1

Extensions of Classical Automatic Control Methods to Fractional Order Systems: An

Educational Perspective

Nusret Tan

Inonu University, Engineering Faculty, Dept. of Electrical and Electronics Engineering,

44280, Malatya, Turkey.

(nusret.tan@inonu.edu.tr)

Abstract

The rapid development of control technology has an impact on all fields of control theory.

This development forced researchers to produce new mathematical control theory, new

controllers and design methods, actuators, sensors, new industrial processes, computer

methods, new applications, new philosophies and solution techniques for new challenges. One

of such new theory is fractional order control system which is based on fractional order

calculus. In recent years there has been considerable interest in the study of feedback systems

containing processes whose dynamics are best described by fractional order derivatives. It

seems that the developments in the area of fractional order control systems can give many

opportunities for new advancements in the control and automation field which is an important

part of science and technology. Therefore, it is important to teach developed results based on

fractional order calculus to see the effects of fractional order integrator and derivative on

control system performance. This can be done by using new and high-quality educational

methods such as advanced computer software programs and interactive tools. The purpose of

this paper is to show how fractional order control methods can be introduced into a first

course on classical control using interactive tools such as Matlab, Simulink and LabView.

Key words: Education, Control Theory, Fractional Order Systems, Interactivity, LabVIEW

1. Introduction

In recent years, fractional calculus has been an important tool to be used in engineering,

chemistry, physics, mechanical, bioengineering and other sciences. It can be seen in the

literature that there have been many publications in this field (Das, 2008; Xue at al., 2007;

Monje et al, 2010). The reason of this amount of interest to this subject is that the future

scientific developments will be based on fractional calculus and the attempts to find solutions

of unsolved complex problems. It is known that the differential operators which we

encountered in mathematics, science and engineering are in the form of 𝑑𝑓/𝑑𝑡, 𝑑2𝑓/𝑑𝑡2,

𝑑3𝑓/𝑑𝑡3.., or in other words, the systems are described by integer order differential

equations, however, is it necessary that the orders have to be integer? Why can it not be a

rational, fractional or complex number? The correspondence between Leibniz and L’Hospital

drawn attention to this subject at the beginning of differential and integral calculus era and

this subject is now called fractional calculus. We have entered the era of fractional

mathematics especially due to the computational facilities provided by technological

developments. Therefore, hereafter developed theories, innovations and applications in

science and technology will be based on fractional calculus. Automatic control expressed as

“hidden technology” by Karl Astrom will be a field which uses fractional calculus the most.

Fractional order control systems are needed for better modelling and performance of

dynamical systems. Therefore, in order to obtain effective solutions to the problems in science

2

and technology, it is necessary to do work on this subject which has a great capacity of

original value.

On the other hand, the subjects being taught to students on automatic control, despite

advances in the field of control theory, seem to have changed little over last many years.

However, developments in computer technology now allow us to use new and high-quality

educational methods such as interactive tools, virtual and remote laboratories to make use of

World Wide Web etc. (Dormido, 2002; Tan et al, 2016a). Today’s computer software

programs such as Matlab and LabVIEW can be used effectively to teach some advanced

subjects to the students without going into details of mathematical derivations. Although the

fundamental control theory concepts known as classical control theory are necessary for

control theory education and because of its many advantages we think this must continue in

the future, it will be important that any developments in control theory which can be linked to

classical control are good candidates for consideration in basic feedback control education by

using interactive tools. The interactive tools are extremely useful and they enable students to

explore changes in system performance as parameters are varied, and to do so looking at

several diagrams simultaneously. This can be done without any programming by just using a

mouse to adjust any parameters and the effects can be immediately seen. Matlab, LabVIEW

and Simulink provide an excellent environment for such studies to teach additional topics

besides subjects of classical control theory. One such topic is the recent development in

methods to analyse systems using fractional order calculus. As summarized above this is an

important topic since most real physical system can be modeled more adequately by fractional

order differential equations. The applications of fractional order differential equations to the

problems in the control theory have been increased in recent years and promising results have

been obtained. It has been shown that there is a strong link between classical control methods

and fractional order approaches. Therefore, from educational point of view, it is important to

teach developed results based on fractional order calculus to see the effects of fractional order

integrator and derivative on control system performance. The purpose of this paper is to show

how the results based on fractional order concepts can be introduced into a basic classical

feedback control theory course given for undergraduate students and the teaching can be

supported by software tools. This will accelerate future developments in the control and

automation field which is an important part of science and technology.

