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Ain Shams Engineering Journal (2014) 5, 1071–1081
Ain Shams University
Ain Shams Engineering Journal
www.elsevier.com/locate/asejwww.sciencedirect.com
ELECTRICAL ENGINEERING
Regression model for tuning the PID
controller with fractional order time delay system
* Corresponding author. Tel.: +91 7588306244.
E-mail address: [email protected] (S.P. Agnihotri).
Peer review under responsibility of Ain Shams University.
Production and hosting by Elsevier
2090-4479 � 2014 Production and hosting by Elsevier B.V. on behalf of Ain Shams University.
http://dx.doi.org/10.1016/j.asej.2014.04.007
S.P. Agnihotri *, Laxman Madhavrao Waghmare
SGGS Institute of Engineering and Technology, Vishnupuri, Nanded, India
Received 5 September 2013; revised 7 April 2014; accepted 16 April 2014
Available online 13 June 2014
KEYWORDS
Regression model;
PID controller;
Iterative algorithm;
Fractional order;
Time delay and system
Abstract In this paper a regression model based for tuning proportional integral derivative (PID)
controller with fractional order time delay system is proposed. The novelty of this paper is that tun-
ing parameters of the fractional order time delay system are optimally predicted using the regression
model. In the proposed method, the output parameters of the fractional order system are used to
derive the regression function. Here, the regression model depends on the weights of the exponential
function. By using the iterative algorithm, the best weight of the regression model is evaluated.
Using the regression technique, fractional order time delay systems are tuned and the stability
parameters of the system are maintained. The effectiveness and feasibility of the proposed technique
is demonstrated through the MATLAB/Simulink platform, as well as testing and comparison using
the classical PID controller, Ziegler–Nichols tuning method, Wang tuning method and curve fitting
technique base tuning method.� 2014 Production and hosting by Elsevier B.V. on behalf of Ain Shams University.
1. Introduction
In different regions, Time-Delay Systems (TDS) encounter
along with engineering, the field of biology, and also econom-ics [1,2]. A time delay is usually a footing regarding unsteadi-ness and also fluctuations within a process [3]. Two sorts of
time delay devices are there as follows: retarded and also fairlyneutral [4]. Throughout TDS, wherever time-delays existingbetween the features regarding input on the process and
also the ensuing result, could be suggested simply by delay
differential equations (DDEs) [5]. Programs along with delaysreveal any category in inexhaustible dimensions mostly
requested the actual modeling as well as the research regardingtransportation and also propagation phenomena [6]. Time-delays on top of things loops normally mortify process display
and confuse case study and also strategy regarding feedbackcontrollers [7].
The actual solidity of time-delay methods is often a prob-lem of chronic interest due to a number of functions of com-
munication systems in the field of biology in addition topopulation dynamics [8]. Generally, solidity review of time-delay methods could be classified straight into two varieties.
First is the particular delay-dependent stability review whichoften contains the data in the length of the particular delay,and one much more is the delay-independent stability review
[9,10]. The actual delay independent stabilization comes witha controller which will temporarily relieve the system in spiteof the length of the particular delay [11]. However, the delay
1072 S.P. Agnihotri, L.M. Waghmare
centered stabilizing controller can be worried using the size ofthe delay in addition to routinely provide the upper bound ofthe delay [12].
