a unit step voltage, though the symbol used in earlier section is
that of Heaviside step input; here it is time dependent voltage excitation. The above time domain
expression of a current and voltage gives an equivalent parallel combination of equivalent resistor and
equivalent capacitor connected in parallel, with
1 1sin2 cos
2
np n p
nn
nC C R nC
Thus with this fractional impedance of super-capacitor we get an equivalent parallel model where pC is
parallel frequency dependent capacity, shunted with frequency dependent resistance pR , that is we get
equivalent p pR C . As 0 1n ; the pC decreases with increase in frequency , and 1
pRincreases
with frequency , so as to make loss tangent independent of frequency. For a parallel circuit
89
with R C , we have loss tangent formula as 1tanRC
, using this we calculate the following
for 1( ) nn
Z ss C
.
1
1 1tan cos2sin
2
cot tan2 2 2
(1 )2
nn
np pn
nCnC R C
n n
n
The above discussion points out to a new treatment towards super capacitor impedance.
Constant Voltage charging & discharging for determining the super
capacitor parameters In the IEC treatment we got a constant current charging case; thereby we observed a voltage which is
not a ramp was due to fractional order loss element. In classical circuit theory, if we charge an ideal
capacitor, C through a resistor R , via a step input voltage RV (figure-12) we get voltage across
capacitor as exponential /0 ( ) (1 )t RCv t V e . Now we will see that when we charge a super-capacitor
with impedance 1( ) nn
Z ss C
through a resistor R the charging profile is not exponential.
Here we will use a constant voltage excitation of RV from time 0t , to time ct T ,(as charging phase,
through a known resistor R ) and thereafter we will switch to discharging phase i.e. voltage source
will be made zero. By this we record the charging and discharging profile 0 ( )v t , and map the curves
with our derived expressions (as we will be shortly doing) to obtain parameters of super-capacitor, nC ,
sR and n .
90
0t
RV
1( )Z s
2( )Z s0( )V s
20
1 2
2 1
0
0
( )( ) ( )( ) ( )1( ) ; 0 1; ( )
(1/ ) 1( ) ( ) ( ) ,(1/ )
( ) ( ) ; ( ) 1 for 0 else ( ) 0
( ) ( )( )
in
nn
nn
in inn nn n
in R
R Rin n
Z sV s V sZ s Z s
Z s n Z s Rs C
s C kV s V s V s kR s C s k RC
V t V u t u t t u tV V kV s V ss s s k
10( )
( )R
n
V kv ts s k
Figure- 12: The constant voltage charging of super-capacitor
We use 1 ( ),
!( )p p p st E ats a
, for 0p , n , 1n to have
11
, 1( )n nn nn
s t E ats a
.
With this we obtain
10 , 1
, 1
( ) ( )( )
n nRR n nn
nnR
n nn n
V kv t V kt E kts s k
V tt ERC RC
The , ( )nE at is two parameter Mittag-Leffler function; as defined below;
, ,( 1)0 0
( ) ( )( ) , ( )( ) ( 1)
l n ln
n nl l
x ktE x E ktl nl n
We have alternate derivation via series expansion
91
01
2 3
1 2 3
2 3
1 2 1 3 1
( )( ) 1
1 ...
...
R Rn
nn
Rn n n n
R n n n
V k V kV sks s k ss
V k k k ks s s s
k k kVs s s
Use Laplace pair 1
1( 1)
n
nt
s n
to invert term by term the above to get following
2 2 3 3
0
2 2 3 3
0
( ) ...( 1) (2 1) (3 1)
1 1 ...( 1) (2 1) (3 1)
( )1 1 ( )( 1)
1
n n n
R
n n n
R
n mn
R R nm
n
R nn
kt k t k tv t Vn n n
kt k t k tVn n n
ktV V E ktmn
tV ERC
Where, ( )nE x is one parameter Mittag-Leffler function used above, with 1( ) xE x e . Therefore for
integer order capacitor with 1n , we have normal exponential charging 0 ( ) (1 )t
RCRv t V e
. For
voltage charging expression for fractional order impedance we have
0 , 1( ) 1n n
nRR n n n
n n n
Vt tv t V E t ERC RC RC
For charging current of circuit of figure-12 with 1Z R and 21
nn
Zs C
, we have
1
( ) 1( ) 1
R nR R
n
nnn
VV V ssI s
Z s R ss R RCs C
Using 1
( )n
nn n
sE ats a
, we get inverse of above as
92
( )n
Rn
n
V ti t ER RC
Clearly for ideal 1n case we get ( )tR RCVi t e
R . The differential equation corresponding to figure-1
for 1n , is ordinary differential equation (ODE)
0 0d ( ) ( ) ( )d inRC v t v t v tt
For 1n we get fractional differential equation (FDE)
0 0d ( ) ( ) ( )d
n
n innRC v t v t v tt
We now consider a lumped ESR sR for super-capacitor, thus for figure-12 we have
21( ) s n
n
Z s Rs C
while charging impedance remains at 1( )Z s R . Therefore for any input
voltage ( )inV s , we write the charging current (in Laplace domain) as
( ) ( )( ) 1 ( ) 1
nin n in
nn s
s nn
V s s C V sI ss C R RR R
s C
Output voltage in Laplace domain is therefore
0 2
1
( ) 1( ) ( ) ( )( ) 1
( ) ( )( ) ( ) ( ) put ( )1( ) 1
( )
11( ) 1
( )( )
n nin n s n
n nn s n
nin in s
nin in s n n s s R
innnn s
n sn
R sR
nn s sn
n sn s
V s s C s R CV s I s Z ss C R R s C
V s V s s RV s V s R C C R R R R VV ss C R R ss
C R RV RV s
C R R R R ss s C R RC R R
Use 1 ( ), ( ) !p p st E at p
s a
; for 1, , 1p n n and 0, , 1p n to get
inverse Laplace of above output voltage expression
0 , 1 ,1( )( ) ( ) ( )
n nn R sR
n n nn s n s s n s
V RV t tv t t E EC R R C R R R R C R R
93
Let us keep the step input from time 0t to ct T , and then at time ct T , the output voltage will be
0 , 1 ,1( )( ) ( ) ( )
n n nR c c R s c
c n n nn s n s s n s
V T T V R Tv T E EC R R C R R R R C R R
We calculate now the voltage profile and then voltage at ct T , for only fractional impedance 1n
ns Cof
the impedance 2 ( )Z s comprising of sR plus this fractional impedance, the voltage is
( )1 1( ) put ( )( ) 1
1( ) 1
( )
nn in R
c s inn n nn n s n
R
n s n
n s
s C V s VV s I V ss C s C R R s C s
VC R R
s sC R R
Using the previous Laplace identity of Mittag Leffler function, we write
, 1( )( ) ( )
nnR
c n nn s n s
V tv t t EC R R C R R
At ct T we thus have the voltage at the fractional impedance
, 1( )( ) ( )
n nR c c
c c n nn s n s
V T Tv T EC R R C R R
This above value of voltage becomes the initial voltage while we discharge the super-capacitor, we call
it (0)cv . Now we see the discharge profile, as the charged fractional order impedance with above value
discharges through R . The discharge current is now
(0) /( ) 1c
s nn
v sI sR R
s C
The voltage output profile of the discharge is thus
0
1
(0) /( ) ( ) 1
(0)1
( )
c
s nn
nc
ns
n s
v sV s I s R RR R
s Cv R sR R s
C R R
Using the same method we get Laplace invert of above and the discharging profile is
94
0 ,1 , 1(0)( ) (0) ( )
( ) ( ) ( )
n nnc R c c
n c c c n ns n s n s n s
v R V T Ttv t E v v T ER R C R R C R R C R R
Here we have discussed alternate approach of constant voltage charging and discharging to get
parameters of super-capacitors.
RV
tcT0
cT
( )inv t
0 ( )v t
t
0 , 1 ,1
0 ,1 , 1
( ) ; 0( ) ( ) ( )
(0)( ) (0) ( ) ;( ) ( ) ( )
n nn R sR
n n n cn s n s s n s
n nnc R c c
n c c c n n cs n s n s n s
V RV t tv t t E E t TC R R C R R R R C R R
v R V T Ttv t E v v T E t TR R C R R C R R C R R
0 ( )cv T
0
( )c cv T ( ) (0)c c cv T v
Figure-13: Constant voltage charging and discharging voltage profile at super-capacitor
The figure-13 displays the curve of voltage profile for a constant voltage charge and discharge case.
Here we point out that the charging curve though similar to exponential charging of a text book
capacitor /0 ( ) 1 t RCv t e , but it is not so. Similarly the discharge profile though similar to exponential
decay /0 ( ) t RCv t e , but is not so. The curves are governed by Mittag-Leffler function instead, where
the fractional order of fractional differential equation i.e. , 0 1n n plays vital role.
With this excitation of constant voltage charging and discharging through a known resistance R one
may gets the output voltage plots and thereby invoking the derived expression it is possible to extract
the parameters of fractional capacitance and the fractional order ( , ,s nR C n ).
95
Conclusion In this chapter we simply developed the charging and discharging profiles what we actually observe in
super-capacitor testing circuit results. These include the effect of fractional order impedance internal
resistance etc. What we pointed out that recommended IEC standard of super-capacitor testing method
does not include the reality of fractional order impedances. It is better method proposed here to deal
with actual waveforms of the test results. The expressions obtained in this chapter therefore can be
used to estimate the parameters of super-capacitors, unlike IEC method where the super-capacitor is
taken as ideal text book impedance. We have also seen the implication of this fractional impedance,
first calculation of energy transfer efficiency and then giving loss tangent definition other that
classically defined one, with the new loss tangent derived as independent of frequency. Here we tried
to relate the roughness of electrode to the order of fractional impedance, and this is new direction in
super-capacitor circuits and systems. This new fractional circuit theory can be extended to battery
dynamics where Warburg impedance is present as fractional loss element.
96
Chapter-3
Application to Real Life Physical Systems
Introduction In the previous chapter, we applied fractional calculus approach to explain the charge-discharge
anomalous behavior observed in basic electrical element as super-capacitors. Now apply the fractional
calculus to few other real life systems, well fractional calculus speaks the language of nature, and to
apply the same to some natural phenomena is thus ‘efficient way of communication’. Several examples
are there in reference, we have taken few new mathematical modeling of natural phenomena here. We
have taken three examples which are described in the following sections. The first one deals with
spreading of viscous fluids and application of fractional calculus, in its dynamics. Here we try to give
memory integral treatment and a suitable fractional differential equation to stress strain relation. For
this example we have extended the interpretation to develop a ‘continuous order’ system where we can
have several memorized relaxation. These particular phenomena we still have to experimentally,
mathematically verify this proposition, though we have developed the theoretical basis of the same-i.e.
continuous order differential equation. The second real life problem is Mechanism of random delay in
networks of computer. This is of particular interest as it not only gives a new direction to statistical
mechanics but also aimed at to be used in systems where computer networks are used in feedback
controllers.
Spreading of viscous fluid and fractional calculus Here we study the spreading of a fluid on a solid substrate under an impressed force. The apparatus
consists of two smooth glass plates with a drop of the liquid sample in between the plates. We use a
CCD camera to record the variation of area (figure-1a), which is a measure of the strain. When the
liquid is Newtonian, the area-time plot shows a gradual increase, with the area reaching a saturation
value after a certain time. However, for a non-Newtonian fluid, the area-time plot shows an initial
increase with a slightly oscillatory nature before saturation is reached, somewhat like the oscillations
of a damped vibrating spring. We attribute this to the visco-elastic nature of the sample. An attempt is
97
made to analyze a visco-elastic system starting from the basic differential equation, and with suitable
boundary conditions, a solution qualitatively in agreement with the experimental graph is obtained.
The approach utilizes fractional calculus which has been found to be extremely useful for non-linear
systems.
A Newtonian liquid film (two experiments were conducted-one with ‘ethylene glycol’ and the other
with ‘castor oil’) is sandwiched between two identical solid plates (‘glass’ and ‘perspex’). A load is
placed on top of the upper plate. The load is varied from 1 to 5 kgs. Here, ‘ethylene glycol’ is a polar
fluid with low viscosity while ‘castor oil’ is a relatively less polar fluid with high viscosity. ‘Perspex’
used has a lower surface energy while glass has a higher energy associated with it. Once the load is put
on the upper plate, the plate separation goes on decreasing with the sandwiched liquid gradually being
squeezed out. Thus, the ‘area’ of fluid in contact with the plate increases with ‘time’. The change in
area is a measure of the strain.
The process is video recorded by means of a CCD camera placed beneath the lower plate. The video
camera is fed to a computer where the captured image (@10 frames/sec) is edited by means of the
Image-Pro software. The area of the liquid film is calculated at different instants of time, using the
software. The area-time plot shows a non-linear nature with the area gradually increasing and finally
saturates to a value depending on the load. The plot, with ‘castor oil’ on ‘perspex’ substrate is shown in
Figure.1. The plot also reveals that with increasing load, the area also increases.
98
Figure-1a: A snapshot of the film (inner blob) superposed on the photograph of the film taken
about 2 s earlier (outline visible along the periphery) shows the shrinking of the film. The colors
have been adjusted for clarity. Courtesy Dept. of Phys; University of Jadavpur Kolkata
99
Figure- 1b : An area-time plot (castor oil on perspex). Courtesy Dept. of Phys; University of
Jadavpur Kolkata
We use “arrowroot” solution as the non-Newtonian fluid with “glass” as the substrate. “A non-
Newtonian fluid is a substance in which the stress is not proportional to the strain-rate and an effective
co-efficient of viscosity varying with strain rate may be defined”. If we plot the change in ‘area’ (a
measure of ‘strain’, once again) with ‘time’, we get a graph which significantly deviates from the one
obtained for Newtonian fluid (figure 1b, and 2). It first gradually increases, but then shows an
‘oscillatory’ nature with time, this is particularly prominent for the loads 3 and 4 kgs.
This oscillatory nature of the ‘area’ (i.e. ‘strain’) is a radically different feature of ‘strain-time’
relationship as we find here, in case of a non-Newtonian fluid. It can well be attributed to the ‘visco-
elastic’ nature of the sample, thereby revealing a fundamental departure from the Newtonian feature.
The anomalous behavior and fractional calculus The non-Newtonian fluid behavior is depicted in figure-2. A non-Newtonian fluid reveals anomalous
visco-elastic properties as compared to Newtonian fluid, the ‘beads-on- a- string’ structure gets formed
in visco-elastic fluid of non-Newtonian in nature. The spreading experiment is conducted where
arrowroot solution is kept between two glass plates and steady load is applied. A camera is kept below
100
to capture snap shots regularly to record the spreading pattern. The area is calculated graphically later,
and its plot with respect to time for various loads is shown in figure-2. Interestingly the observation
reveals oscillatory nature of the spreading. We relate area to strain and following analysis show that
the fractional differential equation gives suitable explanation of this anomalous behavior, of non-
Newtonian relaxation (with memory). Our experiment show that the fractional order corresponding to
this oscillatory relaxation behavior is of fractional order 1.5q . We write for Newtonian fluids a
lumped spring and a lumped dashpot model as
d ( ) E ( ) ( )d
t t tt
(1)
The above equation is generalized to representation of the stress-strain in distributed spring and
dashpot system for a non-Newtonian fluid as a fractional differential equation:
Figure 2: The non-Newtonian area-time plot. Courtesy Dept. of Phys; University of Jadavpur
Kolkata
101
d 1( ) ( ) ( )d
q
q t B t tt
(2)
Where, E /B . The constant of this expression are is generalized viscous coefficient the units of
which are having the non-integer order q imbibed into it, and E , the modulus of elasticity. When, the
order 1q , then normal constant of viscosity is returned. The unit of B for order 1q is per seconds
i.e. 1[ ]s , but for any other order 1q ; the unit modifies as[ ]qs . In figure-1 this relaxation is depicted
with 1q .
Mathematically one has to see the Green’s function for general relaxation in equation given above by
fractional differential equation, so we write the homogeneous equation with RHS equal to zero. To
that, we give delta function stress excitation. The strain built up for any relaxation process may be
treated as convolution integral of a strain variable with integral kernel ( )qK t as
0
d ( ) ( ) ( )dd
t
qt K tt (3)
Well if the memory kernel is 0 0( ) ( )K t B t , we have the above system without memory and the
Green’s function will be 00( ) B tt e , that is the impulse response quickly decays to zero. Here 0 is
initial strain of the system at 0t . Derivation is following:
0
0
0 00
0
( ) ( )
d ( ) ( ) ( )d ( )d
( )
t
B t
K t B t
t B t B tt
t e
(4)
The homogeneous strain relaxation equation for no-memory case is first order Ordinary Differential
Equation i.e. the Newtonian case, with 0 E /B
0d ( ) ( ) 0d
t B tt (5)
If the memory kernel is a constant say 1 1K B , then we will have oscillatory Green’s function, which
never decays to zero.
102
1 12
12
0 1
( )d ( ) ( )d( ) cos( )
K t B
t B ttt B t
(6)
Above is representation of a constant memory system.
The generalized memory integral is as follows and the case is for non-Newtonian fluids. 2
1 1
1
( ) ; 0 2
d 1 d( ) ( ) ; ( 1)d d
qq q
K t B t q
t t B qt t
(7)
Its corresponding generalized differential equation, obtained from above derivation, is the system with
memory with memory index coming as fractional order of the Fractional Differential Equation with,
0 2q , 0d ( ) ( )d (1 )
q qq
q
tt tt q
.
In our experiment, the oscillatory response to a step input we say that the order is between 1 2q
and thus system has memory “long lingering and decaying” memory. The above derived general
equation we say 0 0 , at initial time and the stress be Heaviside’s step input then it modifies to our
original fractional differential equation (FDE), what we had assumed, with q B . The fractional
order of the FDE corresponds to system with memory. The non-Newtonian fluids without oscillatory
behavior will have fractional order 0 1q , and the step-response, for input 0( ) ( )t H t , ( )H t is
Heaviside’s step function (in earlier chapter we used ( )u t ) ; will have monotonically increasing strain
response, given by one argument Mittag-Leffler function, that is 10( ) ( ) [1 ( )]q
q q qt B E B t . Its
impulse response will be having long tailed decay. That is the response will have long-range temporal
correlation. The Newtonian fluid will have integer order in with 1q , the system without memory,
and the step-response will have the monotonically increasing strain as 0( ) 1 exp( )t B t ; where as its
impulse response will decay quickly as 0( ) exp( )t B t . This Newtonian system can be modeled with
a discrete ideal spring and an ideal dashpot. Whereas, the non-Newtonian cases requires a different
representation; like fractal chain of the ideal spring and ideal dashpot. We have observed oscillatory
103
case of strain and thus our fractional order is at 1.5q . This is anomalous result and we observed this
oscillatory spreading at higher loads.
Extension of fractional Calculus to continuous order differential
equation systems The constitutive equations relating stress strain we can write in various forms as follows
0( ) ( )t a t (8)
The (8) is simple spring equation. The pure dashpot equation is (9)
1( ) ( )t a t (9)
The spring dashpot series equation is
1 1( ) ( ) ( )t b t a t (10)
The spring dashpot parallel equation is
0 1( ) ( ) ( )t a t a t (11)
The spring connected to a parallel connection of spring dashpot will have constitutive equation as
1 0 1( ) ( ) ( ) ( )t b t a t a t (12)
We can generalize our observation of stress strain from (8) to (12) as
0 0
d d( ) ( ) ;d d
k km n
k kk kk kb t a t k
t t
(13)
The (13) is generalized integer order differential equation representation. In our observation we have
tried to map the obtained relaxation of strain (fig-1a and fig-2) via equation of Fractional Order
Differential equation (2). Thus in general in (13) we can have k as real number giving fractional
generalization of combination of spring dashpot system (13) as 1 ( )
00( ) ( ) ( )m z k
k tkA t D f t g t
(14)
Where ( )z k are fractional numbers indicating fractional order derivatives, with ( )f t representing strain
and ( )g t representing stress. We are stretching the generalization of (2) to (14) by ‘integrating’ the
order in an interval of interest say ( , ) (0,1)a b ; that is by changing the summation (14) to integral and
we get the following (a further generalization).
104
dd ( ) ( ) ( ); 0 1;d
m zb
m zaz A z f t g t a b m
t
(15)
This is generalization of equation (14) substituting the summation with an integral where ( )A z limited
in the interval ( , ) (0,1)a b , and m is positive integer. Why did we do this? Well, the introduction of
one parameter, rather interval ,a b instead of z renders the fractional derivative operator m zd to
become more flexible; because it includes a variety of memory mechanisms for relaxation! This is
perhaps more apt to represent the dispersion acting with slightly different relaxation. The (15) is
regarded as ‘continuous order differential equation’ where the order is continuous function in the
designated interval. In other terms we can also say that the order has been weighted averaged!
Solving the continuous order differential equation We apply the definition of fractional derivative to (15) and obtain via Caputo’s (1967) rule
( 1)
0
d ( ) [ ( )d ] ( )d ( ) d ( );d (1 ) ( )
m z mb b t
m z za a
f t A z z f uz A z u g t mt z t u
(16)
Taking Laplace of (16) we write ( 1)
0 0
( )d d d ( )(1 ) ( )
mb tstza
A z fe t z u G sz t u
(17)
Interchanging the order of integration in (17), we write (18) and (19)
( 1)
0 0
( ) ( ) d d d ( )(1 ) ( )
mb tstza
A z f ue u t z G sz t u
(18)
1 ( )0
( )d ( ) (0) ( )b mm z z n m n
naA z z s F s s s f G s
(19)
For (19) we write the LHS term by term as following
( ) ( )d ( ) ( ) db bm z m z
a aA z s F s z F s s A z s z (20)
1 ( ) 1 ( )0 0
( ) (0)d (0) ( ) db bm mz n m n n m n z
n na aA z s s f z s f A z s z
(21)
With (20) and (21) we use them in (19) to get and subsequent algebraic arrangement as follows
1 ( )0
( ) ( )d (0) ( ) d ( )b bmm z n m n z
na as F s s A z z s f A z s z G s
(22)
105
1 ( )0
( ) ( )d ( ) (0) ( )db bmm z n m n z
na as F s s A z z G s s f s A z z
(23)
1 ( )0
(0)( )( )( )d
m n m nn
b mm z
a
s fG sF sss s A z z
(24)
In the expression (24) we do simplification of the second term of RHS. Dropping the summation sign
we write the Laplace variables as 1 { ( )}n m n ms s F s s . The m ns we write for the Laplace of ( )m n th
derivative of function f . This we rearrange to get 1 ( )n ns s F s , then 1 ( )n ns s s F s and then 1{ ( )}n ns s F s to which we write, with n th derivative of function ( ) at 0f t t and
obtain 1 ( ) (0)n ns f . Here we apply the Laplace identity ( 1)/ !n nt n s to get
the 1 ( ) ( )(0) ( / !) (0)n n n ns f t n f . We use this long simplification to write (24) in compact way as
follows (the solution in Laplace and then time domain, well in terms of convolution)
1 ( )0
( )( ) (0)( ) d
m n nb nm z
a
G sF s s fs A z s z
(25)
1 ( )0
1( ) ( ) (0)!( ) d
nm n
b nm z
a
tf t g t fns A z s z
(26)
If we consider the derivative order distribution function, ( )A z to be analytic
then,0
( ) ( / !)j jjj
A z A z j
, we can substitute this analytic expansion in (26) and get
1 ( )0
0
1 ( )0
0
( )( ) (0)( / !)d
1( ) ( ) (0)!( / !)d
m n nb nm j z
jj a
nm n
b nm j zjj a
G sF s s fs A z s j z
tf t g t fns A z s j z
(27)
Response function analysis The derivatives of order 1z and 2z with initial condition (0) 0f imply (25)-(27) has ‘filtering effect’,
filtering the function ( )f t with high pass characteristics where response functions are 1zs and 2zs . Since
2 1 1 2 1/ 1z z z z zs s s s is an increasing function of s then the response function 2zs is increasing
106
more severely than 1zs . The high pass filters what we stated has response function as
( ) ( )dbm z
as s s A z z which acts on ( )f t . While we can have 0( ) ( )A z z z , then
0 0
1 1
0
1 1
1 1( ) ( ) ( )( ) d ( ) d
1 1( ) ( )
b bm z m z
a a
z m zm
f t g t g ts A z s z s z z s z
g t g ts s s
(28)
Figure-3 depicts the high pass filter characteristics for various discrete fractional orders 0z .The plot is
between modulus of ( j ) and frequency (X-axis) . The figure-4 depicts the high pass
characteristics of the response function when ( )A z h , for an interval (mentioned near each curve).
Figure-3: Plot show modulus of response function high passes characteristics when the order
distribution function is 0( ) ( )A z z z for 0z as fractional order of 0.2, 0.4, 0.6, and 0.8.
107
Figure-4: Plot show modulus of response function high passes characteristics when the order
distribution function is ( )A z h and with lower and upper limits of integration on the z
This is transfer function of filter associated with derivatives of fractional order. With this type of order
distribution function the constitutive equation for stress strain visco-elastic elements behave as (2).
Now we consider ( )A z kz h ; the response function is
( ) ( )d ( )d
( ) ( ) ( ) / ln / ln
b bm z z
a a
m b a b a
s s s A z z s kz h z
s kb h s ka h s k s s s s
(29)
With 0k the continuous order is a simple case; meaning that the order distribution function of
derivative order continuously placing same weights to all the derivatives (rather infinite numbers) of
fractional orders in the interval ( , )a b . With 0k and 1h ; we obtain the response function as
( )( ) ( )dln
b a mbm z
a
s s ss s s A z zs
(30)
What is of interest to us in (30) is the ‘modulus’ function of ; to get that we put j , j 1s ; a
standard procedure. Now what we see is j / 2 j / 2( j ) ( ) ( ) ( / 2)b b b b b bs e e b , is a vector
with modulus as b an argument (angle) of / 2b . Therefore b as s should be vector subtraction of
two vectors shall give me, resultant modulus as 2 2 2 cos ( ) / 2b a a b b a . Well here we have
used 2 2 2 cos( )A B A B A B B A . For taking modulus of ln s , we just place js to
108
have j / 2ln( j ) ln ln j( / 2)e . From this we calculate 2 2ln (ln ) ( / 2)s . The
modulus of ms is nothing but m . Using all these we get useful expression for modulus of (30) which is
2 2
2 2
2 cos ( ) / 2( )
(ln ) ( / 2)
m b a a b b aj
(31)
For 0m , we get the properties of response function ( )s at s are governed by bs while at
0s are governed by as . This property represents the difference between ( )s of (31) type vis-à-vis
response to single fractional derivative of order say 0z . This new type of response function allows us
to study different behaviors for high frequency and low frequency; since it allows a filter to have
filtering with independent properties at high frequency (early time) and low frequency (late times).
This gives extra freedom to study various complex relaxation processes and dynamic systems of
nature! We have used the term high pass filtering effect; nonetheless the inverse response function
(1/ ( ))s acting on function ( )g t produces low pass filtering action. These are depicted in figure 5 for
discrete and figure-6 for continuous order.
Figure-5 Plot show modulus of response function low passes characteristics when the order
distribution function is 0( ) ( )A z z z for 0z as fractional order of 0.1, 0.3, 0.5, 0.7 and 0.9.