2. Formulization of Fractional Order Derivative and Integrator

Fractional order derivative and integrator can be considered as an extension of integer order

derivative and integrator operators to the case of non-integer orders and it is defined in

general form as the following (Chen et al., 2009),

( )

0

1 0

( ) 0

a t

t

a

d

dt

D

d

(1)

where, a tD represents fundamental non-integer order operator of fractional calculus.

Parameters a and t are the lower and upper bounds of integration, and R denotes the

fractional-order. Two definitions used for the general fractional derivative and integrator are

the Riemann-Liouville definition and the Caputo definition. The Riemann–Liouville

definition for the fractional-order derivative of order R has the following form

3

1

1 ( )( )

( ) ( )

tn

a t n n

a

d f dD f t

n dt t

(2)

where (.) is Euler’s gamma function and 1n n , n N . An alternative definition for the

fractional-order derivative was given by Caputo as

1

1 ( )

( ) ( )

t n

a t n

a

fD d

n t

(3)

where 1n n , n N . The Laplace transform of the Caputo fractional order derivative has

the following result that is particularly significant for fractional order system modeling 1

1 ( )

0

00

( ) ( ) (0)n

st k k

t

k

e D f t dt s F s s f

(4)

When (1) (2) (3) ( 1)(0) (0) (0) (0) ,.., (0) 0nf f f f f is considered, a basic property

facilitating for design and analysis of fractional order systems is expressed for Laplace

transform of fractional order derivative as ( ) ( )L D f t s F s .

In general, fractional order LTI systems were described by the following fractional order

differential equation form as (Vinagre et al, 2000) 1 0 1 0

1 0 1 0( ) ( ) ( ) ( ) ( ) ( )n n m m

n n m ma D y t a D y t a D y t b D r t b D r t b D r t

(5)

By applying ( ) ( )L D f t s F s , a general form of transfer function of fractional order LTI

systems were expressed as, 1 0

1 0

1 0

1 0

( )( )

( )

m m

n n

m m

n n

b s b s b sY sG s

R s a s a s a s

(6)

where ia , jb ( 0,1,2,...,i n and 0,1,2,...,j m ) are real parameters and i ,

j are real

positive numbers with 0 1 n and 0 1 m .

3. Why Fractional Order Control?

Many real systems are known to display fractional order dynamics. For example, it is known

that the semi-infinite lossy (RC) transmission line demonstrates fractional behaviour since the

current into the line is equal to the half derivative of the applied voltage that is

( ) (1/ ) ( )V s s I s . Thus, the significance of fractional order representation is that fractional

order differential equations are more adequate to describe some real world systems than those

of integer order models. Many physical systems such as viscoelastic materials,

electromechanical processes , long transmission lines, dielectric polarisations , coloured noise,

cardiac behaviour, problems in bioengineering, and chaos can be described using fractional

order differential equations. Thus, fractional calculus has been an important tool to be used in

engineering, chemistry, physical, mechanical and other sciences (Magin, 2006; Ppoudlubny,

1999a; Harley et al., 1995; Koeller, 1984; Perdikaris and Karniadakis, 2014; Oldham and

Spanier; 1974).

In feedback control theory the proportional, derivative, and integral control actions affect the

performance of closed loop control system. The closed loop behaviours such as speed of the

response, elimination of steady-state error, relative stability and sensitivity to noise can be

changed by proportional, derivative and integral actions (Monje et al., 2010). By introducing

more general control actions of the form s or 1/ s , R , we can achieve more

4

satisfactory results from closed loop control system. Clearly s represents fractional order

derivative and 1/ s represents fractional order integral.

For modelling dynamic systems, frequency domain experiments are usually performed in

order to obtain equivalent electrical circuits which represent real dynamical systems.

Generally, a frequency domain behaviour of the form / ( )k j , R and in the Laplace

domain which is /k s is required for accurate modelling. /k s is known as Bode’s ideal

transfer function (Monje et al. 2010).

4. Frequency Domain Analysis

The computation of frequency responses of transfer functions plays an important role in the

application of frequency domain methods for the analysis and design of control systems.