Dead time is probably the motivated presentation in addi-tion to firmness involving time-delay devices [13]. Time delaycontinually subsists inside the measurement loop or control
loop, and therefore it is much more challenging to overpowerthis sort of method [14]. In order to enhance the control pre-sentation, a few novel control techniques, like predictive con-
trol and the neural type of an artificial neutral delay in acontrol loop were used [4]. The PID controller is used in anextensive number of difficulties like automotive, instrumenta-tion, motor drives and so forth. [15]. This specific controller
gifts feedback, the idea will be able to take out steady-stateoffsets by means of hooked up actions, and also it can certainlywait for the extracted actions [16]. Due to the simplicity of the
structure and potential to solve many control problems ofthe system [17]. PID controller offers sturdy and trustworthydemonstration pertaining to most of the devices in the event
the PID boundaries tend to be adjusted surely [15]. Manydelay-independent sufficient conditions for the asymptoticsteadiness involving basic delay differential devices are used
[18].As, this long-established PID compensator seemed to be
requested this monitoring and stabilization of the loopsdesign technique [19], considering that the conventional PID
controllers are not appropriate for nonlinear programs andhigher-ordered and time-delayed programs [20]. Below, thisstabilizing PID controller may reduce the working out occa-
sion also it keeps away from this time-consuming stablenessexamination [21] plus the reports of the PID controllersawfully be determined by the complete understanding of the
system time delay [6].Within this report, the regression model based proposed for
tuning proportional integral derivative (PID) controller along
with fractional order time delay system will be recommended.This novelty in this report will be which tuning parameterswith the fractional order time period delay process will be opti-mally expected when using the regression model. Within the
recommended method, this output parameter of the fractionalorder system is utilized to help derive this regression model sit-uation. In this paper, this regression model depends on the
loads with the exponential function. With the iterative criteria,the most beneficial weight to the regression model is decided.With all the regression process, fractional order time delay
system will be tuned plus the stability parameters with the pro-cess will be managed. This detailed justification with the rec-ommended hybrid process will be illustrated inside area 3.Previous those, the recent research works are described in
Section 2. The outcomes in addition to discussion referred inSections 4 and 5 end this report.
2. Recent research works: a brief review
Several associated works are previously accessible to the liter-ature which based on PID controller design for time delay
fractional order system. A few of them are assessed here. Atechnique to compute the entire set of stabilizing PID control-ler parameters for a random (including unstable) linear time
delay system has been offered by Hohenbichler [22]. To handlethe countless number of stability boundaries in the plane for a
permanent proportional gain kp was the most important con-tribution. It was illustrated that the steady area of the planecontains convex polygons for retarded open loops. A phenom-
enon was initiated concerning neutral loops. For definite sys-tems and certain kp, the precise, steady area in the planecould be explained by the limit of a sequence of polygons with
an endless number of vertices. This cycle might be fined fairlyaccurate by convex polygons. Moreover, they explained aneeded condition for kp-intervals potentially containing a sta-
ble area in the plane. As a result, after gridding kp in theseintervals, the set of stabilizing controller parameters could beplanned.
A partial order PIkDl controller of robust constancy areas
for interval plant with time delay has been suggested by Lianget al. [23]. They have explored the problem of computing therobust constancy area for interval plant with time delay. The
partial order interval quasi-polynomial was crumbling into anumber of vertex attribute quasi-polynomials by the lowerand upper bounds. To explain the constancy boundaries of
every vertex attributes quasi-polynomial in the space ofcontroller parameters, the D-decomposition method wasemployed. By overlapping the constancy area of each quasi-
polynomial, the constancy area of interval attribute quasi-polynomial was found out. By choosing the control parame-ters from the constancy area, the parameters of their suggestedcontroller were attained. The vigorous constancy was checked
by means of the value set together with the zero eliminationprinciple. Their suggested algorithm was constructive in exam-ining and planning the robust PIkDl controller for interval
plant.A particle swarm optimized I-PD controller of Second
Order Time Delayed System has been suggested by Suji Prasad
et al. [24]. Optimization was based on the presentation indiceslike settling time, rise time, peak overshoot, ISE (integralsquare error) and IAE (integral absolute error). PID control-
lers and its alternatives are most commonly used, althoughthere are important improvements in the control systems inindustrial processes. If the parameter of controller was notappropriately planned, next needed control output may not
succeed. Compared with Ziegler Nichols and Arvanitis tuning,they have confirmed that their simulation results with opti-mized I-PD controller to be specified enhanced presentations.
A PID controller for time delay systems has been explainedby Rama Reddy et al. [25]. Their suggested technique precisedthe stable areas of PID and a novel PID with cycle leading cor-
rection (SLC) for network control systems with time delay.The latest PID controller has a modification parameter ‘b’.They have obtained that relation of the parameters of the sys-tem. The outcome of plant parameters on constancy areas of
PID controllers and SLC-PID controllers in first-order andsecond-order systems with time delay is moreover precised.Finally, an open-loop zero was introduced into the plant-
unstable second order system with time delay so that the con-stancy areas of PID and SLC-PID controllers get competentlymade bigger.