109
Figure-6: Plot show modulus of response function low passes characteristics when the order
distribution function is ( )A z h and with lower and upper limits of integration on the z
In order to see filtering effect of 1/ ( )s we again assume in ( )A z kz h , with 0k and with initial
conditions ( ) (0) 0jf and write the following
12 2 2
12 2 2
( ) ln( )( )
( j ) (ln ) ( / 2)( j ) ln( j )( j )
( j ) ( j ) 2 cos ( ) / 2
b a m
b a m m b a a b
G s sF ss s h
GGF
h h b a
(32)
This (32) implies low pass filtering action. Taking 1h ; we can invert Laplace and write time response 1
1 1ln 1( ) ( )( 1) ( )
m
a b a
s tf t g ts s m
(33)
From (32) the first expression is Laplace response as low pass filter, we do few algebraic
manipulations on this expression and take invert Laplace in convolution form which gives time
response. While doing so we have used identity 1 1 1/( 1)! / ( )m m ms t m t m . In (33) the third
term in RHS is Laplace invert of higher transcendental function called Robotnov-Hartley function
( , )qF a t we can thus use the expression for the same as follows
110
( 1)( ) 11
( ) 0
1 (1, )1 ( 1)( )
n b a
b ab a n
tF ts n b a
(34)
The Laplace invert for second term of (33) that is 1 ln / as s requires numerical evaluation.
We have considered a uniform distribution of fractional order in the interval, with equal weights
however it can have any functional form. Say if we have the form ( ) nA z Az , then we have solution
for initial conditions zero as:
1
10
1( ) ( )( 1) ( 1).....( 1)( ) /(ln )mm n m n b a m n n
n
f t g tAs m m m n b s s a s
(35)
We notice here that all previously obtained expressions after Laplace Transforms and then integration
with respect to z are appearing as sums of powers of frequency s or , (meaning ( )A z having
dimension of zs ) which have different dimensions and which could be physically unacceptable. Thus
we do the transformation of abscissa (normally frequency) as in figure- 3 to a dimensionless scale. We
assume that ( ) ( )zA z B z , where has dimensions of time. With this change of scale we will get the
expressions obtained earlier as
1 ( )0
( )( ) ( )( ) d (0)b mz n n
m na
G sF s B z s z s fs
For constant ( )B z h , we thus obtain
1 ( )
0
( ) ln( )( ) (0)( ) ( )
m n nm b a n
G s sF s s fhs s s
With 0m , 1h , we obtain relaxation in Laplace Domain with a variety of slight different relaxations
as ( ) ( ( ) ln( )) /(( ) ( ) )b aF s G s s s s , with abscissa as dimensionless s , for figure 4 and 6.
Mechanism of random delay in networks of computer Dynamics of delay in computer based systems demonstrate the stochastic behavior. The delay of
random nature has wide spikes (figure-7a) and if a statistics be taken, it is like a power law, with
pronounced tail (figure-7b). Effect of network delay in control system is very widely researched topic
and has practical relevance to modern computer control industry. A Brownian motion to model the
111
stochastic process of ‘random delay dynamics’ is proposed in this section, through fractional
equivalent of Langevin equation driven by ‘shot-noise’.
The classical method of fluctuation dynamics is by Gaussian assumption of the random behavior, and
dynamics of the same applied to fluctuations in financial assets is widely used in mathematical finance
because of simplification it provides in analytical calculations. This gives integer order differential
equation formulations giving Gaussian solutions. Mandelbrot who introduced the term ‘fractal’
observed that in addition to being non-Gaussian, the stochastic process of financial returns show
interesting property of ‘self-similarity’. That is the statistical dependencies of ‘random phenomena like
financial returns, Brownian motion, have similar functional form, for various time increments t .
Motivated by ‘pronounced tails and the stable functional forms at different time scales, Mandelbrot
proposed that distributions of ‘such’ fluctuating is consistent with -stable Levy distributions that is
the fluctuating processes can be modeled as -stable Levy process. Thus from point of view of the
fractal concept here attempt is made to provide fractality concept for random fluctuating delay
dynamics of computer control system. It is well known that the trajectories of Brownian and Levy
stochastic processes are fractals it means that they are non-differentiable, self similar curves whose
fractal dimensions are different from topological dimensions. Figure-7a depicts LAN delay data
gathered over various packets.
We might expect that network traffic would best be represented modeled or simulated by having some
random source send random sized packets into a network. Following this line of thinking, the
distribution of delay times on network access, would be Poisson distribution. As it turns out this as
naïve model for network traffic seems to be wrong.
Network traffic is best modeled by a process which displays non-random nature of Hurst
parameter H which gives Long Ranged Dependence (LRD) with lingering memory tail (figure-7b), and
a non Gaussian distribution. ‘Indeed Nature’s prediction towards LRD has been well documented in
hydrology, metrology and geophysics’. Here in this section LRD is established for ‘stochastic delay’ of
computer based network system. The estimated Hurst exponent, and then fractal dimension and the
112
Figure-7a: Time domain presentation of the network induced stochastic delay Courtesy Dept. of
PE Jadavpur University Kolkata
113
Power spectral density of random walk-(network delay)
White-noise
( )1
S
0 .7 6
( )S
Equation of motion of this delay dynamics could beFractional Langevin’s equation
1
F B M F B M1
d 0d
x xt
B M B Md 0d
x xt
Figure-7b: Power Spectral Density of Network Delay Courtesy Dept. of PE Jadavpur University
Kolkata
exponent of the power spectral density of figure-7b gives the values as
0.88, 2 1.2, 2 1 0.76fH d H H . This is indicator of fractional Brownian Motion
(FBM), obtained via estimation rule called R/S method. Whereas the standard Brownian Motion (BM)
has 0.5, 2 1.5, 2 1 0fH d H H , the White Noise Case (figure-7b).
Random Delay a Stochastic Behavior Dynamics of delay in any systems demonstrate the stochastic behavior. The delay of random nature
has wide spikes and if a statistics be taken, it is like a power law, with pronounced tail. For example,
effect of network delay in control system is very widely researched topic, and has practical relevance
to modern computer control industry. A Brownian motion to model the stochastic process of ‘random
delay dynamics’ is proposed, through fractional equivalent of Langevin equation driven by ‘shot-
noise’. A shot noise results when a ‘memory-less filter’ is excited by train of impulses derived from a
homogeneous Poisons Point Process (PPP). For simplicity we assume the impulse response of the filter
114
as ( ) exp( / )t t for 0t . Depicted in figure-8 block marked A. Therefore we can represent a shot-
noise as1
( ) ( )nk kk
F t a t t
. We consider the fractional form of Langevin type stochastic
differential equation, and replace standard ‘white noise’ Gaussian stochastic driving excitation force,
by ‘shot-noise’1
( ) ( )nk kk
F t a t t
whose each pulse has randomized amplitude, a ; the pdf of the
amplitude ( )P a , has characteristic function, ( ) exp{ }W k b k , with 0 2 .
The force is acting on a delay generating block where the Fractional equivalent of Langevin equation is
dynamic representation of the system as: ( ) ( ) ( )qtD t t F t , with 0 1q , and 0 representing
fractional initial condition of the delay generating dynamic system, depicted in figure-8 block B. This
equation is detailed in next subsection. The driving force is train of pulses, will give a delay function of
time ( )t , which also may be called fractional stochastic variable, from this above dynamic system
as 0( ; , )t F . The fluctuation of this fractional stochastic variable is, 0( ; , )t F .This fluctuating
delay function then generates spiked delay as 0( ( ; , ))t F .
The physics of the delay generating system can be explained in following manner. That is say for one
moment we have demanded a particular data (information), say the computer gives that data
(information) after time of 500mS; the next moment a different data (information) is called for and the
new data (information) now gets available say at 5000mS. The type of data or information signifies the
amplitude, ka of shot-noise pulse, which after exciting the delay generating system, makes the
particular data or information available. So these 500mS and 5000mS duration becomes the stochastic
variable ( )t with height at the data availability time, gives spike that is ( ( ))t . So a plot of
0( ; , )t F and its delay time 0( ( ; , ))t F , will give spike nature of fluctuating delay dynamics.
Theses spikes are really spiky in nature, deviate and fluctuates a lot. The first type of data again when
asked may take 10mS or even may take 10,000mS; that depends on the load on net-work traffic.
Therefore, the fluctuations are really spiky in real terms.
The classical method of fluctuation dynamics is by Gaussian assumption of the random behavior, and
dynamics of the same applied to fluctuations in financial assets is widely used in mathematical finance
115
Picturing of Network Delay Dynamics
Shot noise driving a delay generation block (fractional Langevin’s equation)
( )t
t e
d ( ) ( ) ( )d
q
q
x t x t F tt
1
( )n
k kk
a t t
1( ) ( )
n
k kk
F t a t t
1( ) ( )
n
k kk
F t a t t
( )x t
time
A
B
Figure-8: Picturing the randon network delay via shot noise driving the fractional Langevin
equation
because of simplification it provides in analytical calculations. This gives integer order differential
equation formulations giving Gaussian solutions.
Here in this section, developed a new extension of fractality concept for dynamics of random delay. It
is proposed here, that a possible fractional calculus approach to model the evolution of stochastic
dynamics of random delay. The proposed fractional dynamic stochastic approach allows obtaining the
probability distribution function (pdf) of the modeled random delay. As an application of the
developed general approach we derive the equation of pdf of increments of random delay , as a
function of increment of time t , ( ) ( ) ( )t t t t where the value of random delay generating
system at any time is ( )t . Statistical properties of incremental delay of computer control system play
important role in understanding the control system for example its stability measures with gain
variations and its robustness. The theoretically predicted pdf of increments of delay as a function of
116
increment of time t is to be experimentally verified, for calibration of numerical parameters of the
proposed dynamical model for random delay.
About Levy distribution A random variable X is said to have an -stable distribution; if and only if its characteristic function
has the following form:
( ) exp i 1 i sgn( ) ( , )f x x x x w x , i 1
1; 0tan ; 12( , ) sgn( ) 0; 0 0 2; 1 1; 0;
2 log ; 1 1 0
xw x x x
x x
Therefore an -stable distribution can be completely determined by four parameters.
1. The characteristic exponent . It is the shape parameter which specifies the thickness of the tail of
the probability density function. Lesser the value more pronounced is the tail, indicating strong
lingering memory. In other words, changes the level of spikiness in the distribution, the larger the
value of , the less likely it is to observe random variable that is distant from its central location. For a
normal distribution 2 , where the tail decays exponentially fast from central ‘dome’, indicating that
there is less likelihood of presences or a random variable at far places, from ‘mean’. This parameter is
also called ‘stability’ parameter. For 2 the distribution of the random variable has no finite
variance, and for 1 , the finite mean does not exist. Generally, 0 2 .
2. The skewness index . Positive values for make the distribution skewed towards the right tail and
negative values make it skewed towards the tail on the left hand side. For normal distribution or any
symmetric distributions 0 . Generally 1 1 .
3. The variable is called the scale parameter or dispersion parameter and it expresses the dispersion
of the distribution. For normal distribution is equal to standard deviation. For non-normal
117
distribution this has a finite non-negative value but it is not the same as standard deviation. For non-
normal stable distribution the standard deviation is infinite. For all cases 0
4. The variable is called the location parameter and it is an expression of the mean or median of the
entire distribution. This is also termed as measure of centrality or mean. For a normal distribution and
other stable distributions for which 1 ; is same as the mean value of distribution. When
1 mean value is not defined, here the value, of the distribution is not the same as mean value.
Generally .
The -stable distributions obey two major properties.
1. The stability property, which states that the sum of weighted independent -stable random
variables is still stable with the same characteristic exponent .
2. The generalized central limit theorem which states that the sum of a number of independently and
identically distributed (i.i.d) random variables, can only be a stable distribution. The generalized
central limit theorem defines the randomness as a result of cumulative effects and these effects are
distributed with heavy-tailed probability density.
A symmetric characteristic function for Levy distribution is ( ) exp( )f b . The pdf of Levy
stable distribution can be obtained by performing numerically Fourier cosine transform giving
0
1( ) d exp( )cosf x k bk kx
.
The delays of computer control system or any other dynamics of fluctuation like in finance, follows a
random behavior, could be explained by the heavy tailed distributions of such type. The delays in
computer control system arises due to forces of random nature are inbuilt into the software delays
caused by ‘multi-tasking programming methods’, and the randomness in ‘Network Traffic’, ‘data bus
traffics’, hardware ‘arbitration logic’ of share resources, ‘decision making algorithms’ etc.
118
Fractional Stochastic Dynamic Model Let us take an example of delays in computer systems. Here it is proposed to describe the dynamics of
random delay of computer control system ( )t by the fractional stochastic differential equation,
representing Brownian motion like system.
d ( ) ( ) ( ) 0 1d
q
q t t F t qt (36)
We have ‘fractional stationary’ condition, as initial condition as 1
010
d ( )d
q
qt
tt
(37)
Where is a constant, expected delay rate (like Poisson’s process formulation) this value could be
zero too, that case is discussed later. ( )F t , is the random forcing function and d / dq qt is Riemann-
Liouvelli (RL) fractional derivative, of order q . Using the definition of RL fractional integration,
d / dq qt of a function ( )f t , that is:
0 0 10
d 1 ( )( ) dd ( ) ( )
tqq qt t q q
f tI D f t tt q t t
, for 0 1q , yields solution to (36) as
1 10 , 0 ,
0
( ; , ) d ( )( ) ( )t
q q q qq q q qt F t E t t F t t t E t t (38)
, 0( ) /k
kE z z k
is two parameter Mittag-Leffler function (we have used this in previous
chapter also).
The fractality index q is related to Mandelbrot’s self-similarity parameter H , also called Hurst Index
where, (1/ 2)q H . The mathematical motivation for applying the fractional stochastic problem (36)
(37) is following. It is easy to see when 1q , the Eq. (36) reduces to the standard (integer order)
Langevin equation with integer order initial condition 00( )
tt
, and Eq. (38) gives standard well-
known stochastic Poisson’s process problem
( )0 0
0
( ; , ) d ( )t
t t tt F e t F t e , because of 1,1( ) zE z e
Thus we see that the fractional stochastic initial value problem (36) and (37) seems to be fractional
generalization of well known Langevin approach to fluctuating phenomena. We define the probability
119
distribution function (pdf) ( , )qP t of the fractional stochastic variable ( )t in the following way, by
Fourier integral of Dirac’s delta function.
0( , ) ( ( ; , )qP t t F
01 d exp i ( ( ; , )
2t F
1 1, 0 ,
0
1 d exp [i ( ( ) )] i d ( )( ) ( )2
tq q q q
q q q qt E t t F t t t E t t
1, 0
1,
0
1 d exp i ( ( ) )2
exp i d ( )( ) ( )
q qq q
tq q
q q
t E t
t F t t t E t t
(39)
Where the brackets ... means the averaging over the all possible realizations of the random
force ( )F t . Thus the ‘averaging operator’ ... is applied to second term, as the first term in (39) is force
free. Here, i 1 . In (39) we have used the Fourier identities, with 2 , for
ˆ( ) d {exp( i2 ) ( )} 1x x x
The inverse Fourier as
i21 d ( )xe x
and orthogonality of Fourier kernel as
1 2 2 1*i2 i2 i2 ( )
1 2d d ( )t t te e t e t
Let the stochastic force ( )F t be a generalized shot noise as defined in statistical communication theory
as follow, and explained in previous subsection as:
1( ) ( )
n
k kk
F t a t t
(40)
Here, ka are the random amplitudes, ( )t is the ‘impulse’ response function of ‘memory-less linear
filter’, and kt are the homogeneously distributed on time interval 0,T moments of time, the number
n obeys the Poisson law. We guess that, defined by Eq. (40), random force ( )F t describes the
120
influence of the different fluctuating factors on the ‘delay generation dynamics’. These forces of
random nature are inbuilt into the software delays caused by ‘multi-tasking programming methods’,
and the randomness in ‘Network Traffic’, ‘data bus traffics’, hardware ‘arbitration logic’ of share
resources, ‘decision making algorithms’ etc. A single-shot-noise pulse ( )k ka t t describes the
influence of a piece of information which has become available at the random moment kt on the
decision-making process at a later time t . The amplitude ka responds to the magnitude of
pulse ( )kt t ; it will depend on type of information and will, therefore, be subjected to probability
distribution. For simplicity we assume that each pulse has the same functional form or, in other words,
one general response function can be used to describe the delay process; in our case it is exp( / )t .
Thus, the averaging procedure includes three statistically independent averaging techniques, which is
used to calculate, the average of (39), that is:
1,
0
1,
10
exp i d ( )( ) ( )
exp i d [ ( )]( ) ( )
tq q
q q
t nq q
k k q qk
t F t t t E t t
t a t t t t E t t
Let 1,
0
d ( )( ) ( )t
q qq q qt t t t t E t t R a function of t and which will be derived later. Using
this we have simplified above expression as
1,
10
exp i d ( )( ) ( ) exp it n
q qq q q k
kt F t t t E t t R a
Now we demonstrate term by term averaging over all possible parameters of force.
1. Averaging over random amplitudes, ka that is ...ka ,
1 1 1 1d ...d ( ,..., )k
n naf a a P a a f (41)
Where 1( ,..., )nP a a is the probability distribution of amplitudes ka .Using (41) we get the following
1, 1 2 1
10
exp i d ( )( ) ( ) d d ...d ( ... ) exp ik
t nq q
q q n n q kka
t F t t t E t t a a a P a a R a
Define a function ( )W is characteristic function of the probability distribution ( )P a , as
121
( ) d ( )i aW ae P a
and pdf ( )P a is distribution function of the amplitudes of the shot-noise pulses. Also for simplicity
assume that;
1 11
( ,..., ) ( ) ( ( ))n
nn k
k
P a a P a P a
Using this simplification we factorize and write the following, for averaging over ka , as:
1 2
1,
0
i i i1 1 2 2
i
exp i d ( )( ) ( )
d ( ) d ( ) ...d ( )
d ( ) ( )
k
q q q n
q
tq q
q q
a
R a R a R an n
n nR aq
t F t t t E t t
a P a e a P a e a P a e
aP a e W R
2. Averaging over kt on time intervalT .
2 1 20 0
1 1d ... d ( )T T
nTf t t f
T T (42)
Above obtained average expression, we now use (42) and write to average over kt
1, 1 2
0 0 0 0,
0
1 1 1exp i d ( )( ) ( ) d d ... d ( )
1 d ( )
k k
t T T Tnq q
q q n q
a t
nT
qn
t F t t t E t t t t t W RT T T
tW RT
3. Averaging over random numbers n of time moments kt
3 30 !
nn
nn
nf e fn
(43)
Where n vT and v is the density of points kt on time interval[0, ]T . Using (43), we get
,
1,
00 0,
0 0
1exp i d ( )( ) ( ) d ( )!
1 d ( )!
k k
nt Tnq q n
q q qnna t n
nTn nvT
qnn
nt F t t t E t t e tW Rn T
v Te tW Rn T
122
Let us calculate the RHS of above, as demonstrated in following steps
(With0
( , ) d 1 ( )T
q qJ t v t W R )
0 0
0
0 00 0
d d { ( )}
d [1 ( )]
( , )
1 1d ( ) d { ( )}! !
T T
q
T
q
q
n nT Tn nvT vT
q qnn n
v t v t W R
v t W R
J t
v Te tW R e v t W Rn T n
e e
e
e
Taking into the account of Eq. (39) and performing the averaging in accordance with (41)-(43), as
described and derived above we obtain, with the time interval [0, ]T is changed as[0, ]t .
1, 0
1( , ) d exp i ( ( ) exp ( , )2
q qq q q qP t t E t J t
(44)
Where the following notation is introduced as derived from above averaging method
1,
0 0
( , ) d 1 d ( )( ) ( )t t
q qq q qJ t v t W t t t t t E t t
(45)
Here the function ( )W is characteristic function of the probability distribution 1( )P a ,
i1( ) ( )aW dae P a
and pdf 1( )P a is a “one-particle” distribution to be introduced into consideration
because of simple assumption. That is we consider a delay dynamics when probability distribution
1( ,..., )nP a a is factorized as product of n equal “one-particle” distribution 1( )P a , Eq. (46).
1 11
( ,..., ) ( )n
n kk
P a a P a
(46)
To evaluate the integrals in Eq. (44) and (45) we should know the type of response function ( )t and
pdf 1( )P a . Let us choose an exponential response function,
; 0( )0; 0
t
e ttt
(47)
123
This means, that the impact has a characteristic-time (time constant) as a number . Evaluate the
integral over dt in Eq. (45),
/1 1,
00
( )d ( )( ) ( ) d ( )( )
t t k qkt tq q q
q qkt
t tt t t t t E t t t e t tqk q
(48)
Expanding in series the /t te and using the formula
1 1 1 ( ) ( )d ( ) ( ) ( )( )
ta b a b
t
a bt t t t t t ta b
makes RHS of Eq. (48) as
/ 1, 1
0 0
( )d ( ) ( )( )
kt k qkqt t q q
q q kk kt
t t t tt e t t t t E t tqk q
.
The function ( , )qJ t as in Eq. (45) then is:
0
( , ) 1 ( ( ; )t
q qJ t v dt W R t (49)
With
, 10
( ; ) ( )k
q qq q q k
k
tR t t E t
(50)
As a second step choose the Levy -stable distribution 1( )P a as “one-particle” probability distribution
function:
i1
1( ) d ( )2
aP a e W
, and ( ) expW b , 0 2 (51)
Where b is scale parameter of Levy -stable distribution. Thus in accordance with Eq. (43) the new
general equation for the pdf of the fractional stochastic delay process ( )t described by Eq. (36) can be
rewritten as:
1, 0
0
1( , ) d exp i ( ) exp d 1 exp ( ; )2
tq q
q q q qP t t E t v t b R t
(52)
With ( )qR t defined as in Eq. (50).
Note that putting 1q in Eq. (50) we get the following (integer order counter-part)
124
/1( ) /( 1) t tR t e e
Fractional Delay Dynamics Let us apply the developed general approach to derive the analytical expression for the pdf of
increments of the delay generating system as a function of time increment t , ( ) ( )t t t
where the value of the instantaneous delay of the delay generating system is ( )t . The pdf of delay
increments fluctuation plays an important role in understanding the delay dynamics, and in overall
system engineering to plant control by computers. As usual we define the pdf ( , , ; )qP t t of the
increments ( ) ( ) ( )t t t t of the delay dynamics system of delay ( )t as a function of
incremental time t by following expression
, 0 0( , , ) ( ; , ) ( ; , )qP t t t t F t F (53)
Where ... means averaging over the all possible realizations of random force ( )F t in accordance with
Eq. (41)-(43). Repeating the same steps used above for derivation of Eq. (52), we find for the pdf
, , ,qP t t as
1, ,
0
1, 0 ,
1( , , ) d cos { ( ) ( )
}exp ( , , ; )
q qq q q
q qq q q
P t t t t E t t
t E t L t t
(54)
With
,0
( , , ; ) d 1 exp ( ; ) ( ; )t
q q qL t t v t b R t t R t
where ( ; )qR t is by Eq. (52). Equation (54) presents a new general expression for the fractional pdf
of increments of delay ( ) ( )t t t fluctuations, when delay ( )t is described by fractional
stochastic differential equations (36).
For a case when 0 the Eq. (54) can be rewritten as
125
1 1, 0 ,
0
1 1( , , ) d cos ( ) exp , ,( )
q qq qP t t t t t L t t
q
(55)
Where
,0
, , d 1 exp ( ) ( )t
q q qL t t v t b r t t r t
and ( )qr t is obtained from Eq. (50) by placing 0 , that is
1,1( ) ( ; 0) ( ) /qq q qr t R t t E t (56)
Further in limit for large time the asymptotic pdf we obtain as:
, ,, ,
0
1, lim , , d cos qL tq qt
P t P t t e
(57)
Here , ,qL t is defined as:
, ,0
, lim , , d 1 exp ( ) ( )q q q qtL t L t t v t b r t t r t
(58)
In Eq. (58) 0 1q and 1 2 The limiting pdf , ( , )qP t is characterized by the fractality index
q and the Levy index . Thus, it is shown how general fractional dynamic approach developed here;
one derives the expression for pdf , ( , )qP t of increments of delay dynamics. The new pdf
, ( , )qP t allows studying any statistical and scaling dependencies of the fluctuating dynamics and
developing the new general approach to deal with random delays in computer controls, and evaluate its
risks and robustness.
In special case of integer order delay dynamics case with 1q the general Eq. (58) can be expressed as
/ /,1
0
( , ) d 1 exp (1 )t tL t v t b e e
(59)
Where we have used (1) 1 and 1,2 ( ) 1 /zE z e z . Then Eq. (57) leads to
/ /,1
0 0
1( , ) d cos( ) exp d 1 exp (1 )t tP t v t b e e
Substituting with u instead of twe have / /(1 )t tu b e e , d d ( / )u t u we get,
126
/(1 )
,10 0
1 d( , ) d cos exp (1 )tb e
uuP t v eu
(60)
The pdf in Eq. (60) is -generalization of Eq. (41) and (42), the Gaussian case will be obtained
for 2 , for Eq. (60), that is 2,1( , )P t .
Obtain the delay values for large number of time, and say we form a set 1 2 3{ } { ( ), ( ), ( ).......}i t t t .
In this set we take differential delay and make the set
as 1 2 1 2 3 2 3 4 3{ } { ( ) ( ), ( ) ( ), ( ) ( ),.......}i t t t t t t . This new set of data has
value zero, positive and negative; showing incremental delay spread. Simply filling of empirical values
for data by ordinary symmetrical Levy -stable is by following expression
LEVY 0
1( / ) d cos( / ) exp( )P c k k c kc
where is Levy index and is scale factor be performed as designed by Famma and Roll in 1972.
Here, c is number in order to normalize the data set. In our delay case this could be one.
For numerical calculations of (58) that is
, ,0
, lim , , d 1 exp ( ) ( )q q q qtL t L t t v t b r t t r t
Write as:
,0
, d 1 exp ( ) ( )q q q qL y K z S s z y s z
where /y t is new dimensionless parameter, signifying change in time, the other dimensionless
parameters are K v and /qqS b c , and the function is 1,1( ) ( )q
q qs z z E z . Then plot the pdf as:
, ,0
1( / , ) d cos( . / ) exp{ ( , )}q qP c y c L yc
for various values of and q starting with 2 , 1q and decreasing the same. By varying these
fractality parameters empirically fitting of the data of delay, the pdf can be seen.
127
The Random Dynamics of computer control system From the basic signals and systems point of view, the random delay of computer control system can be
regarded as an output signal of “a (delay generating) dynamic system” driven by some known signal
say ‘shot-noise’. If we know better about the generating ‘dynamic system’, can we do better job in
compensating it. Well, the answer is yes. This is a new research frontier for exploration which is
especially fundamental and important in the understanding of randomized delay.
The same effect as above cause randomness of the, time variant of ( )t . This implies that the
distribution of such stochastic process follows a stable model. The variation of ( )t , with the bursts
(spikes) implies the heavy tailed behavior in the distribution. -stable distribution is different from
Gaussian in a way that there only exists moments of order less than characteristic exponent . Hence
the variance of a stable (non-Gaussian) distribution is divergent (infinite) unless 2 , in which case
the distribution becomes a Gaussian. This is called the ‘Fractional Lower Order Moment’ statistics. It
has been established that the fundamental solutions of fractional order diffusion equations generate
probability density functions (pdf) evolving in time or varying in space related to stable distributions.
