Optimal fractional order PID controller design using Ant Lion Optimizer

Ain Shams Engineering Journal xxx (xxxx) xxx

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Electrical Engineering

Optimal fractional order PID controller design using Ant Lion Optimizer Rosy Pradhan a, Santosh Kumar Majhi b,⇑, Jatin Kumar Pradhan a, Bibhuti Bhusan Pati a a b

Department of Electrical Engineering, Veer Surendra Sai University of Technology, Burla 768018, India Department of Computer Science and Engineering, Veer Surendra Sai University of Technology, Burla 768018, India

a r t i c l e

i n f o

Article history: Received 8 August 2017 Revised 23 August 2019 Accepted 9 October 2019 Available online xxxx Keywords: FOPID controller ALO Optimal control

a b s t r a c t This paper uses a nature inspired metaheuristic algorithm based on the behavior of ant lions named as Ant Lion Optimization (ALO) to design the fractional order PID controller (FOPID) for controlling first order and higher order systems. The ALO is used to optimize the parameters of the FOPID controllers. This algorithm is based upon the hunting behavior of ant lions which is described by five important phases of hunting the ants: random walk, building trap, entrapments of ants in traps, sliding ants towards the antlion, catching the prey and rebuild the trap. ALO algorithm based fractional order PID controller is proposed for delay system and also for higher order system. To obtain the optimal computation, different performance indices such as IAE (Integral Absolute Error), ISE (Integral Squared Error), ITAE (Integral Time Absolute Error), ITSE (Integral Time Squared Error) are considered for the optimization. As the delay system exhibits non-minimum phase characteristic, to improve gain and phase margin, the ALO based fractional order PID is optimally designed by considering different objective functions such as IAE, ISE, ITAE and ITSE. All the simulations are carried out in Simulink/Matlab environment. The proposed method has superiority value in terms of transient and frequency responses as compared with other methods, which has been demonstrated by illustrative examples. Ó 2019 THE AUTHORS. Published by Elsevier LTD on behalf of Chinese Academy of Engineering and Higher Education Press Limited Company. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction The advances in control engineering have kept momentum since past few decades [1–5]. Despite advances in control technique, the PID controller is still widely used in the industry due to its demonstrated advantages such as easy to understand as it has only three tunable parameters, simple structure and easyness in implementation [6–10]. In recent days, modern control theories have made much advancement in the fields of designing of PID controller by using the idea of fractional calculus to improve its performance in various industrial systems. The fractional-orderproportional- integral- derivative (FO-PID) controller is a generalized controller of the classical PID controller. FO-PID provides a better response and more stability than the integer order PID both for integer order system and fractional order systems in many ⇑ Corresponding author. E-mail address: [email protected] (S.K. Majhi). Peer review under responsibility of Ain Shams University.

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industries [12]. Also, it exhibits better robustness performance as against integer order PID controller [11]. However, tuning of FO-PID poses a challenging problem due to its more parameter, i.e., five parameters to select instead of three parameters in a standard classical PID controller. Therefore many advanced control tuning methods have been developed in recent years to solve the difficulties arises in FO-PID controller design using fractional calculus [11–14]. As evolutionary optimization algorithms based design doesnot depends on the rigorous mathematical model of the plants, recently, many research works have been carried out by using different evolutionary optimization algorithms [13–28]. In this regard, to design FO-PID controller, either Genetic algorithm [37], Fuzzy logic [39], Particle swarm optimization (PSO) [38] or hybrid optimization [40] have been used. Among them PSO based design is widely used due to its faster convergence, however, it often exihibits local solution. Whereas Genetic algorithm achieves global solution at the cost of high computational burden. Therefore, further attempts have been made to propose an alternate algorithm to achieve faster convergence as well as global solution. Recently, a meta-heuristic algorithm inspired from the antlions named as Antlion Optimizer (ALO) algorithm has been proposed by Mirjalili [29]. The ALO has better performance which has been verified by applying 19 benchmark functions. The important factor of

https://doi.org/10.1016/j.asej.2019.10.005 2090-4479/Ó 2019 THE AUTHORS. Published by Elsevier LTD on behalf of Chinese Academy of Engineering and Higher Education Press Limited Company. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Please cite this article as: R. Pradhan, S. K. Majhi, J. K. Pradhan et al., Optimal fractional order PID controller design using Ant Lion Optimizer, Ain Shams Engineering Journal, https://doi.org/10.1016/j.asej.2019.10.005