Luo et al. [26] have suggested a part encloses in selecting twofeasible or achievable patterns, and a FOPI/IOPID controllersynthesis was applied for all the steady FOPTD systems. The
entire possible area of two patterns can be attained andpictured in the plane by means of their suggested plan.Every mixture of two patterns can be confirmed before thecontroller design, with those areas as the previous knowledge.
Regression model for tuning the PID controller with fractional order time delay system 1073
In particular, it was fascinating to compare the regions of thesetwo possible areas for the IOPID and FOPI controllers. A sim-ulation picture demonstrates that their suggested plan has
resulted and their presentation of the designed FOPI controlleris compared to the optimized integer order PI controller and theplanned IOPID controller.
Bouallegue et al. [27] have suggested to a novel PID-typefuzzy logic controller (FLC) tuning approaches from a particleswarm optimization (PSO) strategy. Two self-tuning methods
were inserted so as to develop more the presentation andtoughness properties of the suggested PID-fuzzy strategy.The scaling features tuning problem of these PID-type FLCconfigurations was created and steadily determined, by means
of a suggested limited PSO algorithm. To show the competenceand the supremacy of the suggested PSO-based fuzzy controlstrategies, the case of an electrical DC drive benchmark was
explored, inside an improved real-time framework.Shabib et al. [28] have explained about the power system
nonlinear with frequent changes in operating areas. In excita-
tion control of power systems, Analog proportional integralcopied PID controllers were extensively employed. Fuzzy logiccontrol was frequently out looked as a structure of nonlinear
PD, PI or PID control. They moreover explained the designprinciple, tracking presentation of a fuzzy proportional-integral PI plus derivative D controller. To integrate a fuzzylogic control mechanism into the alterations of the PID struc-
ture, this controller was improved by first explaining discretetime linear PID control law and subsequently more and moreobtaining the steps required. The bilinear transform (Tustin’s)
was applied to discretize the conventional PID controller. Inthat some presentations criteria were employed for comparisonbetween other PID controllers, such as settling times, over-
shoots and the amount of positive damping.Ozbay et al. [29] possess created some sort of traditional
appropriate PID controllers with regard to linear time period
invariant facilities whoever transfer operates were logical oper-ates connected with Sa, exactly where 0 < a < 1, and s is theLaplace transform variable. Effect connected with input–output time period delay about the range of allowed controller
details has been perused. This allowed PID controller detailswere identified from a small acquire style of argument utilizedsooner with regard to specific dimensional facilities.
Zhao et al. [30] proposed an Integral of Time Absolute Error(ITAE) zero-position-error optimal tuning and noise effectminimizing method for tuning two parameters in MD TDOF
PID control system to own sought after regulatory as well asdisturbance rejection overall performance. Your contrasttogether with Two-Degree-of-Freedom control program bymodified smith predictor (TDOF CS MSP) and also the made
MD TDOF PID tuned by the IMC tuning approach demon-strates the potency of the particular described tuning approach.
Feliu-Batlle et al. [31] proposed the latest technique for the
control of water distribution in an irrigation main canal poolseen as substantial time-varying time delays. Time delaysmay perhaps adjust greatly in an irrigation main canal pool
as a result of versions inside its hydraulic operations program.A classic system got its start to development fractional buy PIcontrollers coupled with Smith predictors that produce control
systems which can be robust for the modifications in the pro-cess time delay. The system was given to fix the situationregarding powerful water submission control in an irrigationmain canal pool.
Sahu et al. [32] have outlined around the design and styleas well as effectiveness evaluation regarding DifferentialEvolution (DE) algorithm based parallel 2-Degree Freedom
of Proportional-Integral-Derivative (2-DOF PID) controllerfor Load Frequency Control (LFC) of interconnected powersystem process. The planning issue has been formulated as
an optimization issue and DE has been currently employedto look for optimal controller parameters. Standard as wellas improved aim features have been used for the planning goal.