Thus the randomization dynamics is well defined by fractional differential equations.
Thus basic research question is
1. How to use Fractional Calculus to best characterize the Random Dynamics of physics of delay?
2. How to make use of the acquired knowledge of random dynamics to design a better controller,
possibly also a fractional order controller, to accommodate the randomness in ( )t influence on
control performance of computer control system.
Here we have elaborated a fractality concept and fractional differential equation in mathematics to
represent randomized computer control delay. Our main assumption is that the fluctuating delay can be
adequately described by means of fractional calculus, non-Gaussian, long range dependence, heavy
tailed stochastic process. To describe the dynamics of the random delay, we have introduced a new
fractional stochastic differential equation, driven by a random forcing function ‘shot-noise’, for which
each pulse has random amplitude with -stable Levy distribution. As a result we have obtained the
general expression for the fractional pdf of incremental delays with time.
128
Future studies and experiments on computer control system delay data is required to calibrate the
parameters obtained in this description of pdf. The new fractional pdf has fractality parameters
q and . The parameter q describes the dynamical memory effects in the delay stochastic evolution,
while the Levy index describes the long range dependencies of external impacts on the delay
dynamics, like ‘multi-tasking programming methods’, and the randomness in ‘Network Traffic’, ‘data
bus traffics’, hardware ‘arbitration logic’ of share resources, ‘decision making algorithms’ etc. More
research and experimental data is indeed required to quantify these effects described in these random
delay phenomena of computer control system.
Figure-9: Diverging run-time variance of the network delay data (of figure-7) Courtesy Dept. of
PE Jadavpur University Kolkata
Conclusions Some applications with fractional calculus have been demonstrated here. The system identifications
with a new method of ‘continuous order differential equations’ and its possible frequency domain and
time domain solution have been discussed-in particular to the experimental observations of spreading
of starch samples in visco-elastic experiments. A new extension of a fractality concept and fractional
129
calculus for random delay dynamics in computer control system is described. We have introduced a
new fractional Langevin type stochastic differential equation that differs from standard integer order
Langevin equation, to get the insight of random delay dynamics. The excitation to the fractional
Langevin equation is a stochastic force ‘shot noise’, which has pulse of random amplitude with stable
Levy distribution. This method is developed as the observed delays of computer control system vary
with time in random fashion similar to ‘Brownian motion’. There are large spikes in delay and the
delay of the computer control system varies. Perhaps this is one way to express the dynamics of
random delay in computer control systems. This will help in study of computer based control systems,
where the random delay is important part of control loop.
130
Chapter 4
Solution of Generalized Differential Equation System
Introduction The solution of ‘generalized differential equations’ is what we have derived; a way to solve (solvable)
differential equation systems by physical principles and connected it to the Adomian Decomposition
Method (ADM); especially discovered in 90’s to solve non-linear systems. Mathematical modeling of
many engineering and physics problem leads to extraordinary differential equations (Non-linear,
Delayed, Fractional Order). Let us call them Generalized Dynamic System. An effective method is
required to analyze the mathematical model which provides solutions conforming to physical reality.
For instant a Fractional Differential Equation (FDE), where the leading differential operator is
Riemann-Liouvelli (RL) type requires fractional order initial states which are sometimes hard to
physically relate. Therefore, we must be able to solve these dynamic systems, in space, time,
frequency, area, volume, with physical reality conserved. The usual procedures, like Runga-Kutta,
Grunwarld-Letnikov Discretization with short memory principle etc, necessarily change the actual
problems in essential ways in order to make it mathematically tractable by conventional methods.
Unfortunately, these changes necessarily change the solution; therefore, they can deviate, sometimes
seriously, from the actual physical behavior. The avoidance of these limitations so that physically
correct solutions can be obtained would add in an important way to our insight into natural behavior of
physical systems and would offer a potential for advances in science and technology. Adomian
Decomposition Method (ADM) is applied here in this by physical process description; where a process
reacts to external forcing function. This reactions-chain generates internal modes from zero mode
reaction to first mode second mode and to infinite modes; instantaneously in parallel time or space-
scales; at the origin and the sum of all these modes gives entire system reaction. By this approach
formulation of Fractional Differential Equation (FDE) by RL method it is found that there is no need to
worry about the fractional initial states; instead one can use integer order initial states (the
conventional ones) to arrive at solution of FDE.
131
Generalized Dynamic System & Evolution of its solution by principle of
Action-Reaction General physics law states that a system will react to external stimulus and will have opposition to the
changes; and the process is described by system dynamic equations. Let there be general differential
equation system described as (1).
1 2 11 2 1
11 0
( ) ( ) ( ) ..... ( ) ( )
[ ( )] [ ( )] ... ( ) ( )
m m mx x x m x m
k kk k
D u x a D u x a D u x a D u x a u x
b u x b u x b u x G x
(1)
We can decompose this as linear part (2).
OL ( ) ( ) ( ) ( )u x G x R u N u (2)
The operator OL represents a linear operator representing the highest orders of change in the process
parameter. OL ( ) d ( ) / d ( )m m mxu u x x D u x .This is easily invertible. This order of change )(m could be
one two, or any positive integer or even fractional (say half, one fourth, one and one fourth). This order
of change could be with respect to time, space, space square (area) frequency or time-square-
depending on the process description. Where
1 2 11 2 1( ) ( ) ( ) ... ( )m m
x x m xR u a D u x a D u x a D u x (3)
is the remainder differential operator of order less than m . This R could be of integer or fractional
order. The rest of the terms are put as (4) contains nonlinear as well as the linear terms; and assume
this (4) as analytic function N .
11 0( ) ( ) [ ( )] [ ( )] ... ( )k k
m k kN u a u x b u x b u x b u x (4)
The, )(xG is sum of all external stimulus source/sink. General physics law states, that a system will
react to external stimulus and will have opposition to the changes; by the system reaction terms defined
by R and N defined in the system description equation (1). These R and N generates internal stimulus
when excited by external source/sink as to oppose the cause. The reactions are causal in nature. If the
external stimulus and the internal reactions to the stimulus get balanced then the process parameter
remains static without any growth (accumulation) or decay (loss). Else, the process parameter will
have a solution as infinite (or finite) decomposed modes; generated by system itself to oppose stimulus
generated internally by previous modes.
132
Adomian Decomposition Method (ADM) Computational method yields analytical solution; has certain
advantages over standard numerical techniques. However, the ADM was discovered in mid-late
eighties and utilized to tackle non-linear problems of physics ADM is free from rounding off errors as
it does not involve discretization and does not require large computer memory. ADM is qualitative
rather than quantitative, analytic, requiring neither linearization nor perturbation and continues with no
resort to discretization. ADM splitting gives equation into linear ( OL ), that is containing highest order
of change, the remainder part (R) that has change rates less than ( OL ) and then the non-linear (N) part.
Thereafter, inverting the “highest order” derivative ( OL ) in the linear operator on both sides of the
differential equation is the first step. Second is to identify the initial/boundary conditions and terms
involving the independent variables alone; as initial approximation. Decomposing the unknown
functions (N) into series whose components is to be determined is ADM. The decomposed parts of
ADM method are related physically to system reactions of various modes from zeroth mode to infinity
mode. The sum of all these modes is the solution of Differential Equation (Non-Linear Linear Integer
Order or Fractional Order). Physically the zeroth mode reaction comes from external stimulus plus the
initial integer order states; which instantly generates the internal stimuli of infinite modes-to oppose
this first action (change)-in opposite way, in time or space (at the origin). The exactly ADM
mathematics generates these infinite modes reactions; and therefore ADM is close to physical reality.
The ADM helps to physically visualize the reaction of system by decomposing the total gross reaction
into all these infinite modes. If the differential equation system with ( OL ) is of Riemann-Liouvelli type
fractional operator, then classically one needs the initial states as fractional order like
)0(),0( 1 uu etc; where is not an integer. These states are hard to visualize physically. With this
ADM the RL formulation does not need these fractional initial states instead requires )0(),0( uu , the
integer order states give the solution-and thus physically realizable easily. This new finding too is
highlighted in this chapter along with several other problems to give physical insight to the solution of
extraordinary differential equation systems. This way one gets insight to Physics of General
Differential Equation Systems-and its solution-by Physical Principle and equivalent mathematical
decomposition method. This facilitates ease in modeling systems-close to physical reality.
133
Generalization of Fractional Order Leading terms in differential
equations formulated with Riemann-Liouvelli and Caputo definitions
and use of integer order initial/boundary conditions with decomposition
method In this section merger of two classical definitions of Fractional Derivatives with decomposition,
technique is demonstrated; where only integer order initial/boundary conditions will be employed to
get to the modal solutions-in decomposition method. This generalization and unification is important
as to eliminate the need of much difficult fractional order initial states required classically by RL
formulation of FDE.
Decomposition of Caputo Derivative in Fractional Differential Equations
Let the linear part of the equation OL u Ru Nu G be of Caputo Fractional Derivative represented
as OL CtD , which is composed of integer order derivative of function followed by fractional
integration. That is if an integer m is just greater than fraction 0 ; that is mm )1( , then;
Caputo operator is: mt
mtt
C DDD )( . This gives the differential equation system as:
)()()()( uNuRGuDD mt
mt (5)
Inverting this we get:
)()()( )()()( uNDuRDGDuD mt
mt
mt
mt
(6)
The solution is: 1 1 1
O O O( ) L L ( ) L ( )m m mt t tu t D G D R u D N u (7)
Where 1OL m
tD ; we have used complementation property that is IDD mm .
The is solution to integer order homogeneous condition of 0)( tuDmt ; is same as in the case of
integer order general differential equation solution, described above. Due to this fact, researchers like
to formulate with Caputo derivative.
134
Riemann-Liouville (RL) Derivative and its decomposition for solving fractional differential
equation-with integer order initial condition
The RL derivative operator is )( mt
mtt DDD i.e. the function is first fractionally integrated and then
differentiated by integer order, which is just greater than the fractional order. In this solution, the
homogeneous equation formed by RL operator requires fractional initial states; though sometimes
difficult to interpret physically. However, one can relate these fractional initial states to physical
quantities provided the laws of physics are known (As Ohm’s Law, Stress-Strain relations and flow-
pressure relations etc.). Here in this chapter, it is demonstrated that decomposing fractional derivative
with RL definition by transforming to Caputo expression first then applying decomposition rules, one
can solve the fractional differential equations with RL formulations and with integer order initial
states.
Generally, the Caputo and RL definitions of fractional derivatives are not equal, but are equated by
initial conditions as:
)0()1(
)()( )(1
0
k
m
k
k
tC
RLt fk
ttfDtfD
Where mm )1( ; m 0t (8)
Let the linear part OL be of RL derivative type. Then the formulation with definition of RL derivative
gives the system as: )()()()( uNuRGuDD mt
mt . In the expression with RL we change to Caputo
and relate with RL-Caputo relation and get:
)0()1(
)()()( )(1
0
k
m
k
k
tC u
ktuNuRGuD
(9)
Here the RL differential equation is changed to Caputo formulation. This equivalent to original
differential equation, but with extra power series term with integer order initial conditions appearing as
extra source/sink term. Let us follow the decomposition method, as obtained for Caputo formulation in
the previous section, thereby giving the solution as: 1
1 1 1 1 ( )O O O O
0( ) L L ( ) L ( ) L (0 )
( 1)
kmm m m m kt t t t
k
tu t D G D R u D N u D uk
(10)
Let us examine the bracketed term of the RHS of, the source sink term.
135
1 1( ) ( )
0 0
1( )
0
( 1)(0 ) (0 )( 1) ( 1) ( 1 )
(0 )( 1)
k k mm mm k kt
k k
k mmk
k
t k tD u uk k k m
tuk m
(11)
The expression above contains reciprocal of Gamma function at negative integer points and zero point,
the values of which are zero. The reciprocal of Gamma functions )0())...2(()),1(( mm are
zeros. This reciprocal Gamma function is multiplied by powers of t at 0t ;
as 121 ,....,, mmm . Therefore, at 0t , the inverted source/sink extra term be collapsed to zero,
i.e.
0)0()1(
)(1
0
k
m
k
kmt u
ktD
The fractional differentiation of a power function is given by )1(/)1( ttD ,
Euler’s rule of generalized differ-integration; where , with 1 Let us take monomial of type
say nx , with n as integer. We differentiate this with integer order m such that
( ) 1, 2,3,....;m n m n .Then in the integer orders calculus 0m nxD x . Say a square function
2x differentiated thrice, four times, and so on will give zero. Same is the observation for kmt tD )( ,
returns zero since differential order (fractional) minus the power order that is
kmkm )()( , is 1)......2(),1(, mmm , for )1,.....(3,2,1,0 mk . This is new
observation not used elsewhere earlier in RL fractional calculus context. This new observation and its
application is now useful for solving FDE with RL formulation by decomposition technique where the
extra source sink term appearing in FDE (changing from RL to Caputo) collapses to zero thus giving
ease and uniformity in the two definitions of fractional calculus.
The above argument suggests that with RL derivative formulations too one can have solution in ADM
approach to solve fractional differential equation, with the help of integer order initial/boundary
condition. Therefore the solution of General Fractional Order Differential Equation where the leading
terms are of Fractional Derivative of Caputo or RL type is: 1 1 1
O O O( ) L L ( ) L ( )m m mt t tu t D G D R u D N u (12)
136
Where comes from integer order initial/boundary conditions. This unifies the two definitions of
Caputo/RL to solve FDE with only integer order initial states.
Proposition Classical theory of fractional differential equation solution gives the idea of fractional initial states.
Consider the fractional differential equation system (Homogeneous) as:
0)(][ 021
tybDaDD (13)
The Laplace Transforming the (13) gives:
0)()]0()([)]0()([ 21
21
sbYyDsYsayssY , Gives the following arranged relation, as
0)0()0()(][ 21
21
yaDysYbass (14)
Giving )(/)( sPCsY , with )0()0( 21
yaDyC , and indicial polynomial
as: ))(()( 2 xxbaxxxP , with , as roots of indicial polynomial. Now, we question,
how do we, know that )0()0( 21
yaDyC is finite? How do we physically find the meaning and
value of )0(21
yD that is fractional initial state? If )0()0( 21
yaDyC is not finite then problem is
serious and this approach is meaningless. With partial fraction approach, we get partial fraction of
inverse indicial polynomial and with assumption that C is finite, and nonzero constant, we proceed to
find solution to (13). If 0C , then only solution is trivial solution that is 0)( ty . The partial fraction
of indicial polynomial is
xxxP
111)(
1 (15)
Putting 21
)(sx , then I get
sssP111
)(1 (16)
The inverse Laplace is solution to (13), and in the form of Robotnov-Harley function, as one
possibility we obtain, solution to (13) as
),(),()(
21
21 tFtFCty
137
for , and
1 12 2
( ) [ ( , ) ( , )]y t C F t F t
this is for . The Robotnov-Hartley function, defined as
on
qnn
q qntataF
)}1({),(
1)1(
, with its
Laplace-Transform asas
saF qq
1),( .
In this approach, the fractional initial state is arising from Generalization of Laplace Transform, and
for this example is explained as:
)]([)( 2112
1tyDDtyD
, is Riemann-Liouvelli definition of fractional derivative.
Let )()( 12
1tytyD
.Then Laplace pair is )()( 21
1 sYssY .
Then Laplace of RL
)()( 112
1tyDtyD is )]0()([ 11 yssY , putting values of )(),0()0( 1
21
1 sYyDy , the Laplace of RL
)0()( 21
21
21
yDstyD , requiring fractional initial state.
Had the half-derivative operator in above be of Caputo type, then )]([)( 121
21
tyDDtyDC ,
with )()( 11 tyDty , has )0()()(1 yssYsY . The Laplace of Caputo half derivative
is )0()()]([ 21
21
12
1yssYssYs
requires integer order initial state. Therefore the dichotomy persists
in (13) if formulated by RL scheme then requires )0(y and )0(21y , and if formulated by Caputo
requires )0(y .This is one reason for Caputo derivative being popular. In RL, definition )0(21y is hard
to visualize.
Generally, the Caputo and RL definitions of fractional derivatives are not equal, but are equated by
initial conditions as
)0()1(
)()( )(1
0
k
m
k
k
tC
RLt fk
ttfDtfD
(17)
These two definitions are equal only when, the initial states are zero. The (13) is of first-order
differential equation with remaining lesser orders, classically, should have required only one initial
138
state-yet the treatment is asking for two initial states. However, the definition of order in FDE is not as
simple as Integer Order Calculus-as the first order system with half order element may behave as
classical second order system of Integer Order Calculus, yet the application scientists and engineers
will be comfortable if the RL formulations of FDE requires only integer order states-to get practical
realizable responses to FDE system.
Physical Reasoning to Solve First Order System and its Mode
Decomposition
The reason of application to simple ordinary differential equations the ADM will give much insight
into the action reaction theory of physics-and thus the ADM will try to explain the physical behavior
too. It is demonstrated that ADM is actually translating the physics of the process-where any change is
opposed by the system itself. Consider the first order differential equation:
( ) ( ) ( )x t ax t f t (18)
With initial condition and forcing function (source) as, (0) 0, ( ) ( )x f t K t . Let ( )x t be
instantaneous current of a ‘RC circuit’ connected to battery, by a switch. The circuit equation is (after
switch is closed at time origin)
1 ( )d ( ) BBi t t Ri t VC
or 1 ( )d ( ) BBVi t t i tRC R
(19)
Differentiating (19), we get,
d 1( ) ( ) ( )d
BBVi t i t tt RC R
(20)
The (19) is a voltage/current equation and (20) is current equation, re-written with compliance with
(18). This basic equation like (18) and (20) gives rate of change of current (function) as related to
external stimulus. The current excitation is impulse excitation in (20). The initial current in the system
be zero 0)0( i . This system has characteristic time constant RC seconds; meaning that current in
system changes e times in RC seconds. We shall consider response at larger time scales than RC . In
(18) this characteristic time constant is )/1( a . Here time scales and concept of time-constant is
mentioned. This could be length scale, frequency scale time-square scale area volume or any other
139
scales depending on the units of a in (18)-and nature of (18) too. At, the initial time at zero, the switch
closer gives impulse excitation-of current, and assuming if the capacitance of the circuit were absent-at
this initial instant (capacitance comes into action at a later time), then the current in the resistance is
1
0
( ) ( ) ( )t
BB BB BBt
V V Vi t t dt D t RR R a constant. Here a point is mentioned that integration of the
forcing function comes because the inertial element capacitance is present in the circuit and equation
(20). If the capacitor is completely absent then the current reaction will be simple )()/( tRVBB -
meaning that the current would have vanished instantaneously with the impulse (at 0t ). This initial
moment current in resistor is RVBB / at 0t ; since initially uncharged capacitor acts as short circuit
impedance. The capacitor presence is making the current linger for time greater than zero. The circuit,
as natural reaction to any force, will oppose this flow of current that is the change in current from zero
to RVBB / . This is the capacitor action. Therefore; the foremost reaction comes from the resistive
element (without lag or lead). That is RVRtVDii BBBBt /]/)([)0( 10 , in the absence of the
capacitive element (initially short-circuited)); and this is the first reaction due to external force (plus
initial current if at all be present in the circuit). This sudden, change in charges (Coulombs) cannot
flow into capacitor, as the voltage across it cannot change instantaneously. This constant action of
current gives rise to a rate of change of current in the system (per unit time constant) and is ( RCi /0 )
A/s. The, first reactionary constant current thus is opposed by internal generated current
as ]/[ 01
1 RCiDi t , which is in opposition to this first reactionary current initial reaction, therefore
negative. This action reaction summed up to give iiiti 0)( , the total current. The internally
generated reactionary current gives a rate of change as )/( 1 RCi A/s; which will generate opposition
current to the cause 1i ; as ]/[ 11
2 RCiDi t , which again is added to give total reaction
as 210)( iiiti , as the total current. This way infinite set of stimulus currents are generated as chain
reaction giving the total current as
0
0)(n
iti ; where 0i is the reaction due to external stimulus (and
initial current if present in the circuit) and rest are internally generated modes; acting in opposition to
the rate of change in current.
The reaction )(ti , for (20) can therefore be written as:
140
][)1(1)]/)(()0([)(1
111
n
ntn
BBt iDRC
RtVDiti (21)
This (21) is appearing as physical reasoning as infinite series as.
.....][1][1][1][1)( 31
21
11
01
0 iDRC
iDRC
iDRC
iDRC
iti tttt (22)
In recursion, we obtain
1 10 1
1(0) [ ( ) / ]; [ ]; 1t BB n t ni i D V t R i D i nRC
(23)
Applying (23), we obtain
331
3
21
2
11
0
!31
!211
!211
]/[1/
RCt
RV
RCt
RVD
RCi
RCt
RV
RCt
RVD
RCi
RCt
RVRVD
RCi
RVi
BBBBt
BBBBt
BBBBt
BB
Giving the total reaction of the system (20) as:
RC
tBBBB eR
VRCt
RCt
RCt
RVti ..........
!31
!211)(
32
The physical reasoning logic “opposite reaction to action” gives Mode-Decomposition and addition of
all these modal reactions gives the entire system response. The observation is that zero (foremost)
mode reaction is formed by the external source/sink stimulus plus due to any initial condition. To
oppose that rate of change an opposite internal reaction integral action takes place. This internal action
is the first mode-reaction which causes a rate of change; and again integral action to this first mode, in
opposition makes the second modal reaction. So on and so forth to make sum of “converging”
analytical solution to the system’s differential equation.
141
Physical Reasoning to Solve Second Order System & its Mode
Decomposition
Consider a classical oscillator of integer-second order, mass spring system represented as:
1( ) ( ) ( )kx t x t f tm m
(24)
With initial conditions and forcing function defined as (0) 0, (0) 0, ( ) ( )x x f t t .We can re-write
the equation as
1( ) ( ) ( )kx t f t x tm m
(25)
This equation (25) gives insight into physical aspect of the process. The RHS states the opposing
action to a forcing function, which is manifested as motion given by LHS of (25). At the initial
condition or time, the displacement being zero along with the velocity, implies that the displacement at
just time 0t is due the forcing function alone. This displacement is action is without any opposition.
This, (zero mode) displacement call it 20
1 ( )tx D f tm
. Due to nature of this forcing function as an
impulse, the displacement (zero mode) takes the form as 20
1 1( )tx D t tm m
. This displacement
action would be true, in the absence of any retarding or opposing element say spring or friction. (In
case of (25) it is spring-action). In absence of any opposition, the constitutive equation will
be ( ) ( )mx t f t , and for impulse force, the displacement will be linear function of time ( ) /x t t m ,
with constant velocity ( ) 1/x t m . The presence of spring makes the equation of motion
as 1( ) ( ) ( )kx t f t x tm m
. The external excitation being opposed by the spring action by opposite
spring force and is internally generated 1 ii xmkf . The primary and the zero mode of displacement
are due to external force on the mass that is tm
tDm
tfDm
x tt1)(1)(1 22
0 . This zero mode of
displacement is solely due to external excitation; since the initial conditions are at rest. This
displacement is now opposed by spring. Due to this opposing element, the displacement caused by
142
external force, the spring generates an opposing force (first mode, from zero order mode
displacement), as: 1 0kf xm
, and to this, new (internal force) the displacement would be
32 2 2
1 1 0 2
13!t t t
k k k tx D f D x D tm m m m
(the first order mode reaction-displacement), this
displacement, again generates an internal force; inside the spring as2 3
2 1 3 3!k k tf xm m
, and the
displacement, for second order mode is
3 2 52 2 2
2 2 1 3
13! 5!t t t
k k k t k tx D f D x Dm m m m m
and so on. In the absence of the spring, the opposing forces will be zero. We can call this as
displacement as sum of all the modal displacements from zero to infinity modes, with zero modes
being the only reaction to the bare excitation (and if any initial displacement and velocity be present)
and all other modes are opposing reactions taking place in the spring. The modes can be tabulated as in
Table-1.The process block diagram is represented in Figure-1, with 1k and 1m .Adding up all the
(modal displacements- reactions); the solution to (24) is obtained as infinite series:
3 2 5 3 7 3 2 5
0 1 2 3 2 3 4 2
1 1( ) ..... ....3! 5! 7! 3! 5!
k t k t k t k t k tx t x x x x t tm m m m m m m
Multiplying the above series by mk / and dividing by same we get:
t
mk
kmt
mkt
mkt
mk
kmtx sin1...
!5!31)(
52532
3
This is oscillator with natural frequency mk / radians per seconds. The physical process which
was based on action and opposite reaction law-can be put in symbols as
111
)1()0()0()1()0()0()(
in
n
ni
n xmkxtxfxtxtx
143
3 5 7
( ) ... sin( )3! 5! 7!t t tx t t t
Fig 1: Block diagram showing decomposition and solution of second order differential equation
Adomian Decomposition Fundamentals and Adomian Polynomials
However, we symbolize the general differential equation as:
GFu (26)
:F General non-linear Differential Operator (this can also be Fractional Differential Operator also of
Riemann-Liouvelli (RL) or Caputo type). This operator can be decomposed as:
OLFu u Ru Nu G (27)
OL :Highest Order Derivative (Integer or Fractional Order) which is invertible. :R Linear differential
(remainder) operator of order less than that of OL ; this can also be fractional differential linear
x(0) tx/(0) .f(t)
2d
2d
2d
1
-1
-1
x
144
operator. :N Is the Non-Linear Part which will be decomposed into infinite sum of Adomian
Polynomial. :G Is the source term. The decomposed equation can be re-written as:
OL u G Ru Nu (28)
Applying invert operator on both sides we get:
1 1 1O O OL L [ ( )] L [ ( )]u G R u N u (29)
Where is solution of the homogeneous equation OL 0u ; so that OL 0 , this comes from
initial/boundary conditions. The LHS of (26) physically is reaction of each components of physical
system; with RHS of (26) representing source/sink or forcing term. For example a mass spring and
damper system has the constituent equation as2
2
d d ( )d d
x xm c kx f tt t , the LHS of this is reaction of
each elements, the sum of which balances the RHS and the external force. In terms of (26) and (27) in
this physical system 2OL tD , 1
tDR and xxN )( . The solution to this is
1 1 1O O O
1( ) L L Lc kx t f R Nm m m
.For example if the order of OL is two
then )0()0( utu , assuming the time dependent differential equation system; and the invert
operator in this case is 1 2O
0 0
L (.) (.)d dt t
tD t t . If the order of OL is one then
)0( u and 1 1O
0
L (.) (.)dt
tD t .For decomposition of the )(uN part in the (4) define a “grouping”
parameter close to one as The function u can be expressed as:
.........)( 22
100
uuuuu nn
n (30)
This (30) is Maclurain series with respect to ; with nu ’s as coefficients of the Maclurain series around
0 that is !/)0()( nuu nn . Then )(uN in Maclurain, series with respect to we obtain
nn
n AuN
0
)( (31)
145
Where,
0 0
1 d! d
nk
n knk
A N un
(32)
The parameter is just an identifier for collecting terms in suitable way such that nu depends
on 1210 ,...,, nuuuu , and later on, we will set 1 .Paremetrizing the equation (29) we get:
1 1 1O O OL L [ ( )] L [ ( )]u G R u N u (33)
Expanding with decomposition the (33) we obtain:
1 1 1O O O
0 0 0L L Ln n n
n n nn n n
u u G R u A
(34)
Equating the coefficients of equal powers of in the expression for 0n , to get 0u , then 1n , to get
1u and so on, in (34), we get:
10 O
1 11 O 0 O 0
1 12 O 1 O 1
1 1O 1 O 1
LL ( ) L ( )
L ( ) L ( )..............