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choosing ALO is due to its effective search space using random walk and selection of search agents randomly. Moreover, exploitation of the search space is secured by adjustive limits of traps. The probability of avoidance of local optima is very high as it uses random walks and the roulette wheel in the various stages of the optimization process. The ALO has been considered in many engineering domain due to its proven advantages in designing various PID controllers reported in the literature [23,30–33]. However in [34], the authors has used the ALO-FOPID controller for a first order stable system which in general gives an infinite gain margin and the optimal performance criteria has not considered in designing the controller. This motivates us to degign an optimal fractional order PID controller using ALO for unstable delay systems. In this paper, we implement the social behaviour of ant lions for finding the optimum value of the FOPID controller parameters. By considering the discussed advantages of the ALO, the FOPID controller is tuned by ALO optimization for the different systems. Suitable performance indexes such as IAE, ISE, ITAE and ITSE are considered to design FOPID controller. To show the effectiveness of the proposed controller, a comparative study is carried out for different exampless over the classical PID controller. Moreover, the robustness of the FOPID over PID controller has been verified by simulation. This work is organized as follows. Section 2 presents preliminaries of fractional order PID. In Section 3, the evolutionary algorithm i.e, ALO is briefly discussed along with the new proposed algorithm and the problem formulations are discussed. Section 4 contains the result and analysis of the proposed method. Finally, Section 5 concludes the work followed by references. 2. Preliminaries of fractional order calculus Fractional calculus is one of the branches of math which deals with the definition of non-integer order differentiation and integrator functions. The idea of fractional calculus came into picture when, L’Hopital and Leibniz introduced the half order of derivative function in 1695. After that L. Euler used the half order of derivative function relationship for negative or non-integer order of derivatives which gives the definition of both fractional order differentiation and integrator. In general, fractional calculus represents the generalized expression for conventional differentiation and integration. The concept of fractional calculus involves the differentiation and integrator of fractional order with a fundamental operator aDrt where a and t are the lower and upper limit of operator and r is an order of the operator whose value is not restricted to an integer number. The continuous integro-differential operator is defined in Eq. (1).

aDrt ¼

8 > < > :Rt

dr dtr

/> 0

1

/¼ 0

a ðdtÞ

r

ð1Þ

/< 0

In literature, there are various ways to represent the fractional integro-differential. The most commonly used approximate definitions for fractional integro-differential are Grunwald-Letnikov, Rieman-Liouville and Caputo [28]. Grunwald-Letnikov defination is given in Eq. (2.1).

aDrt f ðtÞ ¼ lim hr h!0

X j¼0

ð1Þj

  r f ðt  jhÞ j

ð2:1Þ

  r is the binomial co-efficient, a and t are the lower j and upper limit of integration. The value of r can be negative or positive according to the integral and differentiation respectively. The simplest and easiest form of Reimann-Liuoville definition is represented in Eq. (2.2). where

aDrt f ðtÞ ¼

 n Z t 1 d f ðsÞ ds rnþ1 dðn  rÞ dt a ðt  sÞ

ð2:2Þ

where ðn  1Þ  a  n; n is an integer, a is a real number and dð:Þ is an Euler functions. Similarly the definition of M. Caputo for integro-differential is defined by Eq. (2.3).

aC Drt f ðtÞ ¼

1 dðn  r Þ

Z a

t

f ðsÞ ðt  sÞrnþ1

ds

ð2:3Þ

where ðn  1Þ  a  n; n is an integer, a is a real number and dð:Þ is an Euler functions. As compare to integer order differential function, it is very difficult to state the fractional order differential function. There is various approximation and numerical methods are implemented in fractional order function as it don’t have accurate solution. For simplification of fractional order function, an infinite number of zeroes and poles are added for converting it into an integer order function. However a logical approximation is possible for fraction order function with finite number of zeros and poles. Oustaloup stated (Oustaloup, 1988) a simplified approximation algorithm for fractional order function by using recursive distribution of poles and zeroes. The Oustaloup simplified approximation definition for fractional order differentiator sa is given in Eq. (2.4).