There are some powerful graphical tools in classical control, such as the Nyquist plot, Bode

plots and Nichols charts, which are widely used to evaluate the frequency domain behaviours

of the systems. The Bode and Nyquist envelopes of a transfer function are important in

classical control theory for the analysis and design. For example, the frequency domain

specifications such as gain and phase margins can be obtained using the Bode and Nyquist

envelopes of a transfer function. The Bode plot of a control system provides a clear indication

of how the Bode plot should be modified to meet given specifications. Therefore, controller

design based on the Bode plot is simple and straightforward.

A transfer function including fractional powered s terms can be called a fractional order

transfer function, FOTF. The frequency response computation of FOTF can be obtained

similar to integer order transfer functions (Tan et al., 2009). For example, with the FOTF 2.3 0.8( ) 1/ ( 4 1)G s s s replacing s by j and using ( ) (cos / 2 sin / 2)j j ,

one obtains

0.8 2.3 0.8 2.3

1( )

(1 1.236 0.891 ) (3.8044 0.4540 )G j

j

(7)

Bode and Nyquist diagrams of this equation can then be obtained as shown in Figs. 1 (a) and

(b).

a) b)

Figure 1: a) Bode plot b) Nyquist plot

10-2

10-1

100

101

102

-100

-50

0Bode diagram

frequency(rad/sec)

gain

(dB

)

10-2

10-1

100

101

102

-300

-200

-100

0

frequency(rad/sec)

phase(d

egre

e)

-0.2 0 0.2 0.4 0.6 0.8 1 1.2-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05Fractional Nyquist diagram

Real

Imagin

ary

5

5. Time Domain Analysis

For time domain computation, there is not a general analytical method for determining the

output of a control system with an FOTF. There have been many studies over the years some

of them are based on integer approximation models and others based on numerical

approximation of the non-integer order operator. The methods developing integer order

approximations can be used for time domain analysis of fractional order control systems

similar to classical control approaches. Some of the well known methods for evaluating

rational approximations are the Continued Fractional Expansion (CFE) method, Outaloup’s

method, Carlson’s method, Matsuda’s method, Chareff’s method, least square methods and

others (Oustaloup et al., 2000; Matsuda and Fujii, 1993; Vinagre et al., 2000).

In the recent papers by the authors (Atherton et al., 2015), two new computational methods to

obtain solutions were given. One was based on the Fourier series of a square wave and is

called the Fourier Series Method(FSM) and the other is based on the Inverse Fourier

Transform Method (IFTM). Following this study, the method has been used to study the time

response of closed loop fractional order systems with time delay and fractional order

controller in (Tan et al., 2016b; Tan et al., 2016c). The results obtained from these methods

are exact, to numerical accuracy in summing infinite series, since they use frequency response

information for an FOTF which can be computed exactly by using the relationship

( ) [cos( / 2) sin( / 2)]j j .

The block diagram of a closed loop fractional order control system with time delay is shown

in Fig. 2

Figure 2: A fractional order closed loop control system with time delay

Here, ( ) ( ) ( )pL s C s G s is the open loop transfer function which is in the form of Eq. (6) such

as 1 0

1 0

1 0

1 0

( ) ( ) ( ) ( ) ( )m m

n n

s sm mp

n n

b s b s b sL s C s G s C s G s e e

a s a s a s

(8)

Then the closed loop transfer function can be written as 1 0

1 0 1 0

1 0

1 0 1 0

( )( ) ( )( )

( ) 1 ( ) ( )

m m

n n m m

s

m m

s

n n m m

b s b s b s eY s L sP s

R s L s a s a s a s b s b s b s e

(9)

Letting

( ) ( )( ) ( )( )

( ) 1 ( ) 1 ( ) ( )

p

p

C s G sY s L sP s

R s L s C s G s

(10)

and replacing s by j in Eq. (10), one obtains

6

( ) ( )( )( )

( ) 1 ( ) ( )

(Re[ ( )] Im[ ( )])(Re[ ( )] Im[ ( )] ( ) ( )

1 (Re[ ( )] Im[ ( )])(Re[ ( )] Im[ ( )] ( ) ( )

p

p

p p

p p

C j G jYP j

R j C j G j

C j j C j G j j G j U jV

C j j C j G j j G j Z jQ

(11)

where

( ) Re[ ( )]Re[ ( )] Im[ ( )]Im[ ( )]p pU C j G j C j G j (12)

( ) Re[ ( )]Im[ ( )] Im[ ( )]Re[ ( )]p pV C j G j C j G j (13)

( ) 1 Re[ ( )]Re[ ( )] Im[ ( )]Im[ ( )]p pZ C j G j C j G j (14)