L ( ) L ( ), 1n n n
u Gu R u A
u R u A
u R u A n
(35)
Finally )()(1
0tut
N
nnN
with 1N ;and exact solution of (26) is )(lim)( ttu NN
.This method is
applied in various problems of physics. The convergence of this method is very well proved by
Adomian et al.. In the ADM method described the expression (35) contains Adomian polynomials nA
as recurring formulations where the invert operator is operational. The finding these nA , from (32) is
demonstrated here in this section. Suppose that the non-linear part of (27) that is )(uN is represented
as nn
n A
0 ; that is ))(( uN is assumed to be analytic in . So we write n
n
n AuNNu
0
))(( .
The nA ’s are polynomial defined in such a way that each nA depends only on nuuuu ,...,, 210 .
146
Thus, ).......,,(),,(),( 210221011000 uuuAAuuAAuAA etc.
Therefore, one possible formulation is listed below in (36).
0
3
3
2
2
2
23
3
3
321033
0
2
22
2
2
21022
01011
0022
100000
361
).,,,(
21),,(
),(
)(...)())(()(
uuNuu
uNu
uN
uuuuAA
uuNu
uNuuuAA
uuNuuAA
uNuuuNuNuAA
(36)
The nA ’s can be re-formatted, from (36) in the following form as:
0 0
1 1 0 1 00
221
22 2 0 00 0
22 0 1 0
32 312 33 3 0 1 2 0 0
0 0 0
31
3 0 1 2 0
( )
d ( ) ( )d
d d( ) ( )d d2!1( ) ( )2!
d d d( ) ( ) ( )d d d3!
( ) ( ) (3!
A N u
A u N u u N uu
uA u N u N uu u
u N u u N u
uA u N u u u N u N uu u u
uu N u u u N u N
0 )u
(37)
In the case where non-linear term is linear, that is say uuN )( ; in that case nn uA , else
)...,,( 210 nnn uuuuAA for all ......3,2,1,0n .For examples if 3)( uuN , then Adomian Polynomials
for this non-linearity are
21032
03
1302
122
0212
013
00 63;33;3; uuuuuuAuuuuAuuAuA and so on. The derivation of
obtaining Adomian Polynomials comes from Generalized Taylor’s series (Maclurain series) of several
variables from linear analysis. This is described as follows:
147
2 2 30 1 2 0 1 2 3
0( ) ( ....) .......n
nn
N u A N u u u A A A A
Put 0 , to get 00 )( AuN
Differentiate once, with respect to we get:
2 20 1 2 0 1 2
22 20 1 2
0 1 2 1 2 320 1 2
d d[ ( ....)] ( ....)d d
( ..) ( ..) 2 3 ...( ...)
N u u u A A A
N u u u u u u A A Au u u
Put 0 in above to get: 01 1 1 0
0
d ( ) ( )dN uA u u N u
u .Differentiating once, more with respect to we
get
20 10 1 1 2 3
0 1
( ..)d d( ..) ( 2 3 ...)d ( ...) d
N u u u u A A Au u
0 1 0 10 1 0 1
0 1 0 1
2 3
20 1 0 1
1 2 0 1 2 320 1 0 1
( ..) ( ..)d d( ...). ( ..)d ( ...) ( ..) d
2 3! ...
( ..) ( ..)( 2 ..) ( ..) (2 3 ..)( ..) ( ..)
N u u N u uu u u uu u u u
A A
N u u N u uu u u u u uu u u u
22 ..........A
Putting 0 in above expression, we obtain 20
022
0
02
21 2
)(2
)( AuuNu
uuNu
, implying
2220 01
2 2 2 0 1 020 0
d ( ) d ( ) 1( ) ( )d 2 d 2N u N uuA u u N u u N u
u u , continuing like this we get set of the Adomian
Polynomials for the function )(uN . If the non-linearity part is
0
21 )(
knAyyN . Then the Adomian
Polynomials are .....22;2;2; 20303202
121012
00 yyyyAyyyAyyAyA
If the non-linearity part is,
0
32 )(
knByyN The Adomian Polynomials are
148
3 2 2 2 2 30 0 1 0 1 2 0 2 1 0 3 0 3 0 1 2 1; 3 ; 3 3 ; 3 6 .............B y B y y B y y y y B y y y y y y
For the linear term,
0
0 )(k
nyyyN , 000 xyA .and nn yA .
For constant KyN )(0 , KyNA )( 000 ; 0........21 AAA .
The series solution
0nnu thus may have finite terms with higher modes as zero; depending on )(uN
Generalization of Physical Law of Nature vis-à-vis ADM
The physical description and then obtaining decomposed solution matches well with the ADM. From
the earlier sections, we generalize the system of General Dynamic System and apply action-reaction
laws to it so that one can obtain the solution by decomposition into finite or infinite modes. Let there
be general differential equation system, describing a General Dynamics as
1 2 11 2 1 0( ) ( ) ..... ( ) [ ( )] [ ( )] ... ( ) ( )m m m k kn
x x x n k kD u x a D u x a D a u x b u x b u x b u x G x (38)
We can write (38) as (39)
OL ( ) ( ) ( ) ( )u x G x R u N u (39)
Where 11
22
11 ... xn
mx
mx DaDaDaR
is the remainder differential operator of order less than m .
This; )(...)]([)]([)()( 01
1 xubxubxubxuauN kk
kkn
contains nonlinear as well as the linear
terms. The, )(xG is sum of all external stimulus source/sink. The OL represents a linear operator
representing the highest orders of change in the process parameter. OL d / dm mx ; which is easily
invertible. This order of change )(m could be one two, or any positive integer or even fractional (say
half, one fourth). This order of change could be with respect to time, space, space square (area)
frequency or time-square-depending on the process description. General physics law states from (38)
that a system will react to external stimulus and will have opposition to the changes; by the system
reaction terms defined by R and N defined in the system description equation (39). These R and
N generates internal stimulus when excited by external source/sink as to oppose the cause. The
reactions are causal in nature. If the external stimulus and the internal reactions to the stimulus get
149
balanced then the process parameter remains static without any growth (accumulation) or decay (loss).
Else, the process parameter will have a solution as finite or infinite decomposed modes; generated by
system itself to oppose stimulus generated internally by previous modes
2 3 10 1 2 3 0 O
2 2 30 1 2 0 1 2 3
1 10 0 1 O 0 O 1
1 11 1 0 2 O 1 O 1
2 1 12 2 0 1 0 3 O 2 O 2
( ) .....; L
[ ( )] ( ...) ....
( ); L ( ) L ( )
( ); L ( ) L ( )1( ) ( ); L ( ) L ( )2
u u u u u u G
N u N u u u A A A A
A N u u R u A
A u N u u R u A
A u N u u N u u R u A
1
1 101 O 1 O 11
0
0
d ( )1 ; L ( ) L ( ); ( ) .
( 1)! d
n kk
kn n n n nn
n
N uA u R u A u x u
n
SDM Applied to First Order Linear Differential Equation and Mode-
Decomposition Solution
Comparing with ADM described by (35) and solution obtained by physical reasoning, to arrive at
solution of (18), (20) gives a similarity. Therefore, the ADM is close to physical system behavior
where the system reacts naturally in opposite way to resist any change-this is physical law, which is
described by ADM, for solving system of differential equations. The initial reaction to the external
disturbances and the complete set of opposing reaction due to self opposed elements to the change,
gets summed up to get the overall reaction yielding solution for (18)
.........)0()( 3210 xxxxxtx
1 1 11 2( ) (0) ( ) ( ) ( ) ...t t tx t x D f t aD f t aD f t
)()0(
)()1()()0()(
10
11
1
1
tfDxf
fDatfDxtx
t
nn
tn
nt
2 3( ) ( )( ) 1 ... exp( )2! 3!
at atx t at at
150
The nf ’s are internal reaction forces due to change in nx ’s. The, action reaction process described is
represented in Figure-2, with parameter 1 Ka , the 1d is anti-derivative operator of unity order is
1
0
(.)dt
tD t
Fig: 2 Block showing solution of first order differential equation by decomposition
In the ADM, as described in (35) we can write the set of modes (reactions) as:
1 1 1 1 10 O 1 O 0 O 0 2 O 1 O 1L ; L [ ( )] L ( ); L [ ( )] L ( )x G x R x A x R x A
In the case of (18) 1 1OL tD , )()( tKtfG , axxN )( , is linear and ,00 Ax with Adomian
Polynomials (35) as: nn axA with no remainder term as 0R . With this decomposition we get
x(0) .f(t)
X0 X1 X2 . . .
1d
1d
1d
1
-1
-1
x(t)
151
1 10 O
0 0
1 1 11 O 0 O 0
21 1
21 1 1 2
2 O 1 O 1
23
2 2
2 31 1 1 3
3 O 2 O 2
L ( )
L [ ( )] L ( ) [ ]
( )L [ ( )] L ( ) [ ]2!
2!( )L [ ( )] L ( )
2 3!
( )
t
t
t
t
x G D K t KA ax aK
x R x A D aK Kat
A ax Ka tatx R x A D Ka t K
tA ax Ka
t atx R x A D Ka K
x t K
2 3( ) ( )[1 ....].
2! 3!atat atat Ke
The infinite currents at instant ( 0t ) are formed. The foremost (or zero) reaction current is due to
initial state of the circuit and solely due to external force-represented by 0x . Then recurring opposite
reactions occur as set of internal forces-due to opposing the changes, giving rise immediately the first
mode second mode (an to infinity-modes) of currents; adding up giving the total current reaction as,
1
0)(n
nxxtx .Therefore the ADM method is related to physical process of physics as to any
“action” there is equal and opposite “reaction”, may be external or internal to the system. In other
words, all system reacts in opposite way to any change (external or internal).
ADM Applied to Second Order Linear Differential Equation System and
Mode-Decomposition
The ADM method for (24) has 1 2OL tD , 0)0()0( xtx , )(1)(1 t
mtf
mG ,
xmkxN )( which generates Adomian Polynomials (35) as 00 x
mkA , and nn x
mkA .Here in (24) the
remainder part is 0)( xR . Using the ADM for (24) one gets the modal displacements as described by
physical reasoning also as:
152
1 20 O 0 0 2
3 2 31 1 2
1 O O 0 1 12 2 3
2 51 1 2 3
2 O O 1 3
1 1L ( ) ;
L ( ) L ( ) [ ] ;3! 3!
L ( ) L ( ) ( / 3!)5!
t
t
t
k kx G D t t A x tm m m m
k k t k k tx R x A D A xm m m m
k tx R x A D tm
And so on, giving the solution to (37) as
3 2 5 3 7 3 2 5
0 1 2 3 2 3 4 2
1 1( ) ..... ....3! 5! 7! 3! 5!
k t k t k t k t k tx t x x x x t tm m m m m m m
This too demonstrates the decomposition by ADM gives the physical modes of reaction process,
generated as infinite series.
Multiplying the above series by mk / and dividing by same we get:
t
mk
kmt
mkt
mkt
mk
kmtx sin1...
!5!31)(
52532
3
This is oscillator with natural frequency mk / radians per second; obtained earlier by physical
law of action-reaction process.
ADM for First Order Linear Differential Equation System with Half
(Fractional) Order Element and Mode-Decomposition
Consider a first order differential equation and presence of a fractional half order element.
12( ) ( ) ( ) ( )tx t D x t x t f t (40)
With initial conditions as (0) 0, ( ) ( )x f t t . The physical explanation was given in previous sections
gives following explanation and its solution
11 1 121 0
1( ) (0) ( ) ( 1) ( ) ( ); (0) ( )n n
t t t n tn
x t x D f t D D f t f x D f t
153
For this system the 21
tDR , 1tDL and nonlinear part is xxN )( ;linear in nature; thus as per
(35) nn xA . This demonstration also shows the fact because the fractional order component is of
lesser order than the main component, which is of, (in this case is integer order) the initial condition
does not depend on the fractional derivative definition. Here the initial states are always of integer
order in nature. The ADM method gives the components as:
1 1 10 O
0 01
211 1 1 121 O 0 O 0
1 111 1 1 12
2 O 1 O 1 1 1
3 3 22 2
L (0) ( ) ( ) 11
L [ ( )] L [ ] [ (1)] [1](1.5)
L [ ( )] L [ ] [ ( )] [ ]
2(2.5) 3 (1.5) 2
t t
t t t
t t
x G x D f t D tA x
tx R x A D D D t
A x
x R x A D D x D x
t t tt
The block is represented as follows in Figure-4, for reaction of first order system in presence of half
order element. An interesting point is mentioned here. The dynamic system, represented by (40) has
leading differential operator as first order. In classical mathematics we call that first order system. The
classical property of first order system, is that it gives response to a step Heaviside forcing function as,
over-damped (i.e. response contains no overshoot or undershoot). Example charging voltage profile of
capacitor having temporal response as [1 exp( / )]V t , where RC , time constant of charging
andV , is the step-height (final value of charging voltage). However, by the presence of a fractional
order element (40) changes that entire classical concept. Meaning a classical first order system may
have under-damped, oscillatory response; and even exhibit instability; and sustained oscillations; this
theory is proven and explained in books of fractional calculus. The (40) is having leading differential
operator as integer order, thereby the requirement of initial conditions to solve this type of system
requires only integer order initial states.
154
Figure: 3 The RC circuit (a first order differential equation), with semi-infinite cable as
fractional half order element.
Figure: 4 Block showing solution of first order differential equation by decomposition in
presence of fractional half order term.
x(0) .f(t)
X0 X1 X2 . . .
1d
11 2d d
11 2d d
1
-1
-1
x(t)
CRO
R
BBV
C
155
The physics of this process may be viewed as, RC circuit reacting to an impulse reaction in the
presence of a semi-infinite RC cable (CRO Probe)-connected to a shunt to measure the current. The
semi-infinite cable acts as half order element and the first order circuit reaction thus will be modified
by presence of this half order element. Refer Figure-3.
MODE FORCE DISPLACEMENT
0 0
1 1( ) ( )f f t tm m
20 0
1tx D f t
m
Higher Modes Higher Modal Internal Forces
Higher Modal Internal displacements
1 1 0 2
k kf x tm m
3
21 1 2 3!t
k tx D fm
2 2 3
2 1 3 3!k k tf xm m
2 5
22 2 3 5!t
k tx D fm
3 3 5
3 2 3 5!k k tf xm m
3 7
23 3 4 7!t
k tx D fm
……. ……. …….
Table-1: Decomposing the action reaction of second order mass spring system
156
MODE FORCE DISPLACEMENT
0 ( )t t
1 t 5 32
(3.5) (4)t t
2 5 32
(3.5) (4)t t
94 52
2(5) (5.5) (6)t t t
3 94 522
(5) (5.5) (6)t t t
1311 6 72 2
(6.5) (7) (7.5) (8)t t t t
….. ….. …..
Table-2 Modal force and displacements for second order system with fractional order damping
ADM for Second Order System with Half Order Element and its Physics
Solution of second order differential equation with presence of half order element [is considered in
(41)
12
0( ) ( ) ( ) ( )tx t D x t x t f t (41)
With the initial condition as (0) 0, (0) 0, ( ) ( )x x f t t .Rearranging the above equation (41), we
rewrite by double integrating both sides as,
32 2 2( ) (0) (0) ( ) ( ) ( )x t x tx D f t D x t D x t
157
20
5 323 221 0
94 523 222 1
( )
( )(7 / 2) (4)
( ) 2(5) (11/ 2) (6)
t
t t
t t
x D t t
t tx D D x
t t tx D D x
The modal displacements are generated after the application of external forcing function is depicted in
the Table-2.The block diagram of the process is shown in Figure-5
Fig: 5, Block diagram showing solution of by decomposition of a second order differential
equation in presence of fractional order term.
The ADM method generates the modes as follows.
x(0) tx/(0) .f(t)
X0 X1 X2 . . .
2d
3 22d d
3 22d d
1
-1
-1
x(t)
158
1 20 O 0 0
5 53 32 231 1 221 O 0 0 0 1 1
94 5231 1 222 O 1 1 1
L ( );
L ( ) ( ) ;(7 / 2) (4) (7 / 2) (4)
L ( ) ( ) 2(5) (11/ 2) (6)
t
t t
t t
x G D t A x
t t t tx R x L A D D x A x
t t tx R x L A D D x
The fundamentals of mode decomposition as explained above in case of second order differential
equation, in the presence of fractional order component may be explained in slightly elaborated way as
follows:
Take a mass spring fractional viscous system as 12 2
122
d d( ) ( ) ( ) ( )d d
x t a x t bx t f tt t
where a is the
constant of half order property and b is the spring stiffness constant for ideal mass less spring. The
above equation of motion is for unit mass attached to an ideal spring with half order visco-elastic
element. The initial condition is: (0) 0, (0) 0, ( ) ( )x x f t t .The above equation can be re-written in
terms of external force and opposing internal forces as:12 2
122
d d( ) ( ) ( ) ( )d d
x t f t a x t bx tt t
.
Decomposing this by modal decomposition, we get. The zero order modes as 20 ( )tx D f t .This zero
order displacement is the reaction without presence of the spring or any other opposing elements. Due
to this zero order displacement, there will be opposing forces as, 1
211 0tf aD x and 12 0f bx . Giving
rise to first order displacements as:32 2
11 11 0t tx D f aD x and 2 212 12 0t tx D f bD x The overall
first modal displacement is, therefore 1 11 12x x x .From this the reaction force for second modes are
generated as:1
221 1tf aD x and 22 1f bx .Giving rise to second modal displacement
as:32 2
21 11 1t tx D f aD x and 2 222 22 1t tx D f bD x .Similarly, we can carry on for infinity as
this self-similar pattern of reactions generated within the system to external stimulus. The observation
is that the half order element adds second force to the ideal spring restoring force as obtained in case of
pure second order classical oscillator. The practical way of explaining the fractional order behavior is
by considering a LC-oscillator and tries to measure the oscillation by CRO probe, which is semi-
infinite cable acting as half order element. The constant k of the half order element is depending on the
159
distributed loss parameter that is per unit series resistance and per unit shunt capacitance. The
constitutive equation for the circuit is:
12
12
1 d d( )d ( ) ( )d d
BBi t t L i t k i t VC t t
or rewritten as
12 2
122
( ) d d( ) ( ) ( )d d
BBi t L i t k i t V tC t t
This is demonstration of the oscillator with fractional order element, shown in Figure-6.
Fig 6: The oscillator circuit (a second order differential equation), with semi-infinite cable CRO-
probe acting as half order element.
Practically in circuit experiments it is observed that a purely oscillating circuit when connected by
shunt to a long CRO-probe, goes to damped oscillations-removal of probe again gives the oscillations.
Use of a very short probe to CRO gives oscillation. This is due to fact the long CRO probe may act as
lossy transmission line, behaving as half-order damping element.
CRO
L
BBV
C
160
Application of Decomposition Method in RL Formulated Partial
Fractional Differential Equations Linear Diffusion-Wave Equation and
Solution to Impulse Forcing Function
Diffusion of arbitrary order is studied at details in fractional calculus books here attempt is made to
obtain series solution with ADM and the physical explanation of several modes generating as reaction.
This problem example elaborates that the time evolution of process parameter takes place without the
forcing function present i.e. 0G ; only the effect is due to initial value (in this case is Dirac’s delta
function as process parameter present at space-origin). The fractional time rate of change of the
process variable is related to spatial double derivative of the same as expressed in (42). In (42) the
formulation of fractional derivative is of Riemann-Liouville (RL) scheme. Let us consider the problem
of fractional time diffusion as:
),(),( 2
2
txux
txut
(42)
With 21 and have the initial condition as )()0,( xxu and 0)0,( xut .The integer order
highest to the fractional order in case of (42) is 2m .Converting the (42) into Caputo derivative
formulation (42), we obtain
),()0,()2(
)0,()1(
),( 2
21
txux
xutxuttxut
C
(43)
Observation here states that the (42) RL derivative of fractional order when changed to Caputo
formulation (43) gives rise to extra source/sink terms of the inverse power function of the independent
variables in the constituent equation. In (43) after applying the definition of Caputo derivative in the
fractional operator we get:
)0,()2(
)0,()1(
),(),(1
2
2
2
2)2( xutxuttxu
xtxu
tDt
(44)
Inverting (44), we get:
161
2 2(2 ) (2 ) (2 ) 1
2 2
( ,0) ( ,0)( , ) ( , ) [ ] [ ](1 ) (2 )t t t
u x u xu x t D u x t D t D tt x
(45)
Applying fractional derivative formula of the power functions in RHS of (45) we get:
)211()11(
)2()0,(
)21()1(
)1()0,(),(),(
21
2
2
2)2(
2
2
txu
txutxux
Dtxut t
(46)
Simplifying (46), we get the following
)0()0,(
)1()0,(),(),(
12
2
2)2(
2
2 txutxutxux
Dtxut t
(47)
The last two terms of (47) gives zero; since reciprocal Gamma function is zero at values zero at
negative integer points giving the modified diffusion equation as:
),(),( 2
2)2(
2
2
txux
Dtxut t
(48)
Taking the (space) Fourier Transform of (48), we get:
),(),( )2(22
2
tkuDktkut t
(49)
With transformed initial condition as2
1)0,( ku ; and 0)0,( kut
where dxtxuetku ikx ),(21),(
, k is the spatial Fourier Transform definition. The parameter k
is “wave-vector”. The system of equations (48) has been transformed to (49) so we solve for
),( tku and write with ADM (35) the solution as:
162
1 1 1O O O
1 2O
2 2 2
2 2 2
( , ) L L ( ) L ( )
( , ) L [ ( , )]( , ) ( ,0) ( ,0) [ ( , )]
( ,0) [ ( , )]
t
t t t
t t
u k t G R u N u
u k t D u k tu k t u k tu k k D D u k t
u k k D D u k t
(50)
In (50) the 1 2OL tD , 0 RG , ),()( tkuuN ; is the linear. Therefore, )0,(00 kuu and
for 1n ; 121
ntn uDLu Following the ADM, (35) we get ),(),(
0tkutku
nn
where the
components are:
0 0 0
21 2 2 2 2
1 O 0 0 1 1
4 22 2 2
2 1 2 2
6 32 2 2
3 2
1 ;2
L [ ] [ ] ;( 1)2
[ ] ;(2 1)2
[ ](3 1)2
t t t
t t
t t
u A u
k tu D A k D D u A u
k tu k D D u A u
k tu k D D u
Therefore we get (51) as
)(21
)1()(
21
...)13()12()1(
121),(
2
0
2
36242
tkEn
tk
tktktktku
n
nn
(51)
The (51) is series solution of (43) in space-Fourier Transformed system, in terms of Mittag-Leffler
function. Taking the Inverse Fourier Transform of (51), we get solution to (42) with impulse excitation
as:
)/(21),( 2
2
2
txMttxu ; for x and 0t ;
Where Inverse Fourier transform is: 1( , ) ( , )d2
ikxu x t e u k t k
, x .
163
The 2
M is special case of Wright function defined as:
0 })1{(!)(
)(n
n
nnz
zM ; where 10
0
2
2
2
21
2!
//
n
n
nn
txtxM
; where 12
0
Application of Decomposition Method in RL formulated Fractional
Differential Equation (Non-Linear) and its solution
So far, we considered linear systems and reasoned out physically the decomposition and the action-
reaction concepts to solve the differential equation systems, by ADM. The non-linear part )(uN in the
earlier cases were of linear in nature and thus the Adomian Polynomials for each mode were same
( nn uA ) for 1n ; for obtaining the subsequent parallel modes-and thereby the solution. The non-
linear part is described gives different Adomian Polynomials for the different modes to get solution of
non-linear systems. Consider RL formulated Fractional differential equation of with non-linearity as:
4d (1 )d
y yt
; With 10 ; and 0)0( y (52)
The nearest integer in this case is one; for the fractional order . The invert operator, 1 1OL tD and
0 C the solution is thus; as in (42)-(48) is
1 1 1 1O
0 0 0( ) Ln t n t n
n n ky t y D A D D A
(53)
The source/sink term that appears in the RL to Caputo change in (52) is )0()1(
yt
.Taking
mtD of this source/sink term gives 0
)0()0(
)1()1()1()0( 1
tym
ty m
; therefore, the
164
decomposed solution of (52) is (53). The non-linearity
0
4)1()(n
nAyyN , and to find the
Adomian Polynomials following are the steps (35) (36)
The iterations are following
00 y , 1)1( 400 yA
)1(
)1()( 110
111
tDDADDty tttt
1 1 0
4 30 0 0
0
1 1
( )d( ) (1 ) 4(1 ) 4
d4( 4)
( 1)
A y N y
N y y yy
tA y
From this, we obtain the next term of solution as:
)12(
4)1(
4)(2
111
112
ttDDADDty tttt
Next step is to obtain 2A as follows:
21
2 2 0 0
24 2
0 0 020
02 22
22 2
( ) ( )2!
d( ) (1 ) 12(1 ) 12d
( ) 4
6 (2 1) 16 ( 1)4 12( 4)(2 1) 2 ( 1) (2 1) ( 1)
yA y N y N y
N y y yy
N y
t tA t
From above we obtain
3
2
2
211
3 )1()13()1(16)12(6)( tADDty tt
The series form solution to (52) is, therefore
165
22 3
2
6 (2 1) 16 ( 1)1 4( ) ......( 1) (2 1) (3 1) ( 1)
y t t t t
(54)
Conclusions A new way to solve extraordinary differential equation systems with principle of natural law of action
and opposite reaction and its relation to Adomian Decomposition method is described here; with a
rider that one may use only ‘integer order initial states’. The several examples of utilization of this new
method is included in this chapter, this is new exposition.
166
Chapter 5
Realization of Fractional Order Circuits and Fractional Order
Control Systems
Introduction Fractional Order Control (FOC) means controlled systems and/or controllers described by fractional
order differential equations. Using the notion of fractional order, it may be a step closer to the real
world life because the real processes are generally or most likely fractional. However, for many of
them, the fractionality may be very small. In particular, it has been seen that materials having memory
and hereditary effects and dynamical processes, such as mass diffusion or heat conduction can be more
adequately modeled by fractional-order models than integer-order models. A typical example of a non-
integer (fractional) order system is the voltage-current relation of a semi-infinite lossy RC line or
diffusion of the heat into a semi-infinite solid, where the heat flow in nature is equal to the semi-
derivative of the temperature. However, the fact that the integer-order dynamic models are more
welcome is probably due to the absence of solution methods for fractional-order differential equations
(FODEs). Recently, some progresses in analysis of dynamic systems modeled by FODEs have been
made. For example, PID controllers, which have been dominating industrial controllers, have been
modified using the notion of fractional-order integrator and differentiator. It is shown that the extra
degrees of freedom from the use of fractional-order integrator differentiator made it possible to further
improve the performance of traditional PID controllers. By using FOC approach, control system's
phase and gain responses can be easily offset to any desired amount. In theory, control systems can
include both the fractional order dynamic system or plant to be controlled and the fractional-order
controller. However, in control practice, it is more common to consider the fractional order controller.