sa ¼ K

M Y 1 þ ðs=wz;n Þ ;a > 0 1 þ ðs=wz;n Þ m¼1

ð2:4Þ

where K is the gain of function which is maintained in such a manner that the function will give unit gain at 1 rad/sec. M is the numbers of poles and zeroes. wz;n and wp;n are the approximate frequencies of poles and zeros at nth instant whose values are valid in the desirable range of wl (lower frequency) and wh (high frequency) of system. The approximated value of frequency range of poles and zeroes are calculated on the basis of recursive equations which are stated in Eq. (2.5)–(2.9).

pffiffiffi wz;1 ¼ wl g

ð2:5Þ

wp;n ¼ wz;n c

ð2:6Þ

wz;nþ1 ¼ wp;n ; g

ð2:7Þ

c¼

 Ma wh wl

ð2:8Þ

g¼

 ð1MaÞ wh wl

ð2:9Þ

The approximation algorithm can be implemented to design a fractional order system as well as the fractional order controller. The most commonly used fractional order controller is fractional order proportional-integrator-derivative (FOPID) controller. FOPID controller consists of five parameters such as proportional gain (Kp ), integral gain (Ki ), derivative gain (Kd ), integer order (k) and derivative order (l). Eq. (2.10) represents the mathematical representation of FOPID where an order of integrator and derivative value lies between 0 and 1.

Pc ðsÞ ¼

UðsÞ Ki ¼ K þ k þ Kd S l EðsÞ S p

ð2:10Þ

As fractional order PID controller is generalized representation of integer order controller. Therefore FOPID with k ¼ 1 and l ¼ 1 is represent the standard PID controller. The time domain expression for FOPID controller is given in Eq. (3).

uðtÞ ¼ Kp eðtÞ þ Ki Dk eðtÞ þ Kd Dl eðtÞ

ð3Þ

Please cite this article as: R. Pradhan, S. K. Majhi, J. K. Pradhan et al., Optimal fractional order PID controller design using Ant Lion Optimizer, Ain Shams Engineering Journal, https://doi.org/10.1016/j.asej.2019.10.005

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3. The proposed algorithm 3.1. Overview of ant lion optimization (ALO) Ant Lion Optimization technique [29] is the one of the nature inspired optimization algorithm for solving the unidimensional as well as multidimensional optimization problem. This algorithm was proposed by Mirjalili [29]. This algorithm is inspired by the hunting behavior of the grey antlions in nature and basically their favorite preys are ants shown in Fig. 1. An antlion dig a cone shaped pit and hide its larvae under beneath the bottom of cone shaped pit to trap the ant. The edge of cone is so sharp such that the ant can easily fall into the bottom. Once the prey trap into the cone, then the antlion throw the sand towards the edge of the cone which make the prey incapable to escape from the trap. After that the antlion consumed the prey and prepare another pit to trap next prey. The above described hunting nature of antlion is explained in five stages, which includes (i) the random walk of ants, (ii) building traps, (iii) entrapment of ants in traps, (iv) catching preys, and (v) re-building traps. As ALO mimics the hunting activity of antlion, so this optimization governed by certain conditions 1. The movement of Ants around search space is done by random walks. 2. The traps of the antlion influences the random walk. 3. The pits are built by the antlions as per the fitnessvalue. The pit size is directly depend on the fitness value. 4. The probability to catch the ants depend on the higher fitness value of the antlion. In turn the larger pit can have maximum probability to capture the ants. 5. In each iteration, an ant is captured by a fittest antlion. 6. In order to trigger the sliding behavior of the ants in the direction of antlions, the limits of random walk of ant is adaptively cutback. 7. When the fitness value of ant is more than an antlion, this means that it is captured by the antlion. 8. An antlion change its position and build a new improve pit for trapping another prey after each hunt. The five major stages for the method and its mathematical description are mentioned below (Mirjalili, 2015) 1. The random walks of ants

To descibe the hunting behavior of the antlion, the antlion and ant should have interaction with each other. To perform this interaction, the ants need to move in the search space for food and shelter and antlions are hunted them by using their traps. Since the stochastic movement of ants for searching of foods, a random walk is pick out for the modeling of ant movements. The mathematical model is presented in Eq. (4).