( ) Re[ ( )]Im[ ( )] Im[ ( )]Re[ ( )]p pQ C j G j C j G j (15)

and

Re[ ( )] Re[ ( )]cos( ) Im[ ( )]sin( )pG j G j G j (16)

Im[ ( )] Im[ ( )]cos( ) Re[ ( )]sin( )pG j G j G j . (17)

Thus, ( )P j can be written as

2 2

[ ( ) ( ) ( ) ( )] [ ( ) ( ) ( ) ( )]( )

( ) ( )

U Z V Q j V Z U QP j

Z Q

(18)

so that the real part and imaginary parts of the closed loop transfer function ( )P j are

2 2

[ ( ) ( ) ( ) ( )]Re[ ( )]

( ) ( )

U Z V QP j

Z Q

(19)

2 2

[ ( ) ( ) ( ) ( )]Im[ ( )]

( ) ( )

V Z U QP j

Z Q

(20)

The Fourier series for a square wave of -1 to 1 with frequency 2 /s T can be written as

1(2)

4 1( ) sin( )s

k

r t k tk

(21)

where T is the period of the square wave. If ( )r t passes through the transfer function ( )P s

then the output, which is the unit step response if T is sufficiently large, can be written as

1(2)

2 21(2)

4 1( ) Re ( ) sin( )

[ ( ) ( ) ( ) ( )]4 1sin( )

[ ( ) ( ) ]

s s s

k

s s s ss

k s s

y t P jk k tk

U k Z k V k Q kk t

k Z k Q k

(22)

Similarly, the impulse response, which is the derivative of the step response is given by

1(2)

2 21(2)

( ) 4( ) Re ( ) cos( )

[ ( ) ( ) ( ) ( )]4cos( )

[ ( ) ( ) ]

si s s s

k

s s s ss s

k s s

dy ty t P jk k t

dt

U k Z k V k Q kk t

Z k Q k

(23)

This method is called FSM.

For IFTM, the impulse response, ( )p t , corresponding to the transfer function ( )P s of Eq. (9)

is given by 1( ) ( )p t L P s where 1L denotes the inverse Laplace transform. The impulse

response can be computed by

7

2 2

0 0

2 2 [ ( ) ( ) ( ) ( )]( ) Re[ ( )]cos( ) ( ) cos( ) ( )

[ ( ) ( ) ]

U Z V Qp t P j t d t d

Z Q

(24)

or alternatively by

2 2

0 0

2 2 [ ( ) ( ) ( ) ( )]( ) Im[ ( )]sin( ) ( ) sin( ) ( )

[ ( ) ( ) ]

V Z U Qp t P j t d t d

Z Q

(25)

Example 1: In this example ( )L s of Fig. 2 was taken as 1.15

5.3 3.65 2.82 1.7 1.42

0.5 0.4( ) ( ) ( )

3 7 4 1.4p

sL s C s G s

s s s s s

(26)

Thus, the closed loop transfer function is 1.15

5.3 3.65 2.82 1.7 1.42 1.15

( ) 0.5 0.4( )

1 ( ) 3 7 4 1.4 0.5 0.4

L s sP s

L s s s s s s s

(27)

The step responses of ( )P s using the FSM and the impulse responses of 1

( )P ss

which are the

step responses of ( )P s using the IFTM for the selected range of frequencies are plotted in

Figs. 3 (a) and (b). Step responses obtained from FSM and IFTM are given in Fig. 4.

a) b)

Figure 3: a) Step responses obtained from FSM for different frequency ranges b) Step

responses obtained from IFTM for different frequency ranges

8

Figure 4: Step responses obtained from FSM and IFTM

6. Fractional Order Control and Implementation

Fractional order functions are infinite dimensional functions and hence it is very difficult to

implement them practically or simulate them numerically (Vinagre et al. ,2000). Therefore,

several integer order approximation methods were proposed for realization of fractional order

systems by replacing them with integer order approximations. Oustaloup presented an

approximation method based on recursive distribution of poles and zeros in a limited

frequency (Oustaloup et al, 2000). Another approximation method using the gain of the

fractional order transfer functions at certain frequencies was suggested by Matsuda (Matsuda

et al., 1993). Recently, SBL fitting approximation method has been presented for fractional

order derivative and integrator operators (Deniz et al., 2016). The integer order

approximations model obtained by SBL fitting method is given in Table 1. Rational

approximations for 0.5( ) 1/G s s obtained using different methods are shown in Table 2.