Standard aim features currently employed, which were Integralof Time multiplied by Squared Error (ITSE) and Integral ofSquared Error (ISE). To be able to additionally raise the effec-tiveness in the controller, some sort of improved aim operate is
derived making use of Integral Time multiply Absolute Error(ITAE), damping ratio of dominant eigenvalues, settling timesof frequency and peak overshoots with appropriate weight
coefficients. The particular fineness in the recommended tech-nique has become confirmed by simply contrasting the resultswith a lately published strategy, i.e. Craziness based Particle
Swarm Optimization (CPSO) for the similar interconnectedelectric power process. Further, level of sensitivity evaluationhas been executed by simply varying the machine details as
well as managing load conditions off their nominal valuations.It is really observed which the recommended controllers arequite powerful for many the system parameters as well as man-aging load conditions off their nominal valuations.
Debbarma et al. [33] have suggested a new two-Degree-of-Freedom-Fractional Order PID (2-DOF-FOPID) controllerended up being suggested intended for automatic generation
control (AGC) involving power systems. The controller endedup being screened intended for the first time using threeunequal area thermal systems considering reheat turbines
and appropriate generation rate constraints (GRCs). Thesimultaneous optimization of several parameters as well asspeed regulation parameter (R) in the governors ended up
being accomplished by the way of recently produced metaheu-ristic nature-inspired criteria known as Firefly Algorithm(FA). Study plainly reveals your fineness in the 2-DOF-FOPID controller regarding negotiating moment as well as
lowered oscillations. Found function furthermore exploresthe effectiveness of your Firefly criteria primarily based mar-keting technique in locating the perfect guidelines in the con-
troller as well as selection of R parameter. Moreover, theconvergence attributes in the FA are generally justify whencompared with its efficiency along with other more developed
marketing technique such as PSO, BFO and ABC. Sensitivityanalysis realizes your robustness in the 2-DOF-FOPID con-troller intended for distinct loading conditions as well as largeimprovements in inertia constant (H) parameter. Additionally,
the functionality involving suggested controller will bescreened next to better quantity perturbation as well as ran-domly load pattern.
Better performances of the closed-loop can be achieved byusing such kinds of controllers in fractional order time delaysystems. The PID controller is used due to the simplicity of
the operation, design and low cost in many industrial applica-tions. However, for making stability system the tuning processof PID controller is one of the complexes. Many tuning meth-
ods PID gains have been developed, but the Ziegler Nicholsmethod is one of the classical PID gain tuning technique.But, the Ziegler–Nichols method is still widely used fordetermining the parameters of PID controllers. In addition,
1074 S.P. Agnihotri, L.M. Waghmare
construction of a mathematical tuning model design becomessometimes costly and these tuning rules often do not give suit-able values because of modeling errors. The linear program-
ming method is efficiently for tuning the PID gains. But, it isan iterative feedback tuning; so, the computational time andthe difficulty of the tuning are high. Thus, the linear program-
ming method is suitable for the specific case of tuning that is alinear objective function. In this paper, iterative algorithm andcurve fitting technique proposed to tune the fractional order
time delay system with PID controller.
3. Fractional order PID controller
Fractional-order controller design has attracted considerableattention in the recent years due to a number of applications.Better performances of the closed-loop can be achieved by
using such kind of controllers. In many industrial applications,the PID controller is used due to the simplicity of the opera-tion, design and low cost. The PID controllers are describedby a fractional order mathematical model. The fractional order
system is designed on control theory based techniques. So theclosed loop control parameters are changed sensitively andtheir performance is enhanced. The fractional order PID con-
troller block diagram structure is given in the following Fig. 1.The above figure describes the combination of the PID con-
troller GC (s) and fractional order system GC(s) and e�hs. The
inputs, error and output of the present closed loop control sys-tem are r E and Y respectively. Here, the output of the frac-tional order PID controller can be described in the followingEq. (1).
YðsÞ ¼ EðsÞ � K � e�hs Kpsþ Ki þ Kds2
ðTs2 þ sÞ
� �ð1Þ
whereY(s) is the output equation,E(s) is the error of the system,
T is the time constant, h is the time delay andK is represented asthe steady state gain of the system, Kp, Ki and Kd are theproportional, integral and derivative gains respectively. The
better tuning parameters are achieved by the proposed tech-nique, which is briefly explained in the following Section 3.1.