This is due to the fact that the plant model may have already been obtained as an integer order model
in the classical sense. In most cases, our objective is to apply the fractional order control (FOC) to
enhance the system control performance. The main advantage of using fractional-order controllers for
167
a linear control system is that the time and frequency responses can be shaped using functions rather
than of exponential type (power series type) and as a consequence, the performance of the feedback
control loop can be improved over the use of integer-order controllers. Iso-damping is one of the basic
advantages, where the close loop overshoot becomes independent of system parameters gains. This
iso-damping is key concept of obtaining fuel efficient control systems. A Fractional slope on the log-log Bode plot has been observed in characterizing a certain type of
physical phenomena and is called the fractal system or the fractional power pole (or zero). In order to
represent and study its dynamical behavior, a method of singularity function is discussed in this
chapter, which consists of cascaded branches of a number of poles-zero (negative real) pairs.
Moreover, the distribution spectrum of the system can also be easily calculated and its accuracy
depends on a prescribed error specified in the beginning. This method would thereafter be used widely
in approximating fractional order transfer functions for the discussed Lead Compensators as well as the
PIλDµ controllers. This chapter presents an effective method for the approximation by a rational
function, for a given frequency band, of the fractional-order differentiator sm and integrator s-m (m is a
real positive number), and the fractional PIλDµ controller). First, the fractional-order integrator s-m (0 <
m < 1) was modeled by a fractional power pole (FPP) in a given frequency band of practical interest.
Next, this FPP is approximated by a rational function, using the method of singularity function
approximations). The above idea was used to model the fractional-order differentiator sm (0 < m < 1)
by a fractional power zero (FPZ). Then, the approximation method of the FPP was extended to the FPZ
to obtain its rational function approximation. Therefore, with this method, one can achieve any desired
accuracy over any frequency band, a rational function approximation of the fractional-order
differentiator and integrator. The rational function approximation of the fractional PIλDµ controller is
just an application of the above method.
Singularity Structure for a single Fractional Power Pole (FPP) A single FPP system can be modeled in the frequency domain as follows:
1( )
1m
T
H ssp
(1)
168
where 1/pT, is the relaxation time constant and 0 < m < 1. As shown in Figure1, the line with slope of -
20m dB/decade is approximated by a number of zigzag lines connected together with alternate slopes
of 0 dB/decade and - 20 dB/decade. The high and the low-frequency properties of the magnitude of the
transfer function with a single-fractional power pole suggest that the lowest and the highest
singularities of the transfer function approximation must be poles.
Figure.1 Bode plot of an FPP with slope of -20mdB/dec and its approximation as zigzag straight
lines with individual slopes of -20dB/dec and 0dB/dec.
169
Figure.2 Choosing the singularities for approximation by assuming a constant error between the
-20 dB/dec line and the zigzag lines.
Thus, we can rewrite (1) as represented by its pole-zero pair as: 1
0
0
11( ) lim
11
N
i im NN
i iT
sz
H ssspp
(2)
where (N + 1) is the total number of the singularities. Hence, for a finite range of frequency, (2) can be
truncated to a finite number N, and the approximation becomes: 1
0
0
11( )
11
N
i im N
i iT
sz
H ssspp
(3)
A way to choose the singularities (the pole-zero pair) for the approximation is developed as follows.
Let us assume that the maximum discrepancy or the error between the zigzag lines and the desired line
is chosen to be y dB, as shown in Figure 2. Then the poles and zeros of the singularity function can be
obtained as follows:
the first pole, [ / 20 ]10 y mo Tp p
170
the first zero, [ /10(1 )]0 10 y m
oz p
the second pole, [ /10 ]1 10 y m
op z
the second zero, [ /10(1 )]1 110 y mz p (4)
……
……
the Nth zero, [ /10(1 )]1 110 y m
N Nz p
the (N+1)th pole, [ /10 ]110 y m
N Np z
where pT is the comer frequency and is determined at a point of -3m dB from the original transfer
function as shown in Figure 1, po is the first singularity and is determined by the specified error, y
decibel, and pN is the last singularity and is determined by N.
Now, let /10(1 )10 y ma and /1010 y mb then /10 (1 )10 y m mab (5)
Therefore, we can obtain the distribution of these poles and zeros as:
11 1 2
1 1 1 1
................ & .................o N N
o N o N
z z pz p pa bp p p z z z
(6)
In addition, the location ratio of a pole to a previous pole is equal to the location ratio of a zero to a
previous zero and it is equal to ab; i.e.:
11 2 1 2
0 1 2 0 1 1
............ ..........N N
N N
z pz z p pab abz z z p p p
(7)
From the above relation we can also generate these poles and zeros from the first pole po using the
following algorithm as:
( ) & ( )i ii o i op ab p z ab ap (8)
It is interesting to note that both pi and zi are in geometrical progression form with ratio equal to ab.
The approximated transfer function can be written as follows: 1 1
0 0
0 0
1 1( )1( )
1 11 ( )
N N
ii ii o
m N N
ii ii oT
s sz ab ap
H ss ssp ab pp
(9)
171
To draw this rational approximation up to maximum frequency max , which can be100 H , we need
N pairs of interlaced poles and zeros. In the logarithmic scale the frequency spread is max 0log log p .
The ratio between subsequent poles are as given in (7) is 1 2
0 1 1
... N
N
pp p abp p p
, therefore the in the
logarithmic scale the distance between successive poles
are 1 0 2 1 1(log log ) (log log ) ... (log log ) log( )N Np p p p p p ab . Therefore, we have the
following expression
max
0max 0
1
loglog log1log log log( )N N
ppNp p ab
This can be approximated to nearest integer as:
max
0log
1log( )
pN Integer
ab
Geometrical Derivation of recurring relationship of Fractional Power
Pole for fractional integration Refer Figure 3 for expansion of Fractional Power Pole with N stage interlaced poles and zeros. In this
figure only two pairs of pole zeros are shown.
172
Figure.3: Showing expanded view of shaping of fractional pole by series of poles and zeros, with
in y dB error.
From the fractional power pole Tp , move in the X-Axis to the point when the inclined line of slope
20m is y dB away ( BC y ), here we get the first integer order pole 0p . That is point B in the X-
Axis. From this pole we draw a slope of 20 and stop at point D on the X-Axis where the segment
BE is at y dB away from the original 20m line, that is ( EL y ). At this point we place a zero, 0z on
the X-Axis, that is at point D . This zero makes the line straight EG , with total slope zero, and we stop
at G , which is y dB away from the original 20m line ( EK y ). The process gets repeated, so that
the fractional power pole line of slope 20m dB/decade is y dB away from the zigzag approximated
expansion. The relation between the fractional pole (relaxation time constant) Tp and the first pole is
found from triangle ABC .
20BC mAB
BC y , and 0log log TAB p p
Thus
log Tp 0log p 0log z 1log p 1log z
A B
C
D
E
F
G H
I
J K
L
2y b
y
y
b
y
y
b
2y
173
0
20log log T
y mp p
, giving 0log log 20Tp p m
Therefore,
200 10
ym
Tp p
The relation between 0p and 0z is obtained from triangles BDE and triangle BDF .From triangle
BDF we get,
0 0
20log log
DF b mBD z p
, giving 0 020 (log log )b m p z . From triangle BDE , we have
0 0
2 20log log
ED y bBD z p
Giving0 0 0 0
2 20log log log log
y bz p z p
, here substitute the value of 0 020 (log log )b m z p , we
get: 0 0
2 20 20log log
y mz p
, giving: 0 02log log
20(1 )yz p
m
or 10(1 )
0 010y
mz p
The relation between 1p and 0z is obtained from triangle JKL , where:
1 0
2 20log log
KL yJK p z
, giving 101 010
ymp z
The relation between 1z and 1p ,is obtained from the triangle GHI . Observing the triangle GHI , which
is equal to triangle BDE . Thus ,GH BD HI DE
1 1
2 20log log
HI y bGH z p
, gives: 10(1 )
1 110y
mz p
Singularity Structure for a Single Fractional Power Zero (FPZ) The singularity structure can be obtained on similar lines as that of an FPP, as:
0 0
0 0
1 1( )
( ) 11 1
( )
N N
m ii ii oN N
Ti
i ii o
s sz ab zsH s
p s sp ab az
(10)
Note that the structure has N zeroes and N poles.
174
The method discussed above would be repeatedly illustrated in the next section, when we will
approximate the fractional order transfer functions of a Lead Compensator and a fractional order PIλDµ
controller.
Fractional Order Integrator and its Rational Approximation The transfer function of the fractional-order integrator is represented in the frequency domain by the
following irrational transfer function:
1( )I mG ss
(11)
where s =jω is the complex frequency and m is a positive real number such that 0 < m < 1. In a given
frequency band of practical interest (ωL, ωH), this fractional-order operator can be modeled by an FPP
whose transfer function is given as follows:
The modulus of ( j ) ( j ) mIG at frequency L is
2
1m
L.The logarithmic value of the modulus at
the frequency L , is therefore, 210 log( )Lm
( )1
Im
c
KG ss
(12)
Suppose that for ω є (ωL, ωH), ω>>ωc. Therefore
1( ) ( )m
I cIIm m m
c
KKG s G ss ss
(13)
The modulus of j( j ) ( ) 1m
mC
CG
at frequency L is2
2
1( )[( ) ] 1 ( )
m LmCC
. The
logarithmic value of this modulus at L is thus,2
220 log( ) 10 log LC
C
m m
. The difference in
logarithmic values of gain modulus at L , be denoted by ,also called slope error,
is log ( ) log ( )I LG j G j .
175
Expanding this by putting the logarithmic modulus value we will have: 2
22
( )10 log( ) 20 log 10 log( )
LL C
C
m m m
22 2
2
( )10 log( ) 10 log( ) 10 log( )
LL C
C
m m m
2 2
2 2
( ) ( )log 1( ) ( ) 10
C L
L C m
2
1010 1mC
L
1010 1mC L
We choose the frequency of interest from 0.1 Hz to 10 Hz, and to realize fractional integrator of order
0.6 with 510 as error in the slope of this realized transfer function with the ideal fractional integrator,
the, Cf , the -3m dB corner frequency is:
51010 0.6 40.1 10 1 2 10Cf
Hz, with KI = (1/ωC m); ωC is the -3m dB frequency corner of the FPP,
which is obtained from the low frequency ωL, as: /1010 1mC L
where ε is the maximum
permitted error between the slopes of the fractional-order integrator of (11) and the FPP of (12) in the
given frequency band of interest (ωL, ωH).
In order to represent the FPP of (12), and consequently the fractional-order integrator, by a linear time-
invariant system model, it is necessary to approximate its irrational transfer function by a rational one.
The method of approximation consists of approximating the -20m dB/dec slope on the Bode plot of the
FPP by a number of alternate slopes of-20 and 0 dB/dec corresponding to alternate poles and zeros on
the negative real axis of the s-plane such that p0 < z0 < p1 < z1 < …….. < zN-1 < pN .Hence, the
approximation is given by:
1
0
0
1( )
11
N
iiI
Im N
ic i
s zKG s Ks s p
(14)
176
Using a simple graphical method that began with a specified error y in decibels and frequency band
ωmax which can be 100 ωH, the parameters a, b, p0, z0 and N can be calculated as:
/10 1 /10 / 20 maxlog /10 , 10 , 10 1
log( )y m y m y m o
o C o o
pa b p z ap N Integer
ab
We find the poles (pi) and the zeros (zi) of (14) are found to be in a geometric progression form. They
can then be derived from the above parameters as pi = po(ab)i for i=0,1,…..N and zi = zo(ab)i for
i=0,1,…..N-1. Hence, the fractional order integrator can be approximated by a rational function in a
given frequency band of interest as:
1
0
0
1 ( )( )
1 ( )1
Ni
oiI
Im Ni
oc i
s z abKG s Ks s p ab
(15)
Fractional Order Differentiator and its Rational Approximation The transfer function of the fractional-order integrator is represented in the frequency domain by the
following irrational transfer function:
( ) mIG s s (16)
where s =jω is the complex frequency and m is a positive real number such that 0 < m < 1. In a given
frequency band of practical interest (ωL, ωH), this fractional-order operator can be modeled by an FPP
whose transfer function is given as follows:
( ) 1m
DC
sG s K
(17)
Suppose that for ω є (ωL, ωH), ω>>ωc. Therefore
( ) ( )m m
mDD Dm
c c
K ssG s K s G s
(18)
with KD = (ωC m); ωC is the -3m dB frequency corner of the FPZ, which is obtained from the low
frequency ωL, as: /1010 1mC L
where ε is the maximum permitted error between the slopes
of the fractional-order differentiator of (16) and the FPZ of (17) in the given frequency band of interest
(ωL, ωH). In order to represent the FPZ of (17), and consequently the fractional-order differentiator, by
a linear time-invariant system model, it is necessary to approximate its irrational transfer function by a
177
rational one. The method of approximation consists of approximating the -20m dB/dec slope on the
Bode plot of the FPZ by a number of alternate slopes of -20 and 0 dB/dec corresponding to alternate
poles and zeros on the negative real axis of the s-plane such that z0 < z0 < z1 < p1 < …….. < zN < pN
Hence, the approximation is given by:
0
0
1( )
1
N
m ii
D D Nc
ii
s zsG s K K
s p
(19)
Using a simple graphical method that began with a specified error y in decibels and frequency band
ωmax which can be 100 ωH, the parameters a, b, p0, z0 and N can be calculated as:
/10 1 /10 / 20 maxlog /10 , 10 , 10 1
log( )y m y m y m o
o C o o
za b z p az N Integer
ab
The poles (pi) and the zeros (zi) of (19) are found to be in a geometric progression form. They can then
be derived from the above parameters as pi = po(ab)i for i=0,1,…..N and zi = zo(ab)i for i=0,1,…..N-1.
Hence, the fractional order integrator can be approximated by a rational function in a given frequency
band of interest as:
00
0
1 ( )( )
1 ( )
Ni
mi
D D Nc i
oi
s z absG s K K
s p ab
(20)
The Phase Shaper of fractional order This section deals with the study of a generalization of the traditional lead compensator. This is
obtained by introducing as a new parameter the fractional order, α, of the structure. By doing so the
proposed fractional order lead compensator (FOLC) has the form
1 1( ) 1 1c c
s sC s k kx ss x
(21)
In this work, the above controller is used just as a lead compensator, that is, its purpose is not to ensure
a constant phase (α π/2) in a frequency interval, but to ensure the fulfilling of the design specifications
traditionally used for a lead compensator. By providing the designer with greater flexibility in shaping
178
the frequency plot of the compensator, due to the slope modification factor α, the fractional lead
compensator enjoys a distinctive edge over its integer order counterpart.
Frequency characteristics of the Lead Compensator
For the conventional lead compensator
1 1*( ) 1 1c c
s sC s k kx ss x
(22)
the zero frequency is given by ωzero = (1/λ) and the pole frequency by ωpole. = (1/x λ). The key idea in
the design of a lead compensator is to increase the phase margin of the open loop system, by adding
phase in the neighborhood of the gain crossover frequency, ωc. The bode plots of (2) are shown in
figure 4, where it can be observed that the maximum phase фm is given at frequency ωm, the geometric
mean of the corner frequencies ωzero and ωpole.
Figure-4: Bode plots of the transfer function C*(s)
Therefore,
1 1sin1m
xx
(23)
1m x
(24)
179
*( j ) j 1 1j 1
m m
c m
Ck x x x
(25)
It must be taken into account that this structure modifies the magnitude curve, moving the frequency
ωc to the right and producing a decreasing on the obtained phase margin. In order to maintain the
specification of the phase margin, фm, this phase lag must be compensated by increasing in a few
degrees the maximum phase фm that the compensator must give (over phase). This over phase is
estimated by trial and error method. However, analytical methods can be used for the design of the
integer compensator in order to guarantee the desired фm at the desired ωc without a trial and error
process, though the over phase is always present.
Now, considering the fractional lead compensator of (21), as can be seen from figure 4, the parameter x
sets the distance between the fractional zero and pole, and the parameter λ sets their position on the
frequency axis. The choice of these two parameters depends on the value of α. Higher the value of α,
the higher the slope of the magnitude of C(s) and the higher the maximum phase фm that the
compensator can give.
Figure-5: Bode plots of the transfer function C(s)
At the frequency ωm
2
2
( j ) ( ) 1 1'( )( ) 1m
m m
c m
C C sk x x x
(26)
180
1 1arg( '( )) arctan( ) arctan( ) sin1m m m m
xC s xx
(27)
Compensation using a fractional Lead Compensator
In the most general case, first of all, the value of compensator gains k’= kcxα can be set in order to
fulfill a static error constant specification for the compensated system. For a general plant model of a
form (system type n)
( 1)( )
( 1)
ii
nj
j
k sG s
s s
(28)
the static error constant kss , has the expression:
0 0 0
( 1)1 'lim ( ) ( ) lim ' lim '1 ( 1)
i nn n i
ss n ns s sj
j
k ss s k kk s C s G s s k k k
x s s s s
(29)
That is, k’ = kcxα= kss/k, setting the relation between parameters kc, x, α. Knowing the value of k’, the
bode plots of the system G’(s) =k’G(s) are obtained, in which the static error constant specification is
already fulfilled. Now, specifications of gain crossover frequency and phase margin must be achieved.
Through the Bode plots of the plant G’(s) the maximum phase (фm) and the magnitude that the
compensator C’(s) must give to fulfill these two frequency specifications is observed. Then, the
relations for the parameters of the fractional structure are given by the equations:
1
c x
(30)
1 1'( )' ( )m c
c
C sk G sx
(31)
1 1arg( '( )) sin arg( ' ( ))1m c cm m
xC s k G sx
(32)
The above expressions come from the fact that open loop transfer function after compensation is / /( ) ( )G s C s must have at gain cross over point ( c ) the phase as m for stability.
Thus / / / /@arg[ ( ) ( )] arg[ ( )] arg[ ( )]
c mG s C s C s k G s . So, a set of three nonlinear equations
181
(30) to (32) and three unknown parameters (x,λ,α) is obtained . Solving these equations, the value of
parameter kc can be easily obtained, since
ssc
kkx k (33)
Therefore we have a total of four non linear equations and four unknown parameters to fulfill three
design specifications, independently, ensuring the maximum phase, фm, at the frequency ωc. One of the
major issues in obtaining an analytical design structure for the fractional lead compensator is the
solution of the above mentioned set of non linear transcendental equations, whose solution and
convergence is strongly governed by factors such as selection of starting point (initial solution guess)
and the number of iterations.
Illustrative Example We aim to fulfill the design specifications for the given plant G(s), using both integer and fractional
order designs, and then compare the performances.
2( )(0.5 1)
G ss s
Velocity error constant, kV=20.
Gain crossover frequency, ωc=10 rad/sec.
Phase margin, фm=0.27π=50°.
For both the compensators, from the velocity error constant specifications, it is obtained
k’=kv/k=10. Now for the system G’(s) =k’G(s), it is observed that at the gain crossover
frequency, ωc=10 rad/sec, the magnitude of G’(s) is -8.16 dB and the phase is -168.7°. Thus we
must design the requisite compensators, to fulfill the conditions given
by '( ) 8.16dBc dB
C s
and ( '( )) 38.7c
C s , The compensator designs obtained are as
follows:
Integer order Compensator: 0.2082 1( )0.048 1C
sG ss
182
Fractional Order Compensator: 0.550.5266 1( )
0.0178 1CsG ss
Figure 6: Compensated Gain plots
From the figures 6 and 7 we observe the following
Evidently, we can observe that the fractional order design enjoys distinctive advantages over its
integer order counterpart by providing greater gain and phase margins throughout the frequency
band of interest, and thus greatly enhancing the system stability.
Apart from the example mentioned above, several other examples were considered which
resulted in similar results as discussed above. The only issue which has to be dealt carefully is
the issue of convergence of solutions of the non linear equations.
The introduction of parameter α, fractional order of the structure, allows flexibility on the fulfillment
of specifications of phase margin, фm, at the gain crossover frequency, ωc, and the static error constant
kss. The proposed method of design is based on the condition of null over phase, forcing the
compensator to give it maximum phase at the gain crossover frequency. The validity and effectiveness
of the method proposed have been shown in the frequency domain with the illustrative example.
183
As mentioned at the beginning of the section, the examples validate the distinct advantages of the
fractional order design over the integer order structure, by providing greater Gain and Phase margins
throughout the frequency range of interest. However, design issues such as accuracy of fractional order
devices and hardware requirements for integer order approximations for implementing the fractional
order design, need to be thoroughly analyzed. Analogously, lag and lead-lag fractional order
compensators can be designed based on similar non linear relations and compared with their integer
order counterparts.
Figure 7 Compensated Phase plots
Fractional Order Lead Compensators and its rational approximation Salient feature of the Lead Compensator is a provision which enables the designers to obtain a rational
function approximation of the fractional order transfer function generated by the described above
method. This approximation is carried out on the principles of singularity function approximation as
184
discussed in previous sections. After obtaining the fractional structure, the user can enter the error
tolerance for approximation in dB, and generate the rational approximation. The rational function is
displayed in the Bode plots for the same. For the example discussed in earlier section on setting the
error tolerance as 2 dB, the results obtained are as follows:
Figure8: Bode response for rational approximation of the fractional Lead Compensator
Evidently enough, greater the error tolerance specified, lesser would be the order of the approximated
polynomial. The table-1 below enlists the specified error against the order of the approximation
polynomial for the above example.
185
Error Tolerance (dB) Order of the Approximated Polynomial
1.0 22
1.5 16
2.0 12
3.0 9
5.0 7
7.0 5
10.0 4
Table 1 Error tolerance and Order of Rational Approximation
Fractional PIλDµ Controller and its Rational Approximation The fractional PIλDµ controller is a generalization of the PID controller, as we have elaborated earlier
in this chapter. The transfer function of this controller is given in the frequency domain by the
following irrational function:
( ) IP D
TC s K T ss
(34)
where s = jω is the complex frequency, KP is the proportional constant, TI is the integration constant,
TD is the differentiation constant and λ and µ are positive real numbers. In general, these real numbers
are such that 1 < λ, µ < 2. Hence, (34) can be rewritten as
1( ) D
I
mIP Dm
TC s K T s ss s
(35)
where (TI/s) is a first-order integrator, (1/smI) is a fractional order integrator with 0 < mI < 1, (TDs) is a
first-order differentiator and (smD) is a fractional-order differentiator with 0 < mD < 1. In order to
represent the fractional PIλDµ controller of (35) by a linear time-invariant system model, it is necessary
to approximate its irrational transfer function by a rational one. Hence, in a given frequency band of
practical interest (ωL, ωH), the fractional-order integrator can be modeled by an FPP and the fractional-
order differentiator by an FPZ. It has also been shown how the FPP and the FPZ can be approximated
by rational functions. Hence, (35) becomes
186
1
0 0
0 0
1 1( )
1 1
I D
i i
I D
i i
N N
I Di iI
P I D DN N
I Di i
s z s zTC s K K T s Ks s p s p
(36)
The poles (pIi), the zeros (zIi) and the parameters KI and NI of the rational function approximation of the
fractional order integrator can be calculated as described in earlier section . Also, the zeros (zDi), the
poles (pDi) and the parameters KD and ND of the rational function approximation of the fractional-
order differentiator can be calculated as described in previous section.
Illustrative Example of Rational approximation of Fractional Order PID
To obtain a rational approximation for the controller 1.71.7
0.18( ) 2.4 6C s ss
, in the frequency band of
0.1 rad/sec to 10 rad/sec, with error 510 and 1dBy , the MATLAB, the rational approximation
as, and its Bode plot is following (Courtesy: Dept of PE, Jadavpur University)
Figure 9 Bode Response for error tolerance = 1dB for fractional order PID
187
Quite evidently, the greater the error tolerance, lesser would be the order of the approximating
polynomial. For the discussed example, for y=1dB, the order is 34, while for y=3dB, the order is 14.
The bode plot for y=3dB is as shown in figure-6.
Hardwire Circuit Technique to realize Fractional Order Elements Here we presents a “new “and simple approach to realize a two port network, having transfer functions
to approximate a fractional order differ integrals (s±α, where, − 1 < α< 1) by a rational function, by
“shaping of phase” as contrary to our earlier findings in previous section of this chapter. This method
is practically tried, to have a similarity with ‘constant phase element’ CPE; that a fractional differ-
integrator is.
The network is implemented by cascade connection of finite number of basic two port networks each
consisting of an operational amplifier, a capacitor and resistors. The objective is to get a transfer
function such that its Bode phase plot remains constant over a desired band of frequency. This is done
by placing poles and zeros alternately along the negative real axis of s-plane i.e. interlacing of poles
and zeros (like we did in magnitude shaping). The underlying theory is discussed and the design
procedure is developed here. Hence the basic idea is, approximate the irrational transfer function s ,
where (0, 1), by rational transfer function ( ) / ( )P s Q s such that over the desired frequency band
gain roll off is 20dB/dec and the phase angle is constant at 900.
Getting the Rational Approximation of s
Depending on the error bound () around required phase angle req = 900 and the frequency band of
interest (l, h), the nth order approximation is obtained, that is n pole-zero pairs. Here the algorithm
proposed in patent is used.
The asymptotic magnitude plot is generally used to calculate the position of poles and zeros. In this an
algorithm is derived for calculation of poles and zeros using Bode asymptotic phase plot. The phase
plot is nonlinear; it is plot of log vs. – 1tan / p , where p is the single pole for which the phase
plot is drawn. For the first order pole the phase variation from low frequency to high frequency is 0o to
-90o and at frequency equal to the pole value the phase value is -45o. For asymptotic phase plot, the
188
entire frequency range (0,) is divided in three parts, low frequency: the phase is constant at 0o,
medium frequency: the phase varies from 0o to -90o and high frequency: the phase is again constant
equal to -90o. Hence the bode phase plot for a single pole (p) is approximated by a straight line
asymptote in each of the three frequency region as given below,
(37)
where λ is the factor deciding the boundaries between the frequency ranges, and hence the slope of the
medium frequency asymptote. For asymptotic Bode phase plot of single zero all the points discussed
are same except the sign of the phase angle is opposite, i.e. z()= - p().
Pole zero calculation The basic idea of getting constant phase is by slope cancellation of asymptotic phase plots for zeros
and poles. Any req (0◦, 90◦) or α (0, 1) is possible as the middle frequency asymptote varies from
0◦ to 90◦. The basic idea of the solution is illustrated in fig. 7. Pole p1 is selected such that its
asymptotic plot passes through (ωl, req), calculated using 1, then z1 and subsequent poles and zeros are
selected so as to keep the asymptotic plot constant at φreq.
189
Figure 10 Asymptotic and exact phase plots illustrating the basic idea is drawn for req=-45o
,
l=1rad/sec and h=1000rad/sec.
With above selection of poles and zeros the asymptotic phase plot is straight line at req. But actual
phase plot is oscillating about asymptotic phase plot, with RMS Error, which we demonstrate below.
Fractional Order (FO) Impedance The fractional order impedance is realized with two port network having passive components like R
and C.
RI
Zf
RI
Zf Zf
RI-
+
-
+
-
+
Figure-11 Fractional Order Impedance
190
-+ +
-
Figure-12 Practical circuit for semi-integrator.
Schematically figure-12 shows electrical components that has fractional order FO electrical
impedance. Here Zf is the series combination of parallel RC network as shown in Figure-15 .The RC
networks represent this impedance and are connected across the operational amplifier. The circuit has
the multiple impedances connected in cascade form with operational amplifier as shown in figure-12.