The exact Bode plots of 0.5( ) 1/G s s and its approximations are shown in Fig. 5 where one

can see that the approximations are good for a range of frequency but fail to give an exact

matching. It is known that the exact step response of 0.5( ) 1/G s s is equal to

0.5

( ) 2 /sy t t and its impulse response equals ( ) 1/iy t t . The exact step response and

the approximations by different methods are shown in Fig. 6 (a), where it can be seen that the

approximations are worse as time increases. Fig. 6 (b) shows the errors in the approximations.

0 10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

Time(sec)

Outp

ut(

ste

p r

esponse)

12 13 14 15 16 17

1.35

1.36

1.37

1.38

1.39

1.4

FSM

IFTM

9

Table 1. Lists of fractional order derivative operator approximations determined using

proposed method for 0.01,1.2 .

s Fractional Order Derivative Approximations by SBL

fitting 0.1s

4 4 3 2

4 4 3 2

5532 1.338 10 4147 191.5 1

4392 1.353 10 5063 284.5 1.894

s s s s

s s s s

0.2s

4 4 3 2

4 4 3 2

7718 1.714 10 4974 214.2 1

4810 1.743 10 7372 470.2 3.587

s s s s

s s s s

0.3s

4 4 4 3 2

4 4 3 4 2

1.087 10 2.227 10 6052 242.6 1

5224 2.263 10 1.083 10 783.7 6.829

s s s s

s s s s

0.4s

4 4 4 3 2

4 4 3 4 2

1.553 10 2.949 10 7507 279.8 1

5622 2.972 10 1.615 10 1324 13.16

s s s s

s s s s

0.5s

4 4 4 3 2

4 4 3 4 2

2.271 10 4.012 10 9566 330.7 1

5987 3.978 10 2.46 10 2283 25.87

s s s s

s s s s

0.6s

4 4 4 3 4 2

4 4 3 4 2

3.447 10 5.677 10 1.268 10 405.6 1

6300 5.493 10 3.88 10 4074 52.66

s s s s

s s s s

0.7s

4 4 4 3 4 2

4 4 3 4 2

5.564 10 8.562 10 1.791 10 528.2 1

6537 8.007 10 6.487 10 7704 113.8

s s s s

s s s s

0.8s

5 4 5 3 4 2

4 5 3 5 2 4

1.008 10 1.452 10 2.844 10 770.1 1

6666 1.298 10 1.213 10 1.628 10 275.5

s s s s

s s s s

0.9s

5 4 5 3 4 2

4 5 3 5 2 4

2.428 10 3.28 10 6.012 10 1488 1

6648 2.773 10 3.006 10 4.561 10 887.3

s s s s

s s s s

Table 2: Integer approximations of 0.5( ) 1/G s s using different methods

Method Integer approximations of

0.5( ) 1/G s s

CFE high frequency

Method

4 3 2

4 3 2

0.3513 1.405 0.8433 0.1574 0.008995( )

1.333 0.478 0.064 0.002844cfe

s s s sH s

s s s s

Carlson’s Method 4 3 2

4 3 2

36 126 84 9( )

9 84 126 36 1car

s s s sH s

s s s s

Matsuda’s Method 4 3 2

4 3 2

0.08549 4.877 20.84 12.995 1( )

13 20.84 4.876 0.08551mat

s s s sH s

s s s s

Least-Squares

Method

4 3 2

4 3 2

0.1002 4.011 11.26 5.076 0.3694( )

8.654 9.364 1.771 0.03744ls

s s s sH s

s s s s

Charref’s Method 4 3 2

5 4 3 2

6.3 74.84 121.1 29.79 0.9986( )

29.85 121.8 76.85 7.497 0.1cha

s s s sH s

s s s s s

Oustaloups’s Method 5 4 3 2

5 4 3 2

74.97 768.5 1218 298.5 10( )

10 298.5 1218 768.5 74.97 1ous

s s s s sH s

s s s s s

10

Fig. 5: Exact Bode plots of 0.5( ) 1/G s s and its approximations

a) b)

Fig.6: a) Step responses of 0.5( ) 1/G s s b) Errors in the approximations

An example is provided below relating to SBL fitting method. For this case, the values in

Table 1 are used to obtain integer order approximation model of fractional order derivative

operator. Also, these values can be used as 1/ s for integer order approximation model of

fractional order integrator operator.