3.1. Over view of the proposed method
In this paper, a regression analysis for tuning the PID control-ler with fractional order time delay system is proposed. The
regression analysis tuning is one of the statistical models,which are used to describe the relationship between two andmore variables. It identifies the best fit parameters from theavailable system dataset. Here, the tuning process of the
proposed model depends on the weight of the exponentialfunction. From the iterative algorithm best weights are deter-mined. It has been allowed to evaluate the best fit parameters
Figure 1 Structure of the fractional order PID controller.
of the system and it makes the exact system response. By usingthe evaluated parameters the fractional order system steadystate response is maintained and the best PID controller gains
were achieved at the time of system parameter variations.
3.1.1. Regression model
The proposed regression model is the widely used prediction
and forecasting technique, which is used to discover the bestfit parameters from the available system data points. Here, itforms the regression equation with the use of system output
parameters. In the regression equation the best values aredetermined with the required adjustments of the systemparameters. The general equation can be described [5,8] in
the following,
znðxÞn ¼ z0 þ z1x1 þ z2x
2 þ z3x3 þ � � � þ zn�1x
n�1 þ znxn ð2Þ
where zn(x)n is the required system data, z0, z1, z2 . . .zn�1
and zn are the system data points at various conditions. Thesystem data are depending on the response of Y(s), whichare varied from different conditions, i.e., 0ton values. Inregression analysis the system response at n value can be
described as the following,In regression analysis condition, xn ¼ 1
YðsÞ, zn ¼ 11þexpwn
Then the regression value zn(xn) is rewritten as follows,
znðxÞn ¼1
1þ expwn EðsÞ � K � e�hs KpsþKiþKds2
ðTs2þsÞ
� �� �n ð3Þ
where wn is the weight of the exponential function, the equa-tion shows the system data at zn(x
n) value. By using the Eq.(3), the following equations are derived from different condi-
tions. Then the following equation can be described at the ini-tial data points and weight. If, xn = x0 and zn = z0, then theEq. (3) is written as follows:
znðxnÞ=n ¼ 0 ¼ z0 ð4Þ
If, xn = x1 and zn = z1 then Eq. (3) is written as follows:
Figure 2 Structure of the proposed method.
Figure 3 Performance of the system (a) PID controller (b) Ziegler–Nicholas method (c) curve fitting technique (d) Wang tuning
controller (e) regression method.
Regression model for tuning the PID controller with fractional order time delay system 1075
Figure 4 (a–c) Comparison of the tuning performance of the system at different time delay instants.
1076 S.P. Agnihotri, L.M. Waghmare
znðxnÞ=n ¼ 1
¼ 1
1þ expw1 EðsÞ � K � e�hs KpsþKiþKds2
ðTs2þsÞ
� �� �1 ð5Þ
If, xn = xn-1 and zn = zn�1 then Eq. (3) is written as follows:
ZnðxnÞ=n ¼ n� 1
¼ 1
1þ expwn�1 EðsÞ � K � e�hs KpsþKiþKds2
ðTs2þsÞ
� �� �n�1ð6Þ
If, xn = xn and zn = zn then Eq. (3) is written as follows:
znðxnÞ=n ¼ n
¼ 1
1þ expwn EðsÞ � K � e�hs KpsþKiþKds2
ðTs2þsÞ
� �� �n ð7Þ
The above mentioned equations at different conditions (4) to(7) are substituted in Eq. (2) and the regression based system
equation is derived. Finally, the data points are combinedwith the standard fractional order system which is derived interms of a regression equation, which is in the form of each
data point and the derived system equation is described asfollows:
znðxnÞ ¼ z0 þ 1
1þexpw1 EðsÞ � K � e�hs KpsþKiþKds2
ðTs2þsÞ
� �� �1þ � � �
þ 1
1þexpwn�1 EðsÞ � K � e�hs KpsþKiþKds2
ðTs2þsÞ
� �� �n�1
þ 1
1þexpwn EðsÞ � K � e�hs KpsþKiþKds2
ðTs2þsÞ
� �� �n
ð8Þ
The best fit data point of the system depends on the weight ofthe exponential function. The iterative algorithm is used to
determine the best weights; it means the certain weight con-tains reduced overshoot time, settling time and steady stateerror. The identification of the best weight using the iterativealgorithm is briefly explained in the following section.