XðtÞ ¼ ½0; cusumð2rðt1 Þ  1Þ;    cusumð2rðtn Þ  1Þ

ð4Þ

where cusum calculates the cumulative sum, n indicates the maximum numbers of iteration, t is the steps of random walk and rðtÞ is the stochastics function defined by the in Eq. (5).



rðtÞ ¼

1 ifran num > 0:5

ð5Þ

0 ifran num  0:5

where ran num presents a random number generated by the uniform distribution within a interval 0 to 1. The mathematical representation of normalized random walk of ant is given in Eq. (6).

 Xi ðtÞ ¼

 t Xti  ai  ðdi  Cti Þ þ Cti ðbi  ai Þ

ð6Þ

Here, ai and bi indicates the minimum and maximum value of t

the random walk of ith variable respectively, Cti and di represnts the minimum value and the maximum value of the ith variable at tth iteration. 2. Building trap The mathematical modeling of antlion’s hunting capability is influenced by a roulette wheel. The ALO algorithm exploit the roulette wheel to search the fittest antlion during the optimization process. This process filters the best antlion with higher probability for catching the prey. 3. Entrapments of ants in traps As per the above discussion, the traps of the antlion influences the random walk. Therefore the mathematical relationship for this assumption is expressed by the Eqs. (7) and (8). t

Cti ¼ Antlionj þ Ct t

t

t

di ¼ Antlionj þ d

ð7Þ ð8Þ

Fig. 1. Hunting nature of antlion [29].

Please cite this article as: R. Pradhan, S. K. Majhi, J. K. Pradhan et al., Optimal fractional order PID controller design using Ant Lion Optimizer, Ain Shams Engineering Journal, https://doi.org/10.1016/j.asej.2019.10.005

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where vector C and d are the hypershere of randomly walked ant t

and around the selected antlion respectively. Ct and d indicates the minimum and maximum value of all the variable at tth iteration

better competitive results in comparison with other metaheuristic algorithms. The flowchart of the ALO algorithm is presented in Fig. 2.

t

respectively. Similarly Cti and di are minimum and maximum of the ith variable at tth iteration respectively. 4. Sliding ants towards the antlion Once the antlion realized that the ant is in the trap, it throw the sand toward the edge of the pit through the its mouth which make the ant trappers inside the pit. This behavior of antlion is model by the Eq. (9) and Eq. (10).

Ct ¼

Ct I

ð9Þ

3.2. Performance criteria The objective functions for the proposed algorithm are the various time domain integral performance indices which are represented by the Eqs. (14)–(17). Optimum values of the controller can be calculated by minimizing the indices functions. The objective function is chosen for minimizing the time response characteristics due to the dependency of error on time. Fig. 3 show block diagram of proposed algorithm based FOPID controller.

Z

1

J1 ¼ IAEðIntegral Absolute ErrorÞ ¼

jeðtÞjdt

ð14Þ

0 t

t

d ¼

d I

ð10Þ t

t T

e2 ðtÞdt

5. Capturing the prey & rebuilding the trap

Z

t

1

tjeðtÞjdt 0

ð16Þ Z J4 ¼ ITSEðIntegral with Time Square ErrorÞ ¼

1

te2 ðtÞdt

ð17Þ

0

The problem can be represented as Minimize J Subjected to

Kpmin < Kp < Kpmax

The hunting final stage is reached when the ant is caught by the antlion. This behavior of antlion is described by the Eq. (12) where the fitness value of ant is more than the fitness value of antlion. In this stage ant is consumed by the antlion. Then the antlion is updated its position or build a new trap to catch a new prey.

Antlionj ¼ Antti iffðAntti Þ > f ðAntlionj Þ

ð15Þ

J3 ¼ ITAEðIntegral with Time Absolute ErrorÞ ¼

ð11Þ

Here, t is the current iteration, T indicates total number of iteration and w is the constant depend on the current iteration. The above two mathematical equations represent the sliding process of ant into the pit.

t

1

0

t

where I indicates the ratio described Eq. (11), C and d are the minimum and maximum value of all the variable at tth iteration respectively.