Example 2: Consider the fractional order transfer function

1.2

1( )

1G s

s

(28)

For the fractional order derivative 0.2s , SBL fitting method, Matsuda’s method and

Oustaloup’s method provide the integer order approximation model in Table 3. Also,

equivalent integer order transfer function ( )TG s of fractional order transfer function ( )G s can

10-4

10-2

100

102

104

-50

0

50

Frequency(rad/sec)

Magnitude(d

B)

10-4

10-2

100

102

104

-60

-40

-20

0

Frequency(rad/sec)

Phase(d

eg)

Hcfe

Hcar

Hmat

Hls

Hcha

Hous

Exact

0 20 40 60 80 100 120 140 160 180 2000

2

4

6

8

10

12

14

16

Time(sec)

Outp

ut(

Ste

p R

esponse)

Analy tical Function

CFE Method

Carlson's Method

Matsuda's Method

Least-Squares Method

Oustaloups's Method

Charref 's Method

0 20 40 60 80 100 120 140 160 180 200-2

0

2

4

6

8

10

12

14

Time(sec)

Err

or

CFE Method

Carlson's Method

Matsuda's Method

Least-Squares Method

Oustaloups's Method

Charref 's Method

11

be seen in Table 3. Fig. 7 shows a comparison of the amplitude and phase responses. Here, the

exact solution for 1.2( ) 1/ ( 1)G s s was obtained by calculating phase and amplitude of

1.21/ [( ) 1]j .

Figure 7: Amplitude and phase responses of ( )G s and approximate integer order models,

SBL fitting method, Matsuda’s method and Oustaloup’s method

Table 3: Equivalent integer order transfer function ( )TG s of fractional order transfer function

( )G s

Met

hod

Integer order approximation models for 1.2

1( )

1G s

s

SB

L

4 4 3 20.2

4 4 3 2

7718 1.714 10 4974 214.2 1

4810 1.743 10 7372 470.2 3.587

s s s ss

s s s s

4 4 3 2

5 4 4 4 3 2

4810 1.743 10 7372 470.2 3.587( )

7718 2.195 10 2.241 10 7586 471.2 3.587T

s s s sG s

s s s s s

Mat

suda

4 3 20.2

4 3 2

3.357 161 453.9 95 1

95 453.9 161 3.357

s s s ss

s s s s

4 3 2

5 4 3 2

95 453.9 161 3.357( )

3.357 162 548.9 548.9 162 3.357T

s s s sG s

s s s s s

Oust

aloup

5 4 3 20.2

5 4 3 2

2.512 98.83 531.7 442.3 56.87 1

56.87 442.3 531.7 98.83 2.512

s s s s ss

s s s s s

5 4 3 2

6 5 4 3 2

56.87 442.3 531.7 98.83 2.512( )

2.512 99.83 588.6 884.5 588.6 99.83 2.512T

s s s s sG s

s s s s s s

Fractional order PID controller (PIλDμ) which has five tuning controller parameters including

an integrator of order λ and an differentiator of order μ provides a better response than the

10-3

10-2

10-1

100

101

102

103

-80

-60

-40

-20

0

20Bode diagram

frequency(rad/sec)

gain

(dB

)

10-3

10-2

10-1

100

101

102

103

-120

-100

-80

-60

-40

-20

0

frequency(rad/sec)

phase(d

egre

e)

exact

SBL

Matsuda

Oustaloup

exact

SBL

Matsuda

Oustaloup

12

integer order PID controller when used both for the integer-order systems and fractional-order

systems. PIλDμ is described in time domain in (Podlubny, 1999b) as follows;

( ) ( ) ( ) ( )p i du t k e t k D e t k D e t

(29)

where pk is the proportional gain, ik is the integral gain, dk is the derivative gain; and

are positive real numbers. The frequency domain formula is given as

( )( )

( )

ip d

kU sC s k k s

E s s

(30)

In Equation (30), classical PID controller can be obtained for 1 and 1 .

Example 3: In this example, SBL fitting method is used to simulate the closed loop control

system with PIλ controller as shown in Fig. 8 in MATLAB. Consider the fractional order PI

controller

( ) ip

kC s k

s (31)

To use fractional order controller ( )C s for simulation of the closed loop control system in

MATLAB, an integer order approximation method has to be applied for fractional operator

s . When an equivalent integer order approximation model of ( )C s is determined using an

integer order approximation method, the closed loop control system as shown in Fig. 8 can be

simulated to obtain the closed loop response. For this, fractional order operator is replaced

with its equivalent integer order model in closed loop control system.