This interconnected network has been designed to generate the pole-zero pairs by way of R-C
components designed for a given fraction. The optimal algorithm is developed to determine to find the
actual values of resistor and capacitor components.
Algorithm for Calculation for Pole-Zero position of FO-impedance The impedance of the FO is realized by using the lumped RC network or using the operational
amplifier with n passive components. The realizable values for different fractions of the impedances
are obtained if the design is firmed around the operating bandwidth and gain of the impedance. This
sets the values of phase slope for a given fraction and needs a proper choice of the poles and zeros.
These calculations are obtained by using the following algorithm.
Given: α or req = α90o on (ωl, ωh) with error e ≤ ε.
191
1. Initialization
a. eRMS=2
k ,where k (0, 1] is a factor selected to improve the robustness of the algorithm with respect
to error. b. Expand frequency band (ωl
*,ωh*) =(ωl/103,102ωh).
c. Select µ, from figure-10 corresponding to eRMS.
d. Set =1, new= req/
2. Hunt for µ:
a. Calculate, pi and zi using new, µ and (ωl*,ωh
*).
b. Calculate, avg, on (ωl,ωh), = avg/ req and new= new/
i. If =1, go to 3c
ii. Else, go to 3a
c. Calculate, e*RMS, RMS error in phase value from avg, on (ωl,ωh),
i. If e*RMS> eRMS , set µ= µ+0.01, go to 3a
ii. Else go to 4
3. Frequency band Adjustment
a. Find out the frequency ω'l, at which the first maximum of phase plot occurs after it enters in to
req±ε band.
b. Find out the frequency ω'h, at which the last minimum of phase plot occurs before it leaves the
req±ε band.
c. Calculate, '
*
ˆl
lll
and
'
*
ˆh
hhh
i. If )ˆ,ˆ(),( **hlhl , set )ˆ,ˆ(),( **
hlhl
eRMS=2 , go to 3.
ii. Else go to 5.
4. Verification: Calculate the poles and zeros finally, µ obtained in 3c in
latest iteration, new obtained in 3b in latest iteration and
)ˆ,ˆ( hl , obtained in 4c.
192
a. If phase plot is not satisfactory go to 2, and adjust k, ),( **hl
b. Else stop.
The error ε considered is of 10, Hence the approximate value of µ can directly obtained from Fig-14.
Here the error ε decreases with increasing µ for various fractions.
Figure-13 Error decrease with increasing µ for various α.
Design and Performance of FO Design Example:
Based on the above algorithm the design example is illustrated assuming value of α to be +0.5.
1. Given: α=+0.5 or req = 450 on (ωl, ωh) = (100, 10000) rad/sec with error ε=10.
2. Initialization
eRMS=2
15.0 =0.3536, where k (0,1]
Corresponding µ from figure-10 is 1.1 (ωl*,ωh
*)=(ωl/103,102ωh).
193
3. Finding µ: about 10 iterations are required to get final value of µ, p-z are calculated in each
iterations and, eRMS are computed to make decisions.
Finally using µ=1.15, and frequency range (100, 10000) rad/sec, the rational approximation for s+0.5 is
obtained, and the Bode plots for the same is shown in Figure-16. On the final plot various parameters
are eRMS =0.3361,avg =45, minimum value in (ωl,ωh)= 44.055, maximum value in
(ωl,ωh)=45.8575,and six pole zero pairs are used for achieving this.
-50
-40
-30
-20
-10
0
Mag
nitude
(dB)
100
102
104
106
0
45
90
Phas
e (deg
)
Bode Diagram
Frequency (rad/sec) Figure-14 Bode Plot for α=+0.5
-1 0 1 2 3 4 5 6 70
5
10
15
20
25
30
35
40
45
50
Frequency
Pha
se(d
B)
Figure-15 Phase plot for req=45o on (100, 10000) rad/sec, with ε 1o
194
Implementation of FO impedance The FO impedances as shown in figure-12 are implemented using resistive and capacitive elements.
The values of these elements depend upon the location of desired poles-zeros. The calculated values of
R-C element for different poles-zeros are listed in table 2. In table 1 Zi & Pi are the desired zeros &
poles respectively, where i is the count for zeros & poles. The input resistance and feedback resistance
are denoted by Rii & Rfi respectively as shown in Figure-17.
i Zi Pi Ci Rfi= Rii Rzi
TP TP
1 2.25
37 6.0406 1µ
264.07
k 500k 443.71k 500k
2 15.9
55 42.764 1µ 37.30k 50k 62.67k 100k
3 112.
95 302.75
680n
f 11.21k 20k 18.83k 20k
4 799.
65 2143.3 68nF 10.94k 20k 18.39k 20k
5 5661
.1 15173 10nF 10.51k 20k 17.64 20k
6 4007
8
10742
0 1nF 14.85k 20k 24.95k 50k
Table 2- Calculated values of R-C components
The calculated number of poles and zeros pair from the MATLAB® program are six. Thus for
implementation of α=+0.5 with six pole-zero pair, six FO impedance circuit are connected in series,
where one pole-zero pair denotes one FO circuit.
195
Figure-16 Circuit diagram for implementing series FO Impedance
0 1 2 3 4 5
x 10-3
-0.4
-0.2
0
0.2
0.4
Time(Sec)
Am
plitu
de
Figure-17 Input and output waveform for α=+0.5
For the circuit shown in Figure-17, the input-output wave form at different frequency is obtained. Here
it is observed that the input-output phase remains constant at 45o for α=+0.5 for the desired frequency
band. The waveform shown in Figure-18 is for frequency f=500Hz. Thus by obtaining the input output
wave form we can analyze the bode plot for α=+0.5.For the circuit shown in Figure-17
196
Realization of impedance function by Analogue Network The realization of any rational function can be achieved by using lumped linear circuit components
mainly resistor and capacitors. The most important consideration in implementation is the availability
of the components of the required values; very large capacitance and low resistance are generally not
available. Here the design of simple one port network and of proposed two port network is given.
Impedance function of Single Port Network One port network consist of series connection of R-C parallel, hence problem is to find the values of
resistance and capacitance. Partial fraction gives
011
( )( )( )
n ni i
ii i i
s z kF s ks p s p
(38)
And the impedance function of the R – C parallel is
1
( ) 1RCCZ s
sRC
(39)
Comparing (38) and (39) for the ith R - C parallel segment we get 1i
i
Ck
and ii
i
kRp
A resistance R0 = k0 and n R - C parallel segments connected in series, will give the one port network
with impedance function equal to F(s). This implementation is possible only when pi < zi or α < 0 ,
therefore for α > 0 case the network is implemented for 1( )F s
and is connected in place of Zi(s). The
problem with this implementation is that there are unique values for Ri and Ci for fix pi and zi. Another
limitation is that any value can come out of the capacitances which may not be available. Also after
calculation very high capacitance values for low poles and very low resistance values for high
frequency poles are observed.
197
Impedance Function for a Two Port Network Therefore another idea, that is cascade connection of two port networks each having first order transfer
function with zero, was tried. In the circuit shown in figure 20, select R – C parallel for both Z1(s) and
Z2(s), giving
1 1 1
2
2 2
1
( ) 1
sC R CG sC s
R C
(40)
Now selecting C1i = C2i = Ci , I have 11
ii i
Rz C
and 21
ii i
Rp C
. Hence easily available value for Ci
can be selected independent for pi and zi and whatever values come out for resistances can be achieved
using appropriate trim – pots. But when the circuit was made and tested, the output showed a lot of
noise riding on actual signal; however the phase angle was constant at required value, on the desired
frequency range. Apart from that, one capacitor can be saved as any first order system has one energy
storing element.
Improved Two Port Network Further improvement is achieved by combining above two ideas. The rational approximation transfer
function is implemented by the cascade connection of the basic two port networks shown in figure 19,
in which the one port network having first order impedance function with a zero, shown in (figure-19),
is used. Partial fraction expansion gives
( ) 'news z kZ s ks p s p
(41)
This means that a resistance (R’) in series with R – C parallel, shown in figure 19 will give the
impedance function
1
( ) ' 1p
new
sR C
Z s Rs
R C
, where '
'pRRR
R R
(42)
198
Figure 18 Single port first order network with one zero.
In this impedance function the zero is always greater the pole. Hence as referred to figure 20, for FO
integrator Z1(s) = R’ and Z2(s) = Znew(s) ( pi = p and zi = z), and for FO differentiator Zi= Znew(s) and
Z2(s) = R’ (pi = z and zi = p). For implementation of ith two port network segment select any available
capacitor (Ci) value and then the resistance values are calculated as
1i
i i
Rp C
; 1'( )i
i i i
Rz p C
, for Fractional Order Integrator (42)
1i
i i
Rz C
, 1'( )i
i i i
Rp z C
, for Fractional Order Differentiator (43)
These values of resistance can again be implemented using appropriate trim-pots. Also generally the
preferred resistance value is closed to 10K, so as to avoid t the loading of previous stage.
Figure-19: Two port network and its transfer function
199
Figure 20: CRO testing results of implemented Fractional Integrator Circuits Courtesy Dept. of
EE VNIT Nagpur
In this chapter, a simple and effective way is presented to implement fractional order differentiator and
integrators by a two port network. The ratio of consecutive poles and zeros is same as that found by
various analytical methods presented earlier in the chapters. The availability of very precise component
is major hindrance in achieving the perfect response, however by very fine tuning of variable resistors
to compensate for the slight deviation in the fixed capacitor value, can further tighten the response
around the desired value.
200
Bode’s Ideal Loop Ideal loop transfer function (TF) is ( ) ( / )gcL s s where gc the ‘desired’ gain cross-over frequency
and is ‘slope’ of ‘ideal cut-off characteristics’. Phase margin (PM) is (1 { / 2})m ; for all
values of the gain. The ideal amplitude margin (or gain margin GM) is . The constant phase margin
PM is 0 0 060 ;45 ;30 for 1.33, 1.5, 1.67 respectively. Nyqust curve for ideal Bode’s TF is simply
a straight line through origin with ( j ) / 2L . Figure-22 gives ideal Bode’s loop, figure-23
depicts the Bode’s magnitude and phase plots for the ideal loop.
Figure-21 Ideal Bode’s loop
Figure-22 Gain and Phase plots for open loop ideal TF ( )L s
201
Bode’s TF can be used as reference system in the following forms, we get from figure-22
( )CLKG s
s K
, and ( )OLKG ss
, (0 1)
General characteristics of Bode’s TF are
A) Open Loop ( )OLG s
(i) Magnitude plot has constant slope of 20dB / decade
(ii) Cross-over frequency is function of loop gain K
(iii)Phase is horizontal and at constant angle of / 2
(iv) Nyquist plot a straight line with angle ( j ) / 2L
B) Close Loop ( )CLG s
(i) Gain Margin GM is mA
(ii) Phase Margin PM is (1 / 2)m
(iii)Step response , 1( ) ( )y t Kt E Kt [17]. Where ,E represents Mittag-Leffler
function of two parameters (is higher transcendental function ).
Integer Order PID controllers The classical PID controllers can be considered as particular form of ‘lead-lag’ compensator in the
frequency domain and its TF is
( ) ic p d
kG s k k ss
Therefore, the contributions of the controllers depend on the gains ; ;p i dk k k . In the frequency response
of the controller, the selection of these gains or PID parameters is equivalent to the selection of the
position, smoothness and minimum value of the magnitude curve, and the slope of the phase plot at the
frequency of minimum magnitude. This is classical method of PID tuning. However at the high and
low frequencies the value of the slopes in the magnitude curve and the values of the contributions in
the phase curve are though fixed. Figure-24 gives plot of PID for all the parameters equal to unity
202
Figure-23 Magnitude and Phase plot of PID controller ( )cG s with 1.0p i dk k k
Figure 24: Gain and phase plot of PID controller ( )cG s with 1pk , 0.5ik and 1dk
203
Comparing the figures 24 and 25, it is desired that values and the position of the magnitude minimum
and the inflection point of the phase plot are modified by the values of gain ik while the slope and
asymptotic phase angles are same.
Fractional Order PID Controllers The transfer function of Fractional Order PID controller is
( ) ic p d
kG s k k ss
Figure 26 gives the plot of Fractional Order PID with 1p i dk k k and 0.5 . We can thus
have to make circuits for half order integration and half order differentiation. From the figure-26 we
observe that Fractional Order Controller allows selecting both the slope of magnitude curve and the
contribution of phases at high and low frequencies. The figure-27 depicts the degree of freedom one
gets in fractional order PID. Whereas the figure-28 explains one of the structure of Fractional Order
Controller. For figure-28 we write the controller output in time domain with controller TF as,
following
1( ) ( ) ( ) ( ); ( ) ( ) / ( )p t d t c p i di
u t k e t D e t T D e t G s C s E s k k s k sT
204
Figure-25: Magnitude and phase of Fractional Order PID controller with 1p i dk k k and
0.5
Figure 26Comparison of PID and Fractional Order PID for degrees of freedom (a) Integer
Order PID and (b) Fractional Order PID
205
Figure-27 Structure of Fractional Order PID connected to a plant
Parameters for Tuning of Controllers In this part, we talk about all the parameters and methods employed in several papers as listed in
reference regarding fractional order control systems.
Process
We assume process transfer function is given, also we assume that it be linear, analytical with finite
poles and shall exhibit essential singularity at infinity.
The Controller
The structure of controller is given in figure-28, gives controller output as
( ) ( ) ( ) ( )p i t d tu t k e t k D e t k D e t
Design Goal
The primary aim is to obtain a robust control system against plant gain variations. Other specifications
include noise and disturbance rejection.
Model Uncertainty
All control systems design is based on a model of plant which is approximation of true dynamics. The
uncertainty is due to
(i) Un-modeled (high frequency) dynamics.
(ii) Neglected non-linearity.
(iii) Effect of deliberate reduced order modeling
(iv) Plant parameter perturbation due to environment factors temperature aging etc.
206
Isodamping
Suppose the phase Bode’s plot of a system is made flat (figure-29), i.e. the phase derivative w.r.t. the
frequency is made zero, then the system is robust to gain variations. This property is called iso-
damping and the frequency at which the phase derivative becomes zero is called the tangent
frequency c . At the “tangent frequency”, the Nyquist curve tangentially touches the sensitivity circle
(figure-30) and Bode phase plot is locally flat, implying that system is more robust to gain (parametric)
variations. For systems that exhibit iso-damping property the overshoots of the closed loop responses
will remain almost constant for gain (parametric) spreads. This gives enhanced robustness.
Figure-28: Bode plot showing Iso-damping
207
Figure-29 Nyquist Plot showing Isodamping
We may express isodamping process or property by following statement
j
d ( ) 0d
cs
G ss
, or equivalently j
j
d ( ) ( )d c
c
ss
G s G ss
We derive the above statement by letting ( j ) ( ) j ( )G x y ; nevertheless the gain function is
function in complex frequency and is complex number. This complex number (function) representation
gives us 1( j ) tan ( ) / ( )G y x . From here we carry on the following differentiation as:
2
2 2 2 2
2 2
d 1 d( / ) (d / d ) (d / d )( j )d 1 ( / ) d
1 d d 0d d
y x x y y xGy x x y x x
y xx yx y
The above implies that
d d (d / d )0 i.e.d d (d / d )
y x y yx yx x
Since tan d ( j ) / d (d / d ) /(d / d )G y x and tan ( j ) /G y x , we can write
208
jj
d ( j ) ( j )d c
c
ss
G Gs
The statement of iso-damping.
Relative Stability
This is measured by parameters percentage maximum overshoot %rM , and damping ratio. In
frequency domain the resonant peak overshoot rM can be used to indicate relative stability. They
provide an approximate indication of the closeness of Nyquist plot of the systems open loop transfer
function ( j )L , or loop TF to the point 1 j0 , in the complex plane. A proper TF is called minimum
phase if all its poles and zeros remain in open Left Half Plane in complex frequency. For a minimum
phase ( )L s is stable if Gain Margin (GM) > 0 dB, and unstable if GM < 0 dB. Generally a minimum
phase system has positive phase margin (PM) and it becomes unstable if PM < 0 dB. For non-
minimum phase care must be taken in interpreting stability based on signs of GM and PM. For
satisfactory performance PM is about 30 to 60 degrees and GM is more than 6 dB.
Sensitivity Function
Referring to figure-30 we have ( )d t a disturbance signal at the load point, and ( )n t representing
measurement noise. Let us call sensitive function as ( )S s and complementary sensitive function as
( )T s defined as in text books as:
1 ( ) ( )( ) & ( )1 ( ) ( ) 1 ( ) ( )
C s G sS s T sC s G s C s G s
The output is ( ) ( ) ( ) ( )[ ( ) ( )]Y s S s D s T s R s N s , and the error
is ( ) ( ) ( ) ( )[ ( ) ( )] ( ) ( )E s R s Y s S s R s D s T s N s . The S(s) is sensitive function and this determines
the suppression of the disturbance at the load, while ( )T s is the complementary sensitive function and
it determines robustness measurement noise and un-modeled system dynamics. We can always have
from above ( ) ( ) 1T s S s .
209
Figure-30 Control system representation for sensitivity functions definitions
In real controllers it is observed that load disturbance ( )d t and reference signal ( )r t are generally of
low frequency and measurement noise is generally at high frequency. Hence to ensure good reference
tracking and rejection of load disturbance at the lower frequency ( ) 0S s , this implies that ( ) 1T s . At
the higher frequencies we thus need to ensure, that noise due to measurement method is rejected, so
that ( ) 0T s implies that ( ) 1S s . Clearly there is design tradeoff between the two functions.
Tuning of Fractional Order PID The objective is to design a fractional order controller so that the system fulfills different specifications
regarding robustness to the plant uncertainties, load disturbance and high frequency noise rejection. So
the specifications related to PM ( )T s ( )S s and robustness constraints are considered in the design
method. So the design problem we should formulate as below.
Phase Margin PM m and gain cross over frequency gc we must specify. GM and PM have always
served as important measures of robustness. It is known from classical control theory that PM is
related to damping of the system and therefore can also serve as performance measure.
1. Equation that defines PM and gain cross over frequency we write as:
( j ) ( j ) 0dB
(j ) ( j )
gc gc
gc gc m
C G
C G
2. The robustness to variation of plant gain we write as
d [ ( j ) ( j )] 0d
c
C G
210
This condition forces the phase of ‘loop-TF’ to be flat at the desired tangent frequency c is for ‘iso-
damping’.
3. To have good output disturbance rejection we write
dB
1( j ) or ( j )1 ( j ) ( j ) s sS B S B
C G
With B the desired value of ( j )S for s . The frequency s is chosen below gc .
4. High frequency noise rejection we take care by the following equation
dB
( j ) ( j )( ) ;1 ( j ) ( j ) t t gc
C GT j AC G
5. Steady state error cancellation is by the fact that fractional integrator s is as efficient as error
nullifier of integer order integrator. Thus the specification of zero steady state error is met by
having the fractional integrator.
Using fractional order PID controller up to five of the above design specifications can be met since the
controller has five parameters ; ; ; ;p i dk k k . Thus for general tuning of PI D controllers design,
problem is basically solving system of five non-linear equations given by the above rules to determine
five controller parameters.
In the entire controller tuning papers mentioned in references. The function FMINCON (.) is used for
the purpose, which finds constraint minimum of a function of several variables. It solves problem of
the form XMIN F(X) subject to:C(X) 0 , eqC (X) 0 , with lb X ub ; where F is function to
minimize, C and eqC represents the non linear inequality and equality constraints. In the cases of
papers of references for controller tuning, F: ( j ) ( j )gc gcC G the main function to be minimized,
with constraints as follows (as an example):
1. 0.4rad / sgc , the gain cross over frequency
2. 0.6rad / sc , the tangent frequency.
3. 045m , the phase margin
4. s( j ) 20dB, < ; 0.011rad / ssS
211
5. ( j ) 20 ; ; 10rad / st tT dB
Isodamping a plant having integer order PID tuned system by topping
with ‘fractional’ phase shaper Feed back control system is one the major area where concept of fractional calculus should be applied
to obtain efficient system. This concept gives overall efficiency (in terms of energy) also longevity and
freedom to control engineer to compensate any shifts in the transfer function due to parametric spreads
aging etc. A system is efficient if the controller were of similar order to that of a plant (system) being
controlled. In reality the systems are fractional order therefore to have fractional order controller will
be efficient. Even for integer order systems the fractional controls give better freedom to achieve what
is “isodamping”. Meaning, to achieve overall close loop behavior of overshoot independent of feed
forward gain (pay-load, amplifier feed forward gain, in power systems the load current/load
resistance). H W Bode envisaged this concept of having fractional integrator circuits to achieve
overshoot independent of the amplifier gain in 1945. He proposed a fractional order controller, the
purpose of which is to have a feedback amplifier of good linearity and stable gains even though the
amplifier show non-linear characteristics and variable gain over ambient and time. Bode proposed a
feedback amplifiers, whose open loop frequency characteristics 0 ( j )G is such that the gain is
constant for 00 and phase is constant or )1( y radians for 0 . The suggested value
was 6/1y , which guarantees a phase margin (PM) of 030 . The open loop transfer function is given
as
0
0 2(1 )2
0
( j )1 / j /
y
o
AG
meaning 0 0( j )G A for 0 and angle i.e. 0arg ( j ) (1 )G y radians for 0 . This is
early development of fractional order controls. Thus it was recognized that the open-loop transfer
function of a good control system show a fractional order integral form with a fractional order between
1 and 2 (between totally being first order and second order). Meaning that open loop transfer function
should be like kssG /1)(0 . This gives close loop transfer function as
212
11
)(11)(
0
kCL ssG
sG , js .
In close loop transfer function 1
1)(
kCL ssG expression, we put for js , j cos jsin
2 2
then
cos jsin2 2
s
cos jsin cos j sin2 2 2 2
k k k kk k k ks
,
put this value of k in )(sGCL to get:
1 1 1( )1 cos j sin 1 cos 1 j sin
2 2 2 2
CL kk k k k
G s k k k ks
12
cos2
1
2sin
2cos21
2cos
1)(2
5.02222
kkkk
sGkkkkk
CL
rM is maximum value of )(sGCL at r when denominator 12
cos22 kkk is minimum.
Therefore, we do
2d 2 cos 1 0d 2
k k k
Which gives us 02
cos22 112 kkk kk , meaning at 2
cos kk the magnitude of )(sGCL is
maximized. we get k
rk /1
2cos
and putting this value of r in expression of )(sGCL we get
2cos1
1
12
cos22
cos
1
12
cos2
cos22
cos
1
2222 kkkkkk
M r
2sin
1k
For finding the damping ratio we should find the poles of )(sGCL by transformation to w -plane and
then with respect to s plane we look at the pole location. Putting ksw in the expression of close loop
213
transfer function we obtain: 1
1)(
w
wGCL with poles at jew 1 in w plane. Therefore the s plane
pole is at kw /1 meaning poles at kjk es //1)1( in the s plane. The line with angle )/( k with the
positive real axis of the s -plane is the locus of poles for )(sGCL and are called iso-damped lines for
particular value of k . The damping ratio s
se )( with respect to imaginary j axis. The angle of
the iso-damped line with respect to imaginary axis is
2
kand thus anywhere on this line the pole
is, the damping ratio is
2sin)(
ksse
This close-loop transfer function gives step response properties of controlled system output as
robustness and stability measures.
5.02 12/cos21
2/sin12/cos1)(
kkjksG
kkkkCL
2/sin1k
M r , kr k /12/cos
The amplitude takes the peak value rM at r . The damping ratio can be obtained from the poles of
)(sGCL as
2sin
k
. The phase margin is given by 02902
kkPM . The overshoot
can be expressed as approximate formula as, 6.08.01 kkM P per unit. The feedback controlled
system be represented as following figure-32
214
Figure 31: Plant control system with fractional order integrator (phase shaper) as extra
compensator apart from conventional PID
The open loop response desired is fractional order as:
1( ) ( ) ( ) ( )kOL PID PG s H s G s G ss
The plant is second order say DC motor
2
1( )PG ss as b
21( )PID
s as bG s s a bss
, with PID constants: 1, ,d i pk k b k a The phase shaper
fractional order element is put before PID in order to have open loop response 2
2 1
1 1 1 1( )OL k q q
s as bG ss s s s as b s
Say we choose 1.6k , for particular 0.408 40.8%PM , means the choice of fractional order phase
shaper 0.6q , meaning putting an additional fractional integrator at the front of the PID block as
demonstrated above. With this choice the closed loop transfer function is:
( ) 1( )1 ( ) 1
OLCL k
OL
G sG sG s s
Let us put conformal translation to wplane, ks w , gives 1( ) (1 )CLG w w , the roots in conformal
wplane is at 1.00arg( j ) kw s . Transforming back we get poles at s plane at
PID ( )PIDG s
PLANT ( )pG s
1( ) qH ss
215
1 1(1.00) arg jk ks w
k
. For 1.6k , the ISODAMPING angle is 0( /1.6) 112.5 from
positive real axis, or 022.5 from the imaginary axis in the s domain. The following picture depicts
the process and the iso-damped lines, with variation in gain.
Im( )s
Isodamded line 1.6k
at angle 022.5
Gain K=1.00
Gain K=2.00
( )e s
Figure: 32, Isodamping lines in complex plane, with gain variation
The open loop gain 1.6( )OLG s s as obtained by putting the phase shaper of fractional order gives a
constant phase angle plot of 0 090 1.6 144 , for infinite band of frequency (ideally). This is ideal
though, but for practical implications the phase angle will have band limit say from 2 radians per
second to 200 radians per second. Therefore practical implementations for iso-damping will be limited
to particular spread in band of gains. The band-limited iso-damping for practical gain variation can be
got from BODE plot for the open loop transfer function as demonstrated below. The shifting the gain
curve up or down in magnitude will move the zero cross over point and the phase for which the phase
angle remains constant will give the gain values of the iso-damping. This is demonstrated in the
following figure. The open loop transfer function plots says that the phase angle remains constant at -
1440 for gain values K=0.25 to K=10.00, meaning the overshoot remains fixed at 40.8% for this spread
of gain values parametric spreads.
216
GAIN
K=0.25 K=1.00 K=10.00
2 200 Frequency (radians/second)
-1440
-1800
PHASE
Figure: 33 Bode plot showing Isodamping possible for few gain spreads in actuality
Conclusions One can use fractional order PID controllers and tune the system to achieve efficient controls where
peak overshoot remains invariant with parametric or gain spreads. Well one can use fractional order
phase shapers to top a plant with tuned PID, to achieve this too that is ‘iso-damping’. The modified
Ziglers-Nicholes method to apply for Fractional Order PID tuning is under development, so is the
modified Monje method (described) herein, and is too under development. Well the fractional
operators are realizable to desire shape of phase and magnitude of Bode plot. The rational expressions
thus can be used to realize circuits hardwire or via software the Fractional Order Controllers especially
Fractional Order PID systems. The fractional order elements thus realized by the described new
algorithms work with in a band (selected) by designer; and now is reality. The industrial digital
fractional order controllers based on digitization of the obtained method is under development, for
industrial usage.
217
References
1. Abbaoui K and Cherrualt Y. The Decomposition Methods Applied to the Cauchy Problems, Kybernetes. 28, pp
103-108. (1999).