I ¼ 10w

Z J2 ¼ ISEðIntegral Square ErrorÞ ¼

ð12Þ t

where t is the current iteration, Antlionj and Antti deonotes as the position of the selected jth antlion and position of ith ant, respectively. Function fðÞ indicates the fitness value of ant and antlion. 6. Elitism Elitism is the important characteristics which allow the algorithm to obtain finest solution at every phase of the optimization method. The finest antlion is obtained and saved as a elite in each iteration of the ant lion optimization algorithm. Since the elite one is influenced the movement of all the ants throughout the iterations. Hence, it is considered that each ant moves about the chosen antlion randomly and elite simultaneously by the given equations

Kimin < Ki < Kimax Kdmin < Kd < Kdmax kmin < k < kmax

lmin < l < lmax

ð18Þ

Here, J indicates the objective function (J1 ; J2 ; J3 ; and J4 ) and e (t) represtns the error e ¼ maxjrðtÞ  yðtÞj, Here,rðtÞ and yðtÞ are defined as system output and the desired input of the system respectively. Since it is not pacticable to integrate upto infinity, we choose the upper limit of the integration is sufficiently large to make the error zero. The upper limit is considered as the simulation time (see Figs. 4, 6, 9). 3.3. Proposed ALO-FOPID controller design

Here, at tth iteration, RtA is the random walk around the selected

The proposed ALO-FOPID control structure is shown in Fig. 3. The detail steps of the proposed controller is given below. Input: System with adjustable FOPID controller parameter with randomly generated ant and anlt lion size n in 5 dimensions. Output: The optimum value of FOPID controller parameter with the help of best solutions.

antlion,RtE indicates the random walk near by the elite and Antti shows the position of ith ant. The ALO has shown high performance in solving the classical optimization problem. ALO converges rapidly towards the optimum with the help of exploitation. Ant lion optimizer has a high degree of exploration rate which helps the algorithm to explore the favourable regions in the search space. Furthermore, the algorithm provides satisfactory local minima avoidance and preferably better approach towards global optima. The algorithm provides

1. Initializing the population of ant, antlions and number of iterations by considering the boundary values of the controller parameters (KP, KI, KD, k, l). 2. The position of ants is shown in the position matrix (Mant), where n and d indicates the number of ants and the number of variables or dimensions, respectively. As the five control parameters need to be adjusted, the number of variable is five in this case.

Antti ¼

RtA þ RtE 2

ð13Þ

Please cite this article as: R. Pradhan, S. K. Majhi, J. K. Pradhan et al., Optimal fractional order PID controller design using Ant Lion Optimizer, Ain Shams Engineering Journal, https://doi.org/10.1016/j.asej.2019.10.005

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Fig. 2. Flowchart of ALO algorithm.

Fig. 3. Block diagram of proposed FOPID controller tuned by ALO.

2

Mant

A1;1 6 A2;1 6 ¼6 6 .. 4 .

A1;2 A2;2 .. .

  .. .

A1;d A2;d .. .

An;1

An;2



An;d

3 7 7 7 7 5

3. Considering the position matrix, the fitness (objective) function of each ant is evaluated and stored in a fitness matrix named as MFA, where n represents the populations of ants d denotes the number of variables and f is the fitness or objective function.

Please cite this article as: R. Pradhan, S. K. Majhi, J. K. Pradhan et al., Optimal fractional order PID controller design using Ant Lion Optimizer, Ain Shams Engineering Journal, https://doi.org/10.1016/j.asej.2019.10.005

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2

AL1;1 AL2;1 .. .

AL1;2 AL2;2 .. .

  .. .

AL1;d AL2;d .. .

ALn;1

ALn;2



ALn;d

fðAL1;1

AL1;2 AL2;2 .. .

  .. .

AL1;d Þ AL2;d Þ .. .

ALn;2



ALn;d Þ

6 6 Mantlion ¼ 6 6 4 2 MFAL

6 6 fðAL2;1 6 ¼6 .. 6 . 4 fðALn;1

3 7 7 7 7 5 3 7 7 7 7 7 5

Fig. 4. Comparison of step responses for example 1.

Fig. 5. Bode Plot of the example 1.