An illustrative example which contains SBL fitting approximation method is given to clarify

this strategy. Assuming that the plant transfer function is given by 2

1( )

3 1G s

s s

and PI

controller is formed as 0.9

( ) ip

kC s k

s in Figure 2, one can found stability region of the

closed loop control system using SBL analysis (Hamamci, 2008). Then, the controller

parameters pk and ik can be selected in stability region to obtain the step response of

configuration in Fig. 8.

Figure 8: The closed loop control system with PIλ simulated in MATLAB

Fig. 9 shows different step responses obtained for different controller parameters ( , )p ik k with

0.9 selected from stability region as (0.2604, 2.2281), (0.6843, 4.7544) and (0.1129,

( )G sue

ik

s

1

1s

Integer approximation

model by SBL fitting

pk

PI

yr

13

1.2807). Better closed loop step responses can be obtained for different controller parameters

( , , )p ik k selected from stability region.

Figure 9: Step responses of the closed loop control system with PIλ for different values pk

and ik

7. Teaching Fractional Order Control Systems

MATLAB is a high-performance language for technical computing. It integrates computation,

visualization, and programming in an easy-to-use environment where problems and solutions

are expressed in familiar mathematical notation. Simulink can easy analyze model and

simulate control systems (MathWorks, http://www.mathworks.com/). Simulink can work with

the MathWorks Real-Time Workshop. Real-Time Workshop generates and executes stand-

alone C code for developing and testing algorithms modeled in Simulink. The resulting code

can be used for many real-time and non-real-time applications, including simulation

acceleration, rapid prototyping, and hardware-in-the-loop testing (Rodriguez, et al. 2005). So

Simulink is also widely used in simulation of all kind of systems. However, Simulink lack the

imitation of physical instruments or equipment in appearance and operation. That’s why, it is

a good idea to combine LabVIEW and MATLAB in simulation of control system (Xuejun et

al, 2007).

The LabVIEW ( Laboratory Virtual Instrument Engineering Workbench) software is a

graphical programming language and used to develop a virtual instrument (vi) that includes a

front panel and a functional block diagram. User enters input from the front panel of the vi.

LabVIEW has become a vital tool for engineers and scientists in research throughout

academia, industry, and government labs. LabVIEW is taught at many universities in Europe

and USA. LabVlEW has been used in the classroom for teaching of difficult subjects. For

example, the graphical programming approach of LabVlEW has been used to teach computer

science concepts. The graphics make the concepts more intuitive and easier to understand.

LabVlEW is used to teach control systems. The graphical programming approach is based on

dataflow theory which is an ideal platform for learning how signals flow from one function to

another such as from an acquisition function through a filter to a spectrum analysis and finally

0 5 10 15 20 25 300

0.5

1

1.5

t (sec)

y(t

)

kp=0.2604, k

i=2.2281

kp=0.6843, k

i=4.7544

kp=0.1129, k

i=1.2807

14

to a graph. Each function is an icon that is wired to other icons as in the example given in

Figure 10. The wires are the signals flowing from one icon to the next. Research and industry

use LabVlEW for automated test (ATE), medicine, chemistry, process control, simulation,

calibration, and general-purpose data acquisition and analysis (Vento, 1988).

Figure 10: The front and block diagram panel image of LabVIEW

The temperature control system given in Figure 10 performs on LabVIEW environment. The

application includes two windows. The first window is front panel and second is block

diagram panel of the program. The front panel is developed as a graphical interface and it is

performed interactively. The block diagram panel is that provided wire connection and data

flow.

Figure 11: The front panel image of frequency response application of FOTF using LabVIEW

The application given in Figure 11 has been developed in the LabVIEW environment for the

frequency domain analysis of fractional order control systems. The Bode, Nyquist and

Nichols diagrams for any fractional order transfer function can be plotted by using the

program. The program with this properties helps to design suitable controller for a given

control system. The program is especially suitable for use in the field of education since it has

interactive features.

15

Figure 12: The front panel image of PI controller design application for the fractional order

plant using LabVIEW

The application given in Figure 12 runs the stability boundary locus (SBL) method to find all

stabilizing PI controllers in a stability region for closed loop control systems with fractional

order plant transfer function using LabVIEW application. Effect of the parameters of

controllers selected from the stability region can be immediately observed and step response

of the system can be immediately plotted. Thus, the controller which gives the best results can

be designed based on proposed interactive approach.