Figure 5 (a–c) Comparison of the tuning performance of the system at different time delay instants.
Regression model for tuning the PID controller with fractional order time delay system 1077
3.1.2. Best weight determined by iterative algorithm
An iterative algorithm is the act to determine the desired goal,
target or result. In the process a mathematical function isapplied repeatedly, and the output of the one iteration is usedas the input of the next iteration. It can produce the approxi-
mate numerical solutions depending on the mathematical eval-uation. Here, the best weight is determined from certainevaluations. During the iteration process, the selected weight
checks for the convergence condition, i.e., the overshoot time,the settling time and the steady state error is reduced. If theresultant weight satisfies the convergence condition, then it
means that the steady state response of the system has beenmaintained.
wn ¼xn � l1
l2
ð9Þ
where xn is the data points of the fractional order system, l1and l2 are the scaling transformation l1 = mean(xn) andl2 = std(xn).
Steps to determine the best weight
Step 1: Initialize the system data points fromxnmin � xn � xn
max:Step 2: To determine the weight using the Eq. (9) and the
system data points.Step 3: The determined weights are applied into the Eq.(8) and check the output.
Step 4: If the convergence is satisfied, i.e., the overshoottime, the settling time and the steady state error isreduced. Then the selected weight is the best weight(wBF
n ) or else go to the next step 5.
Step 5: To adjust the weights as wn ± 1, which isachieved by changing the data points. Then repeat theprocess until the best weight is achieved (wBF
n ).
The final output of the iterative algorithm is the bestweight, which is used to develop the optimal system response.
The sequences of best weights are applied in the Eq. (1) and itcan be described in the following Eq. (10).
Figure 6 (a–c) Comparison of the tuning performance of the system at different time delay instants.
1078 S.P. Agnihotri, L.M. Waghmare
zBFn ðxÞn ¼ zBF0 þ zBF1
1
YðsÞ
� �1
þ zBF21
YðsÞ
� �2
þ zBF31
YðsÞ
� �3
þ � � � þ zBFn�11
YðsÞ
� �n�1
þ zBFn1
YðsÞ
� �n
ð10Þ
The above equation is the resultant regression equation. The
proposed method operational structure is described in the fol-lowing Fig. 2. The proposed regression model is implementedin the MATLAB platform and the effectiveness is analyzed by
comparing the classical PID controller, Ziegler–Nichols tuningmethod and iterative algorithm and curve fitting techniquebase tuning method.
4. Results and discussion
The proposed regression model is implemented on the MAT-
LAB 7.10.0 (R2010a) platform. The numerical results of the
proposed method are presented and discussed in this section.In the implementation process, the system parameters values
are set manually, i.e., the proportional gain (Kp), integral gain(Ki) and derivative gain (Kd) are 0.5, 0.05 and 3 respectively, timeconstant (T) is 360 s, time delay (h) is 1 s, steady state gain is (Tf)
14.9, the reference step is 10 and the simulated time is 1000 s.The effectiveness of the proposed method is tested and com-pared using the classical PID controller, Ziegler–Nichols tuning
method and curve fitting technique based tuning method. Thefractional order system with various time delays (h = 20, 40,60, 80 and 100) at PID controller, Ziegler–Nichols tuningmethod, curve fitting technique, Wang tuning controller and
regression method are analyzed in the following Fig. 3a–erespectively.
Then the fractional order time delay system of tuning per-
formance of the proposed regression method is compared withthe conventional PID controller, which is illustrated in theFig. 4a–c. It is demonstrated under various time delays such
as h = 20, 40 and 60. In Fig. 5a–c the regression method iscompared to the Ziegler–Nicholas method at time delays
Figure 7 (a–c) Comparison of the tuning performance of the system at different time delay instants.