5. The best fitness function is chosen from MFAL by using roulette wheel selection and then respective antlion is selected as an optimum one. 6. In each iteration, an antlion is selected using roulette wheel. According to the antlion, the boundary positions are updated by using Eqs. (9) and (10) which are proportional to the current iteration. 7. Random walk for each ant is created and normalized it’s step around the search space of selected antlion by using Eq. (4) and (6) for every ant. After that the ants’ positions are updated by considering Eq. (13). 8. Then the fitness values of all the ants are calculated and replaced the antlion with its related ant, if it becomes appropriate as shown in Eq. (12). 9. The position of optimum antlion is updated, if the antlion having highest fitness function than the optimum. 10. The above process is repeated until the maximum iteration is not satisfied. Once all the iterations are over the best or elite solution is calculated. 4. Result and discussion

Fig. 6. Step response of the 2nd order system.

2 MFA

fðA1;1

6 6 fðA2;1 6 ¼6 .. 6 . 4 fðAn;1

A1;2 A2;2 .. .



An;2



 .. .

A1;d Þ A2;d Þ .. . An;d Þ

3 7 7 7 7 7 5

4. The antlions are concealthemselves within the search space, so the position and the fitness value are also stored in the matrices Mantlion and MFAL respectively. Here n and dindicates the number of antlions and the number of variables, respectively.

This section presents the design of Ant lion optimization based Fractional order PID control for three different plants such as 1st order time delay, 2nd order unstable time delay and 3rd order system. For system dynamics analysis solvers plays a mojor role. In literature, solvers like ODE 45, ODE 23, ODE 113, ODE 23t, multistage-Adomian decomposition etc. are reported [41-44]. In this work, we have considered various solvers such as ODE 45, ODE 23, ODE 113, ODE 23t etc. It has been observed through simulation that all the solvers give the same transient and steady state performance. However, ODE23t takes less computational time as compared to other mentioned solvers. The performance of the systems is compared with the classical self tuning method and ALO tuned PID controller. Simulation studies are carried out to make a comparison between different stated objective functions for all the below examples. Example 1. The heat exchange system [35] is considered for the first example and given as:

GðsÞ ¼

9:5 e0:3s 1 þ 30:5s

The system transfer function consists of first order time delay, to make the system simplified there are three different ways to represent the delay i.e. neglect the delay by making h ¼ 0, Taylor series expansion and first order Pade approximation. In this paper, the first order delay with Taylor series expansion is considered. The transfer function with Taylor series expansion is:

GðsÞ ¼

9:5 ð1  0:3sÞ 1 þ 30:5s

Please cite this article as: R. Pradhan, S. K. Majhi, J. K. Pradhan et al., Optimal fractional order PID controller design using Ant Lion Optimizer, Ain Shams Engineering Journal, https://doi.org/10.1016/j.asej.2019.10.005

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The lower bound and upper bound are set five and 20 respectively for proportional gain (K p ), integral gain (K i ), derivative gain (K d ) and for integer order (k) and derivative order (l) the lower and upper bound are set as 0 and 1. During the design of FOPID controller for first order system, the following parameter values have been considered. Population size = 12, Maximum number of iterations = 100, Number of evaluations = 20. To obtain the desired PID parameters, different objective functions such as IAE, ISE, ITAE and ITSE are considered. Simulation is run for the above objective functions and the obtained parameters for the various stated index value for example 1 are given in Table 1. From Table 2, it is cleared that the ITAE index gives better result in comparison with other index for the given system. To show the efficacy of the proposed controller, a comparative analysis is carried out with the welltuned IOPID. Table 3 shows the transient performance (i.e., rise time, overshoot) and steady state performance (i.e., settling time) of the proposed controller and IOPID, fractional order filter PID and fractional order filter with PI. The step response of the above controller is shown in Fig. 4. From Fig. 4 and Table 3, it is clear that the ALO optimized controller performs better in terms of transient and steady state analysis. The bode analysis is presented in Fig. 5 and Table 4. Fig. 4 and Table 4 clearly demonstrate the advantages of ALO FOPID in terms of frequency response. Example 2. The example 2 has been considered from the Ref. [36], where the authors have used ABC algorithm for the designing of FOPID controller. The plant is a 2nd order unstable time delay system. It is presented as:

GðsÞ ¼

400 e0:5s s2 þ 50s

The goal of the optimization is to obtain preferably promising results in terms of settling time, maximum overshoot, gain margin, phase margin etc. The controller is designed by eliminating the delay by using pade approximation and which is given as:

GðsÞ ¼

200s þ 400 s2 þ 50s

The lower bound and upper bound are set 0 and 1, respectively for proportional gain (K p ), integral gain (K i ), derivative gain (K d ), integer order (k) and derivative order (l). During the design of ALO based FOPID controller for 2nd order plant the following parameter values has been considered. Population size = 12, Maximum number of iterations = 100, Number of evaluations = 20. The best parameters of FOPID controller with different objective functions are shown in Table 5. The step response with optimized FOPID controller for the approximated plant is depicted in Fig. 6.

Table 3 Performance of various methods for Example 1. Method

Tr

Ts

Overshoot

Fractional filter PID [35] IMC PID ALO- FOPID-ITAE

– 0.48 0.49

19.6 4.84 1.54

– 0 4.17

Table 4 Bode Analysis of example 1. Algorithms

Phase Margin(deg)

Delay Margin(s)

Fractional filter PID IMC PID ALO-FOPID

35.0428 83.6 96.45

Inf 1.96 Inf

A comparative analysis of proposed ALO-FOPID algorithm with IMC-PID technique is shown in Table 6. The comparative analysis are based on the transient performance, e.g., rise time (Tr ), settling time (Ts ), peak time (Tp ) and maximum overshoot (%Mp ). It is clear from Table 6 that the proposed ALO optimized FOPID controller shows better result as compared IMC-PID method in terms of transient as well as steady state performance. The step response is given in Fig. 7 for IMC-PID and ALO tuned FOPID controller. The frequency response analysis by using bode plot for ALO tuned FOPID for the 2nd order system is shown in Fig. 7. In Table 7, gain margin, phase margin and peak gain are presented for the ALO algorithm. From bode plot, the minimum peak gain, maximum phase margin and bandwidth are obtained. Therefore we conclude that, the ALO algorithm results the best frequency response for the 2nd order plant. In general, as the robustness of the actual delay system is less than the approximated system, we further investigated the robustness of the proposed controller for the actual delay plant by considering input or output channel gain variation of the form (1 ± d). It has been observed from simulation that the integer based compensated system tolerate a gain variation (d = +1.3 to 1.5) whereas the proposed fractional order compensated system tolerate (d=+1.9 to 1.9). Fig. 8 shows the robustness response for d = +1.3. From the above, we conclude that the fractional order compensated system (due to its additional tunable parameters) has more robustness than the integer based compensated system. Example 3. The third order system has been adopted from the reference [18] for example 3. The 3rd order plant is given as:

Table 1 ALO- FOPID controller for various objective functions for example 1. Parameters/Objective functions

IAE

ISE

ITAE

ITSE

KP KI KD k

3.31 0.41 1.13 0.45 0.94

2.81 0.98 1.89 0.30 0.74

3.12 0.28 1.85 0.34 0.31

3.70 0.29 1.83 0.53 0.39

l

Table 2 Performance of various indices. Method

Tr

Ts

Overshoot

Peak

Peak Time

ALO- FOPID-IAE ALO- FOPID-ISE ALO- FOPID-ITAE ALO- FOPID-ITSE

1.30 0.82 0.49 0.58

2.89 2.57 1.54 1.76

0.074 0.009 4.17 3.12

0.98 0.98 1.02 1.01

7.51 8.67 2.23 2.46

Please cite this article as: R. Pradhan, S. K. Majhi, J. K. Pradhan et al., Optimal fractional order PID controller design using Ant Lion Optimizer, Ain Shams Engineering Journal, https://doi.org/10.1016/j.asej.2019.10.005

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Table 5 Performance of various indexes. Parameters/Objective functions