Example 4: Consider the transfer function as below,

2.2 0.9

5( )

7 2G s

s s

(32)

Figure 12: Transfer function input panel for frequency response application

16

Figure 13: Bode Diagram

a) b)

Figure 14: a) Nyquist diagram b) Nichols diagram

Figure 15: Gain and phase margin panel

In this example, frequency response application of fractional order transfer functions is

examined. Transfer function given in Eq. (32) is entered to panel as shown in Fig. 12. Bode,

Nyquist Nichols diagrams are simultaneously plotted as shown in Fig. 13 and Fig. 14. Also,

the application computes gain and phase margin values which are shown in Fig. 15. The

application has interactive features, thus user can try different values, while the application is

performing.

Example 5: Consider the transfer function with time delay as below,

2

2.4 1.2 0.4

2( )

2.5

sG s es s s

(33)

Figure 16: Transfer function input panel for PI controller design application

17

a) b)

Figure 17: a) Selected point in stability region b) Stable step response for closed loop system

In this example, PI controller design for fractional order transfer functions is examined.

Transfer function given in Eq. (33) is entered to panel as shown in Fig. 16. SBL graph is first

plotted when the program is run. Then, user can select any points in the controller parameter

plane which is shown in Fig. 17 (a) by moving mouse on the opened SBL plot window. As

seen in Fig. 17 (a), selected point is 0.262545pK , 0.06iK and this point is in stability

region. Thus, the closed loop system is stable and step response of the system is plotted for

the selected point and the result is given in Fig. 17 (b). According to selected point, the PI

controller is designed as shown in Fig. 18.

Figure 18: Designed PI controller according to selected point

a) b)

Figure 19: a) Selected point on stability curve b) Critical step response for closed loop

system

Now, consider another point as seen in Fig. 19 (a), where selected point is 0.546182pK ,

0.21iK and this point is exactly on SBL curve. Thus, the closed loop system is critically

stable and step response of the system is plotted for the selected point and the result is given

in Figure 19 (b). According to selected point, the PI controller is designed as shown in Fig.

20.

18

Figure 20: Designed PI controller according to selected point

Figure 21: Selected point outside stability region

Figure 22: Stability status panel

Consider another point as seen in Fig. 21, where selected point is outside the SBL curve.

Thus, the closed loop system is unstable and step response is not plotted, because the

apllication uses Inverse Fourier Transform Method (IFTM) for plotting the time response and

IFTM method is defined in stable condition (Atherton et al, 2015). Thus, the application

writes ‘SYSTEM IS UNSTABLE’ on stability status panel as shown in Fig. 22.

8. Stability Analysis

Stability analysis of fractional order system has been an interesting area of research in that,

the theorems proposed for classical systems are usually inadequate for fractional order

system. There can be found numerous stability analysis methods for classical systems in the

literature. For instance, Routh Hurwitz method can be used for stability analysis of

polynomials with integer orders, but does not work for fractional order polynomials. On the

other hand, it is well known that the frequency domain methods of classical control theory are

applicable to the fractional order control systems. Therefore, frequency domain based stability

criterion can be applied to the fractional order polynomials.

The stability test of fractional-order systems is different from the integer-order systems. The

only known stability test method is for commensurate-order systems. For a commensurate-

order polynomial, if the absolute values of the angles of all the poles are larger than απ/2, the

system is stable. The stable region of commensurate-order systems is shown in Fig. 23. The

BIBO(Bounded Input Bounded Output) stability analysis of fractional order control system

can be done by using time domain simulation.

19

Figure 23: Stability region for commensurate-order systems.

9. Conclusions

The objective of this paper has been to draw attention to the new developments in the field of

control systems with fractional order derivative and integrator and to show how these ideas

can be introduced into a first course on classical control theory. The methods are based on

fractional order calculus and allow students to think more practically in terms of

representation of real systems with fractional order differential equations. It has also been

shown that with suitable software, the theoretical results obtained in the field of fractional

order control can be used for the analysis and design of systems. Mathematical concepts of

fractional order calculus, the advantages of fractional order control, time and frequency

domain analysis of non-integer order control systems, teaching and stability issue have been

summarized.

Acknowledgment

This work is supported by the Scientific and Research Council of Turkey(TÜBİTAK) under

Grant no. EEEAG-115E388.

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