Regression model for tuning the PID controller with fractional order time delay system 1079
h = 20, 40 and 60 respectively. The Wang tuning controller is
compared with the proposed method and it is explained in theFig. 6a–c respectively. The tuning performance of our methodeffectiveness is compared to the curve fitting technique at time
delay instants h = 20, 40 and 60, which is illustrated in theFig. 7a–c respectively. The bode diagram of the system is eval-uated and subsequently setting the described time delay values.The bode diagram of the system after setting time delay instant
h = 20 is illustrated in the Fig. 8a. Fig. 8b shows that bodediagram of the system after setting the time delay instanth = 40 and bode diagram after setting the time delay instant
h = 60, which is illustrated in the Fig. 8c.The proposed regression method’s effectiveness was dis-
cussed in this section. Here, the fractional order time delay
system is tested under the conventional PID controller,Ziegler–Nichols method, curve fitting technique, Wang tuningcontroller and our proposed method. These techniques areanalyzed with various time delay instants h = 20, 40, 60, 80
and 100. During the time delay occurrence in the fractional
order time delay system, the output performance was oscil-
lated. Therefore the rise time, overshoot time and settling timehave been increased. Then the output performance of the everytechnique is compared to the proposed technique. By using the
PID controller at different time delay instants h = 20, 40 and60. It takes more time to reach a low value to the highest value(rise time), the required output signal exceeds its target(overshoot time) and the settling time is also underprivileged.
For Ziegler–Nichols method, reaching the high value at the300 s at the time delay instant h = 20. The rise time getsincreased from 320 s and 340 s, due to the time delay becomes
increased at 40 and 60 respectively. The output of the Ziegler–Nichols method exceeds its target at increased time. Also theoutput requires more time to achieve the steady state or set-
tling time, i.e., takes more than 700 s at various time delayinstants h = 20, 40 and 60. Similar to the Ziegler–Nicholsmethod the Wang tuning controller is also inefficient to pro-duce better tuning performance. The curve fitting technique
also shows the following unfeasible solutions at various time
Figure 8 Bode diagram of the system after setting different time delay instants (a) h = 20 (b) h = 40 and (c) h = 60.
1080 S.P. Agnihotri, L.M. Waghmare
delay instants h = 20, 40 and 60. During the time delay occur-rence, the rise time is increased more than 80 s, a peak over-shoot time somewhat is reduced and the settling timebecomes above 100 s. The proposed regression method
achieves the rise time at less than 30 s, peak overshoot timegets reduced and the settling time has been less than 80 s.The comparative analysis shows that, the proposed technique
gets better for tuning the system response when comparedto curve fitting technique, Wang tuning controller, Ziegler–Nichols method and PID controller.
5. Conclusion
The proposed regression technique is used for tuning the frac-
tional order time delay system with PID controller. In theregression technique, fractional order system output parame-ters were utilized to derive the time delay system response in
the form of the regression function. Then, the best weightswere selected by an iterative algorithm for fitting the best sys-tem response. The best weight was applied to the regressionequation and the corresponding time delay system response
was analyzed. The results have been verified through varioustime delay responses and analyzed the stability of the PIDcontroller with fractional order time delay system. The effec-tiveness of the proposed method is analyzed via the compari-
son with the classical PID controller, Ziegler–Nichols tuningmethod and iterative algorithm and curve fitting techniquebase tuning method. The comparative analysis shows that,
the proposed hybrid technique is the best technique for tuningthe fractional order time delay system, which is competent overthe other tuning methods. In future, the proposed method can
be used in the higher order system tuning parameter optimiza-tion. Also the PID controller with fractional order time delaysystem tuning parameters will be predicted by using different
types of optimization techniques.
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Mr. S.P. Agnihotri (Santosh Prabhakar
Agnihotri) obtained his Bachelor’s degree in
Instrumentation and Control from University
of Pone. Then he obtained his Master’s degree
in Instrumentation Engineering and perusing
PhD from SGGS Institute of Engineering and
Technology, Vishnupuri, Nanded, Swami
Ramanand Teerth Marathwada University
his current research interests is time delay
system.
Dr. Laxman Madhavrao Waghmare Presently,
he is working as Director, SGGS Institute of
Engineering and Technology, Vishnupuri,
Nanded, completed B.E. (1986) and M.E.
(1990) from the SGGS Institute of Engineer-
ing and Technology, Nanded and Ph.D.
(2001) from Indian Institute of Technology
(IIT), Roorkee. His field of interest includes
intelligent control, process control and, image
processing. He is recipient of K.S. Krishnan
Memorial national award for the best system
oriented research paper (published in the IETE Research Journal).
Presently working for a research project sponsored by the NRB of
DRDO.