2. Abbaoui K and Cherruault Y. Convergence of Adomian Method Applied to Differential Equations, Computers
Moths Applic. 28 (5) pp 103-109. (1994).
3. Abbaoui K and Cherruault Y. New Ideas for Proving Convergence of Decomposition Methods, Computers Math.
Applic. 29 pp 103-108. (1995).
4. Abhay Parvate, Anil. D. Gangal, Calculus on Fractal Subset on Real line-I: Formulation, Fractals, vol. 17, No. 1
(2009) World Scientific, pp 53-81.
5. Abramowitz. M, Stegun. I.A. (eds.), Handbook of Mathematical Functions, Nat. Bur. Stand. Appl. Math. Ser 55.
U.S. Govt. Printing Office Washington D C (1964).
6. Adam Loverro. Fractional Calculus history definition and application. Department of Aerospace and Mechanical
Engg. University of Notre Damme. Technical Note May 8-2004.
7. Adomian G. Nonlinear Stochastic System theory and Application to Physics. Kluwer Academic Publishers,
Netherlands. (1989).
8. Adomian G. An Analytical Solution of Stochastic Navier-Stokes Systems, Foundations of Physics, 21 (7), pp 831-
843. (1991).
9. Adomian G. Solution of Physical Problems by Decomposition, Computers Math. Appl., 27 (9/10), pp 145-154.
(1994).
10. Adomian G. Solving Frontier Problems of Physics: The decomposition method, Kluwer Academic, Boston.
(1994).
11. Adomian G. Solutions of Nonlinear P.D.E, Appl. Math Lett, 11 (3), pp 121-123. (1998).
12. Adomian G and Rach R. Linear and Nonlinear Schrodinger Equations, Foundations of Physics, 21, pp 983-991.
(1991).
13. Agarwal O P, Solution for fractional diffusion wave equation defined in bounded domain, Non-Linear Dynamics
29 pp 145-155 (2002).
14. Agarwal.O.P. Solution for a Fractional Diffusion-Wave Equation Defined in a Bounded Domain, Nonlinear
Dynamics, 29, pp.145-155. (2002).
15. Al-Alouni M A, A class of second order integrators and low pass differentiators, IEEE Trans. on circuit and
systems 1: Fundamental Theory and Applications, 42(4) pp. 220-223 (1995).
16. Al-Alouni M A, Filling the gap between the bilinear and backward difference transforms: an interactive design
approach, Int. J of Electrical Engineering Education, 34(4) pp.331-337 (1997).
218
17. Alexander S and Orbach R .Density of states on fractals (fractons), J-Physique-Letters 43 (1982) L-625-L-631.
18. Almenas. K, Lee. R, Nuclear Engineering: An introduction, Springer-Verlag Berlin/Heidelberg/New-York (1992).
19. Andrea Roceo, Bruce. J. West, Fractional Calculus and evolution of fractal phenomena, Elsevier Physica A. 265
(1999) pp. 535-546.
20. Anh. V.V et al, Fractional differential equation driven by Levy noise. J. Appl. Math and stochastic analysis 16:2
(2003) pp 97-119
21. Arora H.L and Abdelwahid F.I. Solution of Non-Linear Order Differential Equations via Adomian Decomposition
Method, Appl. Math Lett. 6(1) pp 21-23. (1993)
22. Axtell M, Bise. M E. Fractional calculus applications in control systems, Proc of IEEE Nat. Aerospace and
Electronics Conf, New York 1990, pp.563-566.
23. Bachelier. L. Theorie de la speculation, Ann. Sci. Ecole. Norm. Suppli. 3(17) 1900, pp 21-86. Reprinted Cootner.
P. H. (Ed.). The Random Character of Stock Market Price, Cambridge MA: MIT: Press 1964.
24. Bagley R L and Torvik P J. Fractional calculus in transient analysis of viscoelastically damped structures, AIAA
Journal, vol.23, no.6. Pp.918-925, (1985).
25. Bagley R L. Power law and fractional calculus model of viscoelasticity, AIAA Journal, vol. 27, no.10, 1989, pp.
1412-1417.
26. Bagley. R.L and Calico R.A. Fractional order state equations for control of viscoelastic structures, J. Guid. Cont.
and Dyn, vol. 14, no 2, pp 304-311, March-April (1991).
27. Bagley. R.L. The thermorheological complex materials, Int J. Engg. Sci, vol. 29, No. 7, pp 797-806 (1991).
28. Balakrishnan V. Anomalous diffusion in one-dimension, Physica. A 132 (1985) pp. 569-580.
29. Barcena et al. Discrete control for computer hard disc by fractional order hold devise. University of Pais Vasc
Spain. 2000 (pp 1-34)
30. Batman H. Higher transcendental function, Vol. 3, Mc Graw Hill New-York (1954).
31. Benson. The fractional order governing levy motion. Water resources research, vol. 36 no. 6 pp 1413-1423
June2000.
32. Bhattacharya R, Gupta V K. Application of central limit theorems to solute transport in saturated porous media,
from kinetic to field scales: Dynamics of fluids in hierarchical porous media (Cushman J H Ed.), Academic Press
San Diego California 1990 pp61-96.
33. Bode H W. Network analysis and feed back amplifier design, Van Nostrand New-York 1949.
34. Bohannan G. Analog Realization of fractional order control element revisited Department of Physics Montonna
State Univ. Technical Note Oct 27, 2002.
35. Bohennan Gary et al. Electrical components with fractional order impedances, Patent US 20060267595,
November 2006.
219
36. Bush. V. Operational Circuit Analysis Wiley New-York (1929)
37. Buslowicz M. Stability of linear continuous-time fractional order system with delays of retarded type, Bulletin of
the Polish Academy of Sciences Technical sciences, Vol. 56, No.4 2008, pp. 319-324.
38. Carlsaw H.S, Jaeger J.C. The Conduction of Heat in Solids, Oxford Univ. Press, London New-York (1947).
39. Carlson and Halijak. Simulation of the fractional derivative operator s and fractional integral operator s/1 .
Kansas State University Bulletin, vol. 45, pp 1-22 July 1961.
40. Carlson et al. Approximation of fractional capacitors ns /1/1 by regular Newton process, IEEE Trans. on Circuit
Theory, vol. 11, no.2, 1964, pp.210-213.
41. Carpinteri.A and Mainardi.F. Fractals and Fractional Calculus in Continuum Mechanics, Springer Verlag,
Vienna-New York (1997).
42. Cecilia Reis et al. Synthesis of logic circuit using fractional order dynamic fitness function. Transaction on
engineering computing and Technology ISSN ENFORMATICA VI 2004 1305-5313 pp 77-80.
43. Charef.A, Sun H H, Tsao Y Y, Onaral. B. Fractal systems as represented by singularity function, IEEE Trans. on
Automatic Control, vol.37, no.9, pp.1465-1470, Sept 1992.
44. Chaves A S. A fractional diffusion equation to describe Levy flights, Phys. Letts. A, 239, 1998 pp13-16.
45. Chen, Vinagre. A new type of IIR digital fractional order differentiator, Signal Processing. 83(2003) pp 2359-
2365
46. Chen, Vinagre, Podlubny et al. On fractional order disturbance observer DETC03-VIB48371 ASME Design
Engineering Technical Conference Chicago USA Sept 2-6, 2003. pp. 1-8.
47. Chen, Vinagre, Podlubny. Fractional order disturbance observer for robust vibration suspension, Non Linear
Dynamics 38:pp. 355-367 (2004).
48. Cherruault Y. Convergence of Adomian’s Method, Kybernetes18 pp 31-38. (1989).
49. Chien Chang Tseng (Senior member IEEE). Improved design of digital fractional order differentiator using
fractional sample delay TCAS 1-1815, 2004.
50. Chien-Cheng Tseng, Soo-Chang-Pei, Shih-Cheng Hsia. Computation of fractional derivatives using Fourier
transforms and digital FIR differentiator, Elsevier Signal Processing 80(2000) pp. 151-159.
51. Chie-Yi Yang, Kuo-Chi Hsu, Kuan-Chih Chen. The use of the Levy stable distribution for geophysical data,
Hydrology Journal, 17: pp. 1265-1273, (2009).
52. Churchill Ruel V, Brown James W and Verhey Roger F. Complex Variables & Applications, Third Edition Mc
Graw Hill Kogakusha Ltd. (1974).
53. Churchill. R.V. Operational Mathematics, 2nd Edition, McGraw-Hills New-York (1948).
220
54. Chyi Hwang and Cheng. A numerical algorithm for stability testing of fractional delay systems, Elsevier
Automatca 42(2006) pp 825-831.
55. Compte. A. Continuous time random walks on moving fluid, Phys. Rev. E, 55(6) 1997 pp. 6821-6831.
56. Coronel-Brizo. Hernandez-Montoya. On fitting the Patreo-Levy distribution to stock market index data: Selecting
a suitable cut-off value. Physica A 354 (2005), pp. 437-449, Elsevier.
57. Crank. J. The Mathematics of Diffusion, Oxford Univ. Press London-New-York (1956).
58. D. Middleton. An introduction to Statistical Communication Theory, New York McGraw-Hill, 1988.
59. Daoud. M. Spectral dimension and conductivity exponent of the percolating cluster, J-Physique Letters 44 (1983)
L-925-L-928.
60. Davidson. B. Neutron Transport Theory, Oxford Press London and New-York (1957).
61. Diethelm K. An Algorithm for Numerical Solutions of Differential Equations of Fractional Order, Elec. Transact.
Numer. Anal. 5 pp 1-6. 1997.
62. Diethelm K. An algorithm for the numerical solution of differential equations of fractional order, Electronics
transactions on Numerical Analysis, ISSN 1068-9613, vol.5, March 1997, pp.1-6.
63. Diethelm K, Ford N.J. and Freed A.D. A predictor-Corrector Approach for the Numerical Solution of Fractional
Differential Equations, Nonlinear Dyn29 pp 3-22. (2002)
64. Diethelm Kuo and Ford. Analysis of fractional differential equations University of Manchester. Numerical
Analysis Report No 377 pp 1-18 (2003).
65. Dorcak, Ivo Petras et al. Fractional order state space models. Int. Carpathian Control Conference ICCC
Malenovice Czech Rep. May 27-30 2002 pp 193-198.
66. Dostal.J. Operational Amplifiers, Butterworth-Heinemann, Boston, 1993.
67. Dutta Roy.S.C. On realization of constant-argument immitances of fractional operator, IEEE Trans. on Circuit
Theory, vol. 14, no. 3, 1967, pp. 264-374.
68. E. Fama. Test of the Multi-period Two Parameter Model. J. Business 35 (1963), p. 420.
69. Einstein A. E. Investigations on theory of the Brownian movement, (Translation), Dover, Mineola N.Y. 1956.
70. El Sayeed. A.M.A. Fractional order diffusion wave equation Int. J. of Theor. Phy 1996, vol.35, pp.311-322.
71. Engheta N, Werner D.H (Ed), Mitra.R (Ed). Frontiers in Electromagnetism, IEEE New-York 2000 pp 523-552.
(2000).
72. Engheta. The fractional curl operator in electromagnetic. Microwave Opt.Tech.Letter vol. 17, pp 86-91, 1998.
73. Enrico Scalas. The application of continuous time random walks in finance and economics, Physica-A 362 (2006)
Elsevier, pp. 225-239.
74. Enrico Scalas, Rudolf Gorenflo, Francesco Mainardi, Marco Raberto. Revisiting the derivation of the fractional
diffusion equation, Elsevier ARXIV:COND-MAT/0210166 V2 11.02.2002 (2002).
221
75. Erdelyi A. Asymptotic expansions, Dover (1954).
76. Erdelyi.A. (Ed). Tables of Integral Transforms. vol. 1, McGraw-Hill, 1954.
77. Erdelyi.A. On some functional transformation Univ Potitec Torino 1950.
78. F. Black and M. Scholes. Pricing Options and Corporate Liability. J. Political Econ. 81 (1973), p. 637.
79. Falconer K. Fractal Geometry, John Wiley, New-York 1990.
80. Famma. E, Roll. R, Parameter Estimate for Symmetric Stable Distribution, J. Am. Stat. Assoc. 63: pp. 817-836,
(1972).
81. Feller. W. An introduction to probability theory and its applications. Vol. II, 2nd Edition, John Wiley New-York
1971.
82. Fracncois Dubois Stephanie Mengue. Mixed Collocation for fractional differential equations, Numerical
Algorithms Dec. 2005 vol.34, no.2 pp 303-311.
83. Gel’fand. I.M, Shilov.G E. Generalized functions, Vol.1, Academic Press New-York (1964).
84. George A J and Chakrabarti. A. The Adomian Method Applied to Some Extra Ordinary Differential Equations,
Appl. Math Letts. 8 (3) pp 91-97. (1995).
85. George F Simons, John S Robertson, Differential Equations with Applications & Historical Notes, II-Edition,
McGraw-Hills. Inc. 1991.
86. Glockle W G and Nonnenmacher. T F, Fractional Integral Operators and Fox Functions in the Theory of
Viscoelasticity, Macromolecules, vol.24, pp.6426-6434 (1991).
87. Glockle W G, Nonnenmacher. T F, Fractional integrators and Fox functions in the theory of viscoelelasticity,
Macromolecules, vol.24, pp.6426-6436, (1991).
88. Gopal.M, Modern Control System Theory, Second Edition April, 1993, Wiley Eastern Limited.
89. Gorenflo R, Mainardi F. Fractional oscillators and Mittag-Leffler function, Technical Report Preprint A-14/96,
University of Berlin (1996).
90. Gorenflo R, Mainardi.F. Fractional calculus: integral and differential equations of fractional order, Fractals and
Fractional Calculus in Continuum Mechanics Springer-Verlag, Vienna-New York, 1997.
91. Gorenflo R, Yu. Luchko and Rogosin.S. Mittag-Leffler Type Functions: Notes on Growth Properties and
Distribution of Zeros, no. A-97-04. Department of Mathematics and Informatics, Free University of Berlin, 1997.
92. Gorenflo.R. Fractional Calculus: Some numerical methods, Fractals and Fractional Calculus in Continuum
Mechanics Springer-Verlag, Vienna-New York, 1997.
93. Gradshteyn I.S, Ryzhik I M. Tables of Integral Series & Products, Academic Press, San Diago (1992).
94. Halijak C A. An RC impedance approximation to 2/1)/1( s . IEEE Circuit Theory, vol. CT-11, no 4, pp 494-495
Dec 1964.
222
95. Heaviside. O. (1922). Electromagnetic Theory, Chelsea Edition New-York (1971).
96. Heymans and Podlubny et al. Physical interpretation of initial conditions for fractional differential equation with
Reimann-Liouville fractional derivative. Rheological Acta (online) Nov29, 2005.
97. Hifler R. Applications of fractional calculus in Physics, World Scientific Singapore (2000).
98. Hifler R. Fractional Diffusion based on Riemann-Liouville Fractional Derivative, J. Phys. Chem. B Vol. 104
(2000) p3914.
99. Hifler R, L. Anton. Fractional master equation and fractal random walk, Phys Rev A E51 (1995) R818-R851.
100. I. Koponen. Analytical approach to problem of convergence of truncated Levy flights towards the Gaussian
Stochastic Process. Phys. Rev. E 52 (1995), p. 1197.
101. Ichise.M, Nagayanagi.Y, Kojima.T. An analog simulation of non-integer order transfer function analysis for
electrode process, J. Electroanal. Chem. Vol. 33, no.1, pp. 253-265, 1971.
102. Igor Podlubny et al. Using continued fraction expansion to discretize fractional order derivative, FDTA Non
Linear Dynamic 2003. For submission to special issue on Fractional Derivative and their application.
103. Igor Podlubny, Dorcak, Kostial. J. On fractional derivatives, fractional order dynamic systems and DPI -
controllers, Procc. of the 36th IEEE CDC San Diego December 2004, pp 4985-4990.
104. Igor Podlubny. Fractional differential equations, Academic Press San Diego 1999.
105. Igor Podlubny, Ivo Petras et al. Realization of fractional order controllers, Acta Montanistica Slovak, Rocnik
8(2003) pp 233-235.
106. Igor Podlubny. The Laplace Transform Method for Linear Differential Equations of the Fractional Order, Inst.
Exp. Phys. Slovak Acad. Sci., no. UEF-02-94, 1994, Kocice.
107. Igor Podlubny. Geometric and Physical meaning of fractional derivative. Technical University Kosice
Slovak.vol.5 No.4 (2002) ISSN 1311-045 Fractional Calculus and Applied Analysis 5(4) pp 367-386.
108. Igor Podlubny. Presentation on fractional calculus IEEE chapter control San Diego, September 29, 2005.
109. Igor. Podlubny. Fractional derivatives: A new stage in process modeling and control, 4th International DAAAM
Symposium, Brno, Czech Republic, 16-18 Sept, 1993, pp. 263-264.
110. Igor. Podlubny. Fractional order systems and PI D controllers, IEEE Trans. Auto Cont., vol. 44 no. 1, pp 208-
214 Jan 1999.
111. Igor. Podlubny. Matrix approach to discrete fractional calculus, Fractional Calculus and Applied Analysis vol.3,
no.4 (2000).
112. Ivo Petras .The fractional order controller method for their synthesis and application. BERG University Koissce
Slovak Republic Technical Note 2004.
113. Ivo Petras. Control of fractional order Chua’s system. Univ of Koisscs Slovak Technical Note 2000.
223
114. Ivo Petras, Vinagre. Practical application of digital fractional order controller to temperature control, Acta
Montansitica Slovak 2002.
115. Ivo Petras, Vinagre. Practical Applications of digital fractional order controller to Temperature Control, Acta
Montanistica, Slovaca Rocnik 7(2002), 2 pp 131-137.
116. Jack Leszcynski et al. A numerical method solution for ordinary differential equation of fractional order.
Technical university of Czestochowa Poland Lecture Notes 2005.
117. Jones. H E, Shenoi.B A. Maximally flat lumped element approximation to fractional operator immitance function,
IEEE Trans. on Circuits & Systems, vol. 17, no.pp.125-128, 1970.
118. Juan Bisquert, Albert Compte. Theory of electrochemical impedance of anomalous diffusion, J of Electro
analytical Chemistry 499(2001) pp 112-120.
119. Kapoor V, Gelhar. Transport in three dimensional aquifer: 1. Dynamics of concentration fluctuations. 2.
Predictions and observations of concentration fluctuations. Water Resources. Res. 30(6) 1994, pp. 1775-1801.
120. Karabi Biswas, Siddhartha Sen, Pranab Kumar Dutta. Modeling of a capacitive probe in a polarizable medium,
Sensors actuators: Phys. 120 (2005) pp115-122.
121. Karabi. Biswas, Sen S, Dutta P K. A constant phase element sensor for monitoring microbial growth, Sensors and
Actuators SNB9063 (2005) pp1-6.
122. Karimi.A, Garcia. D, Longchamp. R. PID controller design using Bode’s integral, Proceedings of the American
Control Conference, Volume 6, pp. 5007-5012, 8-10 May, 2002.
123. Kaya D. An Explicit and Numerical Solutions of Some Fifth Order KdV Equation by Decomposition Method,
Applied Math Comp. 144 (2-3), pp 353-363. (2003)
124. Kaya D. A Numerical Simulation of Solitary-Wave Solution of the Generalized Regularized Long-Wave Equations,
Appl. Math. Comp. 149 (3) pp 833-841. (2004)
125. Kaya D and El-Sayeed S M. An Application of the decomposition method for the Generalized KdV and RLW
Equations, Chaos, Solitons and Fractals, 17 (5), pp 869-877. (2003).
126. Kaya D and El-Sayeed S M. On a Generalized Fifth Order KdV Equations, Phys. Lett. A, 310 (1) pp 44-51.
(2003)
127. Kaya D and Yokus A. A Numerical Comparison of Partial Solutions in the Decomposition Method for Linear and
Nonlinear Partial Differential Equations, Math. Comp. Simul. 60 (6) pp 507-512. (2002).
128. Kiran M Kolwankar, Anil D Gangal. Local Fractional Derivatives and fractal functions of several variables, J of
Physics 9801010 vol.1 (1998).
129. Kiran. M. Kolwankar, Anil. D. Gangal. Local Fractional Fokker-Plank Equation, Physical Review Letters (Jan
12, 1998) The American Physical Society vol. 80, No. 2 pp 214-217.
130. Kirayakova.V. Generalized Fractional Calculus and Applications, Pitman Research Notes in Maths. No. 301,
Longman, Harlow, 1994.
224
131. Klafter J, Blumen A, Shlesinger M F. Stochastic pathways to anomalous diffusion, Phys Rev A 35 (1987).
132. Kolwankar and Gangal. Fractional differentiability of nowhere differentiable functions and dimensions, Chaos
6(4) pp. 505-513, (1996).
133. Kostadin Trenceveski et al. On fractional derivative of some function of exponential type. Ser Mat. University
Beograd PUBL. ELEKTROTEHN FAK. Ser. Mat. 13(2002) pp 77-84.
134. Kundert.K. Modeling di-electric absorption in capacitors. The Designer Guide Community Technical Note 2004.
135. Kuo F.F. Network Analysis and Synthesis, Willey 1966.
136. Lankarani H M and Nikravesh P E. A contact Force Model with Hysteresis Damping for Impact Analysis of
Multibody System, J. of Mech. Design 112 pp 369-376. (1990)
137. Launsse P, Oustaloup et al. A restricted complexity controller with CRONE control system design and tuning.
University Bordeaux-ENSEIRB 2003 Technical Note.
138. Lazarevie.M.P. Finite Time Stability Analysis of PD fractional control of robotic time delays systems,
Mechanics Research Communications 33(2006) pp.269-279.
139. LePage. W R. Complex variables and the Laplace transform for Engineers Dover, New-York (1961).
140. Lubomir Dorcak, Ivo Petras et al. Comparison of the methods for discrete approximations of fractional order
operator Acta Montanisca Slovak Rocnik 8(2003) cislo-4, pp 236-239.
141. Lubomir Dorcak. Numerical Models for simulation of fractional order control system Slovak Academy of Science
1994. UER04-94, pp 1-15.
142. Lurie. B.J. Three parameter tunable Tilt-Integral-Derivative TID Controller, US Patent, 5-371-670, 1994.
143. Lynch V E, et al. Numerical methods of solution of partial differential equations of fractional order, J of Comput.
Phy. 192(2003) pp 406-421.
144. Mainardi.F. Fractional relaxation-oscillation and fractional diffusion wave phenomena, Chaos Solitons and
Fractals, vol. 7, 1996, pp.1467-1477.
145. Manabe. S. Early development of fractional order control. ASME DETC2003/VIB-48370 pp 1-8.
146. Manabe. S. The Non-Integer Integral and its application to Control Systems, ETJ of Japan, vol. 6, no. 3-4, 1961,
pp. 83-87.
147. Mandelbrot B. B. The variation of certain speculative prices, J. Business (36), 1963 pp. 394-419.
148. Mandelbrot B.B. The Fractal Geometry of Nature, Freeman, New-York 1977.
149. Mandelbrot B.B. Self-Affine and Fractal Dimension, Physica Scripta Vol. 32, (1985) pp. 257-260.
150. Mandelbrot. B.B, and J.W. van Ness. Fractional Brownian motion, fractional noise and applications. SIAM Rev.
10 4 (1968), p. 422.
151. Mandelbrot. B.B. The Variation of Certain Speculative Prices. J. Business 36 (1963), p. 294.
225
152. Manuel Duarte Ortigueira. An Introduction to the Fractional Continuous Time Linear Systems, the 21st Century
Systems, IEEE Circuits and Systems Magazine, Third Quarter 2008, pp. 19-26.
153. Marco Raberto, Enrico Scalas, Francesco Mainardi. Waiting times and returns in high frequency financial data:
an empirical study, Elsevier ARXIX:COND-MAT/0203596VI 28 (2006) pp. 1-18.
154. Matignon D. Generalized Fractional Differential and Difference Equation: Stability Properties and Modeling
Issues, Proc. of Maths. Theory of Networks and Systems Symposium, Padova, Italy 1998.
155. Mauro Bologna. Short Introduction to fractional calculus. Univ. Tarpeca Chile 2005. Lecture Notes pp 41-54.
156. Meakin Paul and Stanley Eugene H. Spectral Dimension for the Diffusion-Limited Aggregation Model of Colloid
Growth, Physical Review Letters Vol. 51 No. 16 pp. 1457-1460. (1983)
157. Metzler, R and Klafter J. The Random Walk’s Guide to Anomalous Diffusion: A fractional Dynamics Approach,
Physics Report, 339, pp 1-77. (2000)
158. Michelle Caputo and Mainardi.F. A new dissipation model based on memory mechanism, Pure and applied
Geophysics, vol.91, no.8, 1971, pp.134-147.
159. Michelle Caputo. Distributed Order Differential Equation Modeling Dielectric Induction and Diffusion, Fractional
Calculus and Applied Analysis, vol. 4, No. 4 (2001) pp 421-442.
160. Michelle Caputo. Linear Models of Dissipation whose Q is almost Frequency Independent, Geophysics Journal of
Royal Astronomical Society, (1967), 13, pp 529-539.
161. Michelle Caputo. Mean Fractional Order-Derivative Differential Equations and Filters, Ann. Univ. Ferrara-Ser.
VII-Sc. Mat. Vol. XLI, (1995) pp 73-84 .
162. Michelle Caputo. The rheology of an anelastic medium studied by means of the observation of splitting of its
eigenfrequencies, J. Acoust. Soc. Am.vol.86, no.5. 1989 pp.1984-1987.
163. Michelle Caputo. The Riemann Sheet Solutions of Anelasticity, Annali di Matematica pura Ed applicata (IV), vol.
CLXVI (1994), pp 335-342.
164. Miller K S and Ross B. An Introduction to Fractional Calculus and Fractional Differential Equations, Willey
New York 1993.
165. Mohan V Aware, Ashwin Dhabale, Bhandarkar. S, Shantanu Das. Analogue Emulator for DC Motor with Integral
& Fractional Order PID, Patent Reference Application 334/MUM/2010 Intellectual Property Office of India
Mumbai (2010).
166. Mohan V Aware, Ashwin Dhabale, Bhandarkar. S, Shantanu Das. Modular Circuit for realizing any order of
Fractional Differ Integrals, Patent Reference Application 344/MUM/2010 Intellectual Property Office of India
Mumbai (2010).
167. Mohan. V. Aware*, Ashwin. S. Dhabale*, S. G. Bhandarkar+, Shantanu Das. *VNIT-Nagpur, +NPCIL. Design
algorithm for two port fractional order differintegrals-Patent Document at Indian Patent Office.(Mumbai-India)
2010
226
168. Moshrefi-Torbati. M & Hammond J K. Physical and geometrical interpretation of fractional operators, Journal
of Franklin Institute, vol. 335, no.6, pp.1077-1086, 1998.
169. Moutushi Dutta Choudhury*, Subrtata Chandra*, Soma Nag*, Sujata Tarafdar*, Shantanu Das, *Department of
Physics Univ. of Jadavpur Calcutta. Oscillatory Spreading and Surface Instability of a Non-Newtonian Fluid
under Compression. Condense Matter Archives Ref: ARXIV: 1012.0882v2 dated 10 December, 2010.
170. Munkhammar.J.D. Fractional calculus and Taylor-Riemann series. UUDM Thesis Report 2005.
171. Nakagava.M, Sorimachi.K. Basic characteristics of fractance device, IEICE Trans. fundamentals, vol. E75-A, no.
12, 1992, pp.1814-1818.
172. Naqvi Q A and Abbas M. Fractional duality solutions and corresponding sources. PIER vol. 25, pp 290-294
(2000).
173. Oberhettinger.F and Badii L. Tables of Laplace transforms, Springer-Verlag Berlin (1973).
174. Ogata A and Banks R.B. A solution of differential equation of longitudinal dispersion in porous medium, U.S
Geological Survey Professional Papers 411-A, 1961.
175. Oldham and Spanier. Fractional calculus Academic Press San Diego 1974.
176. Oldham and Spanier. The replacement of Fick’s law by formulation involving semi differentiation.
J.Electroanal.Chem, vol.26, pp.331-341 (1970).
177. Oldham. K B. Interrelation of current and concentration at electrodes, J.Appl Electrochem, vol. 21, pp.1068-
1072 (1991).
178. Om Prakash Agarwal. A General Formulation and Solution for Fractional Optimal Control Problem, Nonlinear
Dynamics 38, Kluwer Academic Publishers, pp 323-337.(2004).
179. Oustaloup A. From fractality to non-integer order derivation through recursivity, a property common to these two
concepts: a fundamental idea for a new process control strategy, Proceedings of the 12th IMACS World Congress
Paris July 18-22, 1988, vol.3, pp.203-208.
180. Oustaloup.A et al. Frequency band complex non-integer differentiator characterization and synthesis. IEEE Trans.
on circuits and systems-I: Fundamental Theory and Application, vol. 47, no. 1, pp.-25-39, 2000.
181. P. Gopikrishnan, V. Plerou, L.A. Nunes Amaral, M. Meyer and H.E. Stanley. Scaling of the distribution of
fluctuations of financial market indices. Phys. Rev. E 60 (1999), p. 5305.
182. P. Lévy. Théorie de l'Addition des Variables Aléatoires, Gauthier-Villars, Paris (1937).
183. Poularikas A.D. The handbook of formulas and tables for signal processing, The Electrical Engineering
Handbook Series, CRC Press and IEEE Press, New York 1999.
184. R.N. Mantega and H.E. Stanley. Scaling behavior in the dynamics of an economic index. Nature 376 (1995), p.
46.
227
185. R.N. Mantega and H.E. Stanley. Stochastic Process with Ultraslow Convergence to a Gaussian: The Truncated
Levy Flight. Phys. Rev. Lett. 73 (1994), p. 2946.
186. Rafael Barcena et al. Auto tuning of fractional order hold circuit IEEE Int. Conf on Control applications Sept 5-7,
2001, Mexico City pp 7-12.
187. Ramiro S, Barbosa et al, Time domain design of fractional differintegrators using least square, Signal Processing
86(2006) pp.2567-2581.
188. Rammal R and Toulouse G, Random walks on fractal structures and percolation clusters, J. Physique-Letters 44
(1983) L-13-L-22.
189. Rayes-Melo M E, Martinez-Vega J .J, Guerrero-Salazar C A, Ortiz-Mendez U. Modeling of relaxation
phenomena in organic dielectric materials, applications of differential and integral operator of fractional order,
Journal of Optoelectronics and Advanced materials, Vol. 6, No. 3, September, 2004, pp. 1037-1043.
190. Riew F. Mechanics with fractional derivatives, Phys. Rev. E, vol.55, no.3, pp3581-3592, (1997).
191. Rina Schumer, David A Benson, Mark M Merscaert, Stephen. W. Wheatcraft. Eulerian derivation of fractional
advection dispersion equation, J. of contaminant hydrology 48(2001), Elsevier pp. 69-88.
192. Rina Schummer, David A Benson. Multiscaling fractional advection dispersion equation and their solution,
Water Resource Research Vol. 39, No.1 pp12-1-12-11 (2003).
193. Robotnov Y N. Elements of Hereditary Solid Mechanics, English MIR Publishers Moscow (1980).
194. Roman H E, Alemany.P.A. Continuous time random walks and fractional diffusion equation, J Phys A: Math.
Gen, vol.27, pp.3407-3410 (1994).
195. Rouse Prince E (Jr.). A theory of Linear Viscoelastic Properties of Dilute Solutions of Coiling Polymer, The
Journal of Chemical Physics Vol. 21 No.7 (1953), pp. 1272-1280.
196. Rutuja Dive*, Prof. M.V. Aware*, Prof A.S. Dhabale, Shantanu Das. *VNIT-Nagpur, New Method of Fractional
Order Differintegral Representation with Fractional Approximation, National Symposium of Mathematical
Modeling of Natural Phenomena NMMNP-2010, Calcutta-India. October 29, 2010.
197. Rutuja Dive, Prof. A. S. Dhabale, Dr. M. V. Aware (VNIT Nagpur), Shantanu Das. Calculation of Pole Zero
Position for Realization of Fractional Order Impedance by Interlacing Poles and Zeros, International Conference
of System Dynamics & Controls ICSDC 19-22 August, 2010, Manipal-India.
198. Sabatier J, Agrawal O.P, Machado J. A (Eds.). Advances in Fractional Calculus, Springer-Verlag 2007.
199. Saha Ray S Bera R K. Analytical solution of Bagley Torvik equation by Adomian’s decomposition method, Appl.
Math. Comp. 168 pp 389-410.
200. Saha Ray S, Bera R K. An approximate solution of non-linear fractional differential equation by Adomian’s
Decomposition, Analytical Applied Math Computer 167 pp 561-71.
201. Saha Ray S. Exact solution for time fractional diffusion wave equation by decomposition method. Physics Scripta
(2007) pp 53-61.
228
202. Saha Ray. S. A new approach for the application of Adomian decomposition method for solution of fractional
space diffusion equation with insulated ends, Applied Mathematics and Computations, Vol. 202, No. 2, pp.544-
549.
203. Saha Ray. S. Exact solution for time-fractional diffusion-wave equations by decomposition method, Physics
Scripta, vol. 75, pp.53-61.
204. Saha Ray.S and Bera R K. Analytical solution of dynamic system containing fractional derivative of order one-
half by Admonian decomposition method. ASME J. of Applied Mechanic July 2005, vol. 72/1.
205. Samko S G, Kilbas A A, Maritcheva.O.I. On Fractional Integrals and derivatives: Theory and Applications,
Gordon and Breach New York (1993).
206. Samorodnitsky. G, and Taqqu M.S. Stable Non-Gaussian Random Process, Chapman and Hall, New York, 1994.
207. Sapagovas. M, Vileiniskis. V. The solution of two-dimensional neutron diffusion equations with delayed neutrons,
INFORMATICA, vol. 12, No. 2 pp.337-342.
208. Saptarishi Das*, Sumit Mukherjee*, Indranil Pan*, A. Gupta*, Shantanu Das. *Jadavpur Univ. Calcutta.
Identification of Core Temperature in Fractional Order Noisy Environment for Thermal Feed Back in Nuclear
Reactor, IEEE Student's Technology Symposium IIT Kharagpur India, 14-16 January 2011.
209. Saptarishi.Das*, Basudev.Majumder*, Indranil.Pan*, Amitava.Gupta*, Shantanu Das *PE of Jadavpur Univ.
Calcutta. A New Fractional Fourier Transform Based design of a Band Pass FIR filter for Power Feed Back in
Nuclear Reactor under noisy environment, ICETECT (IEEE Conference Kanyakumari-India) 2011.
210. Sardar T, Saha Ray S, Bera R K, Biswas B.B. The solution of the multi-term fractionally damped Van-der-Pol
Equation, Physica Scripta IOP Publishing Phys. Scr. 80(2009)025003 (66pp) 1-5, 2009.
211. Sardar Tridip, Saha Ray S, Bera R K, Biswas B.B, Shantanu Das. The Solution of Coupled Fractional Neutron
Diffusion Equations with Delayed Neutron, Int. J. of Nuclear Energy Science & Technology,Vol.5, No.2, pp105-
113, 2010, Inderscience Publishers.
212. Saxena. R.K. Fractional calculus and fractional differential equations Lecture Note (chapter-3) pp 1-39, Jai
Narayan Vyas University Rajasthan 2006.
213. Saxena.R.K, Mathai.A.M, Haubold H J. On Generalized Fractional Kinetic Equations. Int J of Math. Phys.
0406046 v1 22-06-2004.
214. Scafetta. N, Griffin. L, West. B. J. Holder Exponent Spectrum for Human Gait, Physica. A 328 (2003) pp. 561-
583, Elsevier.
215. Schneider W.R and Wyss W. Fractional Diffusion and Wave Equations, Math. Phys. 30, pp.134-144. (1989).
216. Scott Blair G W. Measurement of Mind and Matter, Dennis Dobson London 1950.
217. Scott Blair G W. Psychoreology: links between the past and the present, Journal of Texture Studies, vol.5, pp.3-
12, (1974).
229
218. Shantanu Das, N. C. Pramanik, Micro-structural Roughness of Electrodes Manifesting as Temporal Fractional
Order Differential Equation in Super-Capacitor Transfer Characteristics, International Journal of Mathematics &
Computation™ 20 (3), 94-113. 2013
219. Shantanu Das, B B Biswas. Fractional divergence for Neutron Flux profile in Nuclear Reactor, Inderscience
Publishers Europe Int. J. of Nuclear Energy Science & Technology, Vol.3, No.2, 2007 pp.139-159.
220. Shantanu Das, B. B. Biswas. Total Energy Utilization from Nuclear Source, PORT-2006, Nuclear Energy for
New Europe Slovenia 2006.
221. Shantanu Das, B.B. Biswas. Embedding Nuclear Physics Rules & Fuel Chemistry Limits in Control Algorithm for
Fuel Efficient Reactor, Int. J. of Nuclear Governance Economy and Ecology, vol. 1, No. 3, pp. 244-264. (2007).
222. Shantanu Das, B.B. Biswas. Fuel Efficient Nuclear Reactor Controls, ICONE-13 (50843), Int. Conf. on Nuclear
Engineering, Beijing May, 2005.
223. Shantanu Das, B.B. Biswas. Shaped Normalized Reactor Period Function, definitions properties & application.
Int. J. Nuclear Energy Science & Technology, vol. 2, No. 4, pp. 309-327 (2006).
224. Shantanu Das, B.B.Biswas. Controlling Nuclear Plants with Fuel Efficiency, Int.J. of Nuclear Power-atw-Gmbh.
Vol. 2 February 2007, pp. 107-116.
225. Shantanu Das. Convergence of Riemann-Liouvelli and Caputo Derivative for Practical Solution of Fractional
Order Differential Equations, International Journal of Applied Mathematics & Statistics, Vol. 23, Issue No. D11,
pp. 64-73, (2011).
226. Shantanu Das. Efficient control of Nuclear Plants IAEA-TM- Control & Instrumentation 2007.
227. Shantanu Das, et al. Ratio control with logarithmic logics in new P&P control algorithm for a true fuel-efficient
reactor. Int. J. Nuclear Energy Science & Technology. Vol. 3, No.1, 2007 pp. 1-18.
228. Shantanu Das. Fractional Stochastic Modeling for Random Dynamic Delays in Computer Control System,
International Journal of Applied Mathematics & Statistics, 2011, Vol. 21, No. J11, pp131-140.
229. Shantanu Das. Generalized Dynamic Systems Solution by Decomposed Physical Reactions, International Journal
of Applied Mathematics & Statistics, June -2010, J10 (special) issue CESER publications, pp44-75 2010.
230. Shantanu Das. Mathematico-Physics of Generalized Calculus University Course work book for PhD (Applied
Mathematics and Physics) for Department of Applied Mathematics, University of Calcutta and Department of
Physics, University of Jadavpur; in limited prints at University of Calcutta and University of Jadavpur-Calcutta.
Lecture Notes 1-8, www.shantanudas.lecture.com , (2009-2010)
231. Shantanu Das, Shiben Bhattacharya, R.T.Keswani, S. Sunder Rajan. Circuit Theory Approach with Fractional
Calculus to describe half space Geophysical Analysis for Transient Electromagnetic Method, Geophysical Journal
No.2. T-31-2009, pp147-159.
230
232. Shantanu Das. Solution of Extraordinary Differential Equation with Physical Reasoning by Obtaining Modal
Reaction Series, Modeling and Simulation in Engineering, Hindawi Publishing Corp. Vol. (2010), ID-739675, pp.
1-19.
233. Shantanu Das, Functional Fractional Calculus for system identification and controls, Springer-Verlag Germany
2007.
234. Shantanu Das, Functional Fractional Calculus 2nd Edition, Springer-Verlag, Germany, 2011.
235. Shantanu Das, Anomalous diffusion & its electrical response in impedance spectroscopy of polymer electrolytes
via generalized calculus. Current Trends in Theoretical Chemistry CTTC-2013, BARC.
www.shantanudaslecture.com.
236. Shantanu Das, Essence of fractional calculus in applied science, Work-Shop on fractional Calculus FOS08, 2008,
University of Jadavpur, Kolkata, www.shantanudas.lecture.com.
237. Shantanu Das, Capacitor Story, Fractional Calculus & Applied Engineering, STTP, VNIT-Nagpur, 2013,
www.shantanudaslecture.com.
238. Shantanu Das, Fractional cross product and fractional curl with application in vector field of electromagnetic
theory in doubly positive system DPS (n > 0) and doubly negative system DNG (n < 0) Lecture-Notes DIAT Pune,
2013. www.shantanudaslecture.com.
239. Shantanu Das, Fractional Order Controls, IEEE Lecture at Univ of Jadavpur, 2006,
www.shantanudaslecture.com.
240. Shantanu Das, Lecture Notes on Visco-elasticity Part-A/B/C/D. Dept. of Physics, University of Jadavpur,-2013,
www.shantanudaslecture.com.
241. Shantanu Das, Generalized Fractional Calculus Appreciation, IEEE Lecture at IIT-Kharagpur, 2007,
www.shantanudaslecture.com.
242. Shantanu Das, Half and One-Half Derivative in Physics-a reality. Physics Colloquium BARC, 2010,
www.shantanudaslecture.com.
243. Shantanu Das, Non-Linear Dynamics with Fractional Calculus, Interdisciplinary problems in non-linear dynamics,
Dept. of Applied Mathematics, University of Calcutta, 2010, www.shantanudaslecture.com.
244. Shantanu Das, Reality of Fractional Calculus in six different applications, National Work Shop on Fractional
Calculus-Theory and Applications. Dept. of Mathematics, Univ. of Pune, 2012. www.shantanudaslecture.com.
245. Shantanu Das, Reality in Fractional Calculus, National Workshop on Application of Fractional calculus in
Engineering, RAIT, Univ. Mumbai, 2012, www.shantanudaslecture.com.
246. Shantanu Das, Physical Laws & Solving extra ordinary differential equations, National Workshop on Application
of Fractional calculus in Engineering, RAIT, Univ. Mumbai, 2012, www.shantanudaslecture.com.
231
247. Shantanu Das, Physical reality of series solution of generalized differential equation, Frontiers of Mathematics
and Mathematical science, 102nd Foundation day lecture of Calcutta mathematical Society, 2010.
www.shantanudaslecture.com
248. Shantanu Das, Ordering Disordered System by Fractional Calculus, Theoretical Techniques in Disordered
Systems, Condensed Matter Physics Research Centre, Univ. of Jadavpur, 2009, www.shantanudaslecture.com
249. Shantanu Das, Tutorial Notes: L’Hospital’s Question, ‘What is ½ derivative of function, f ( x ) = x , DIAT-Pune,
2013, www.shantanudaslecture.com.
250. Shantanu Das, Why Fractional Calculus Approached Solution for Diffusion Problems? BRNS-TPDM BARC,
2013, www.shantanudaslecture.com.
251. Shantanu Das, Evolution of Temporal Fractional Derivative due to Spatial Stochastic Disorder in Transport
Phenomena, International Journal of Mathematics & Computation 17 (4), 1-20, 2012.
252. Shantanu Das, Fractional stochastic modeling for random dynamic delays in computer control system,
International Journal of Applied Mathematics and Statistics 21 (J11), 131-140 2011.
253. Shantanu Das, Geometrically Deriving Fractional Cross Product and Fractional Curl, International Journal of
Mathematics & Computation™ 20 (3), 6-29, 2013.
254. Shantanu Das, Formation of Fractional Derivative in Time due to Propagation of Free Green’s Function in
Spatial Stochastic Disorder Field for Transport Phenomena, International Journal of Mathematics &
Computation™ 17 (4), 68-92, 2012.
255. Shantanu Das, Multiple Riemann Sheet Solution for Dynamic Systems with Fractional Differential Equations,
International Journal of Applied Mathematics & Statistics 28 (4), 83-89-2012.
256. Shantanu Das, Frequency and time Domain Solution for Dynamic System having Differential Equation of
Continuous Order, Int. J. of Appl. Math. Stat. 29 (2012), No.5, 6-16.
257. Shantanu Das, Generalized Dynamic Systems Solution by Decomposed Physical Reactions, International Journal
of Applied Mathematics and Statistics 17 (J10), 44-76-2010.
258. Shantanu Das, Gramian for Control of Fractional Order Multivariate Dynamic System, International Journal of
Applied Mathematics and Statistics™ 37 (7), 71-96 2013.
259. Shantanu Das, Generalization of Fractional Calculus Operators with Applications: Developments, International
Journal of Mathematics & Computation™ 21 (4), 23-50. 2013.
260. Shantanu Das, Mechanism of Wave Dissipation via Memory Integral vis-à-vis Fractional Derivative, International
Journal of Mathematics & Computation™ 19 (2), 72-83 2013.
261. Shantanu Das, Fractional order boundary controller enhancing stability of partial differential wave equation
systems with delayed feedback, International Journal of Mathematics & Computation™ 19 (2), 42-59-2013.
262. Shantanu Das, Lecture-Notes: Circuit Analysis with Fractional Capacitors, Part-1, Part-2, Part-3, VEC, Univ. of
Mumbai, 2013, www.shantanudaslecture.com.
232
263. Shantanu Das, Fractional Order Charging and Discharging of Super-capacitors, BRNS TPDM BARC, 2012.
www.shantanudaslecture.com.
264. Shawagfeh N T. The Decomposition Method for Fractional Differential Equations, J. Frac. Calc 16 pp 27-33.
(1999)
265. Shequin Zhang. Positive solutions for boundary value problems on non-linear fractional differential equations.
Electronic J. of differential equations 2006 no.36 pp 1-12.
266. Shih-Tong, Srivastava H M et al. A certain family of fractional differential equations, Taiwanese Journal of
Mathematics Sept-2000, vol.4, no.3, pp 417-426.
267. Sierociuk Dominik. Fractional Kalman Filter algorithm for states, parameters and order of fractional system
estimation, XV Krajowa Konferencja Automatyki, Warszawa, Polska (2005) pp 1-14.
268. Sinha S, Dutta T, and Tarafdar S. Adhesion and fingering in the lifting Hele-Shaw cell: Role of the substrate, The
European. Physical. Journal. E 25, (2008) pp. 267-275.
269. Sommer Jens-Uwe and Blumen Alexader. On the statistics of generalized Gaussian structures: collapse and
random external fields, J. Phys. A: Math, Gen. 28 (1995) pp. 6669-6674.
270. Srivastava H M. A certain family of sub-exponential series, Int. J. Math. Educ. Sci. Tech, vol.25, no.2, 00211-216,
(1994).
271. Srivastava H M, Owa S. Some characterization and distortion theorems involving fractional calculus, generalized
hyper geometric functions, Hadamard products, Linear Operators and certain subclasses of Analytical functions,
Nagoya Math J Vol. 106 (1987) pp1-128.
272. Srivastava H M. On extension of Mittag-Leffler function, Yokohama Math. J. 1968, vol.16, no.2 pp.77-88.
273. Stacey. W.M. Nuclear Reactor Physics, Wiley-Interscience Publications, New-York (2001).
274. Stauffer. D. Hausdroff Dimension and Fluctuations for the Largest Cluster at the Two-Dimensional Percolation
Threshold. Z-Physik B 37, (1980) Springer-Verlag pp. 89-91.
275. Subrata Chandra *, Soma Nag *, Sujata Tarafdar *, Shantanu Das. *Department of Physics, University of
Jadavpur, Analysis of the Spreading of non-Newtonian fluid using Fractional Calculus, National Symposium on
Mathematical Modeling of Natural Phenomena NMMNP-2010, 29 October-2010, Bose Institute Calcutta-India.
276. Sugi.M et al. -Variations of AC admittance in the inhomogeneous distributed RC lines. Jpn. J. Appl. Phys.
Vol. 39, no 9A, 2000, pp. 5367-5368.
277. Sugi.M, Hirano Y, Miura Y.F, Saito.K. Simulation of fractal immitance by analog circuits: an approach to
optimized circuits, IEICE Trans. Fundamentals, vol. E82-A, no. 8, 1999, pp. 1627-1635.
278. Suman Saha, Saptarishi Das, R. Ghosh, B. Goswami, Amitava Gupta, R.Balasubramanium, A.K.Chandra,
Shantanu Das. Design of Fractional Order Phase Shaper for Iso-Damped Control of PHWR Under Set-Back,
IEEE Transactions on Nuclear Science, Vol.57, No.3, pp1602-1612, June 2010
233
279. Suman Saha, Saptarishi Das, R. Ghosh, B. Goswami, Amitava Gupta, R.Balasubramanium, A.K.Chandra,
Shantanu Das. Fractional Order Phase Shaper Design With Bode’s Integral for Iso-Damped Control System,
ISA-Transactions 49(2010) Elsevier pp. 196-206.
280. Suman Saha, Saptarishi Das, R. Ghosh, B. Goswami, Amitava Gupta, R.Balasubramanium, A.K.Chandra,
Shantanu Das. Fractional Order Phase-Shaper Design With Routh’s Criterion for Iso-Damped Control System,
IEEE-India Council Gujarat-Section (Paper ID 740) INDICON, December, 2009.
281. Sun H H, Abelwahab, A. A, Onaral.B. Linear Approximation of Transfer functions with a pole of fractional
power. IEEE Trans on Automatic Control vol. AC-29, no.5, pp.441-444, May 1984.
282. Suparna Sinha and Sujata Tarafdar. Viscous Fingering Patterns and Evolution of Their Fractal Dimensions,
Industrial Engineering & Chemical Research, 2009.
283. Tarasov. E. Vasily. Electromagnetic field of fractal distribution of charged particles. Physics of Plasmas 12,
082106 (2005) pp. 1-9. (2005).
284. Tom Hartley and Carl F Lorenzo. Chaos in fractional Chua’s circuit, IEEE Trans. On Circuits and Systems-I:
Fundamental Theory and Applications, vol.42, no.8, 1995, pp. 485-490.
285. Tom Hartley and Carl Lorenzo. Fractional order system identification by continuous order distribution Elsevier
Signal Processing 2003.pp 2287-2300.
286. TsaoY.Y, Onaral.B and Sun. H.H. An algorithm for determining global parameters of minimum phase systems
with fractional power spectra. IEEE Trans. Inst. And Meas (1989), vol. 38, no. 3, pp 723-729.
287. Tustin A, et al. The design of systems for automatic control of the position of massive objects. The proceedings of
Institute of Electrical Engineers, 105C (1), 1958.
288. U.A. Muller, M.M. Dacorogna and O.V. Pictet. In: R.J. Adler, R.E. Feldman and M.S. Taqqu, Editors. A
Practical Guide to Heavy Tails, Birkhäuser, Basel (1998), p. 283.
289. Valimaki et al. Principles of fractional delay filters. IEEE Conf on Acoustic Speech Processing (ICASSP)
Istanbul, Turkey 5-9 June 2000.
290. Varsha Bhambhani, YangQuan Chen, Dingyu Xue. Optimal Fractional Order Proportional Integral Controller
for Varying Time-Delay Systems, IFAC World Congress Seoul Korea July 2008.
291. Varsha Bhambhani, Yiding Han, Shayok Mukhoopadhyay, Ying Luo and YangQuan Chen. Random Delay Effect
Minimization on Hardware-in-loop Networked Control System Using Optimal Fractional Order PI Controllers,3rd
IFAC FDA November 5-7, 2008, Ankara Italy.
292. Vinagre, Ivo Petras. Two direct Tustin discretization methods for fractional order differentiation / integration.
J.Franklin Institute: 340 (2003) pp 349-362.
293. Vingare, Ivo Petras, Podlubny, et al. Using fractional order adjustment rules and fractional order reference
model in model reference adaptive control, Non linear Dynamics 29, pp 269-279 (2002).
234
294. Wallis.M. Computer experiments with fractional Gausian noises PART 3: Mathematical Appendix .Water
Resource research 5, 1969, pp 328-338.
295. Wang Jifeng, Yunkai et al. Frequency domain analysis and application for fractional order control system J of
Phy. Conference Series 13(2005) pp 268-273.
296. Wang. J.C. Realization of generalized Wartburg impedance with RC ladder networks and transmission lines, J. of
Electrochem. Soc., vol. 134, no.8 August 1987, pp. 1915-1920.
297. Wantanbe Hiroshi. Spectral dimension of a wire network, J. Phys A: Math, Gen. 18 (1985), pp. 2807-2823.
298. Wazwaz A. A reliable Modification of Adomian Decomposition Method, Appl. Math. Comp 102 (1) pp 77-86.
(1999).
299. West B.J. & Sheshadri V. Linear Systems with Levy Fluctuations, Physica 113A, pp. 205-216 (1982).
300. Westerlund et al. Capacitor theory IEEE Trans on Dielectric and Electrical Insulation 1994, vol.1, no.5, pp.826-
839.
301. Westerlund. Dead Matter has memory. Physics Scripta 1991, vol.43, pp.174-179.
302. Wyss. The fractional diffusion equation, J. Math. Phys, vol.27, no.11, pp.2782-2785 (1986).
303. Xiaorang Li et al. On concept of local fractional differentiation. Department of applied Mathematics University of
Western Ontario London 2004 Lecture Notes.
304. Yang Quan Chen. Analytical stability bound for class of delayed fractional order dynamic system, IEEE Conf. On
Decision and control Florida WeA05-1 2001 pp 1421-1424.
305. Yang.C et al. A conceptually effective predictor corrector method for simulating fractional order dynamic control
system. ANZIAM. J. 47(EMAC2005) pp C168-C184.
306. YangQuan, Rongtao Sun, Anhong Zhou. An Overview of Fractional Order Signal Processing (FOSP)
Techniques. Proceedings. DET07, ASME Design Engineering Technical Conference, Las Vegas, Nevada, USA,
September, 4-7, 2007, pp. 1-16.
307. Yoshinoi Akoi, Mihir Sen et al. Approximation of transient temperature in complex geometries using fractional
derivatives. Department of Aerospace and Mechanical Engineering University of Notre Damme. Technical Note
Dec 19, 2005.
308. Yu. K. W, Hui P.M and Stroud D. Anomalous Transport in random resistor-capacitor networks: Application to
composite materials. Physics Letters-A. Elsevier Science Publications Vol. 118, No. 6 (1986) pp305-308.
235
I have several light years to travel to
understand fractional calculus…still trying
its physical and geometrical interpretation.
Shantanu Das PhDNo.ENGG01200704021 October, 2013
www.shantanudaslecture.com
http://scholar.google.co.uk/citations?user=9ix9YS8AAAAJ&hl=en