IAE

ISE

ITAE

ITSE

KP KI KD k

0.92383 0.99989 0.69913 0.60115 0.34541

0.9987 0.9995 0.9992 0.5462 0.0011

0.8114 0.8825 0.9999 0.1066 0.1428

0.4777 0.4768 0.6175 0.3782 0.8807

l

Table 6 Performance of various methods for Example 2. Method

Tr

Ts

Overshoot

Peak

Peak Time

IMC-PID ALO-FOPID-IAE ALO-FOPID –ISE ALO-FOPID –ITAE ALO-FOPID –ITSE

1.2623 0.0015 0.0085 0.0028 0.0015

7.7475 1.79 0.161 0.8094 0.1483

11.2263 0.5615 0.3512 0 1.71

1.1123 1.0069 1.0045 1.001 1.0079

3.3015 3.4154 3.1063 6.5359 3.2179

Table 7 Bode Analysis of 2nd order system. Algorithms

Peak Gain(dB)

Phase Margin(deg)

Gain Margin(dB)

IMC-PID ALO-FOPID

797 130

118 101

0 Inf

GðsÞ ¼

0:6s2 þ 3s þ 3:75 0:4s3 þ 1:6s2 þ 3s þ 3:75

The FOPID control parameter optimized by the proposed ALO method for the above example is presented in Table 8. This yield promisingly better result than the classical PID and existing FOPID methods. Fig. 9 and Table 9, clearly presents the advantages of FOPID in terms of transient as well as steady state response. The Fig. 10 and Table 10 presents the bode analysis of the example 3, which clearly states the better result in terms of phase margin and gain margin in comparison with other methods. Similar to previous examples, the parameters used are given as: population size = 100, maximum number of iterations = 100, number of evaluations = 20. The robustness of the third order system has been investigated by considering the parameter variation of the plant. As the zeros of the system limit the performance of the system and LTI controller

Fig. 8. Robustness response of the 2nd order system.

cannot handle it, here, the coefficient of s2 i.e., 0.6 is varied to 0.1. The above yields increase in imaginary part as compared to real part, thereby causing oscillation in the responses. Fig. 11 shows robustness response for the above discussed parameter variation. From the above, we conclude that the fractional order compensated system has more robustness than the integer based compensated system.

Fig. 7. Bode Plot of the 2nd order system.

Please cite this article as: R. Pradhan, S. K. Majhi, J. K. Pradhan et al., Optimal fractional order PID controller design using Ant Lion Optimizer, Ain Shams Engineering Journal, https://doi.org/10.1016/j.asej.2019.10.005

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R. Pradhan et al. / Ain Shams Engineering Journal xxx (xxxx) xxx Table 8 Performance of various indexes. Parameters/Objective functions

IAE

ISE

ITAE

ITSE

KP KI KD k

99.999 99.999 0.0073 0.998 0.9421

99.999 99.999 2.0922 0.9997 0.99937

73.9881 99.9998 6.0 0.9912 0.86635

99.9827 43.0697 4.0886 0.99969 0.72818

l

Table 9 Performance of various methods for Example 3. Method

Tr

Ts

Overshoot

Peak

Peak Time

ZN-PID ALO-FOPID-IAE ALO-FOPID –ISE ALO-FOPID –ITAE ALO-FOPID –ITSE

0.4039 0.0142 0.0012 0.00093 0.0157

3.7074 0.0199 0.0597 0.0884 0.0476

0.3697 0.0887 0.6799 0.4558 0.2253

1.004 1.01 1.0068 1.0046 1.0021

9.8318 1.05 1.1631 1.2858 1.8119

Table 10 Bode Analysis of 3rd order system. Algorithms

Peak Gain(dB)

Phase Margin(deg)

Delay Margin(s)

Gain Margin (dB)

ZN-PID ALO-FOPID

413 140

Inf 89.50

2.14 Inf

7.56 40.45

Fig. 11. Robustness response of the 3rd order system. Fig. 9. Step response of the 3rd order system.

Fig. 10. Bode Plot of the 3rd order system.

Please cite this article as: R. Pradhan, S. K. Majhi, J. K. Pradhan et al., Optimal fractional order PID controller design using Ant Lion Optimizer, Ain Shams Engineering Journal, https://doi.org/10.1016/j.asej.2019.10.005

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5. Conclusion This research paper proposes an ALO based algorithm for tuning FOPID controller by optimizing the defined performance indices of different systems. The systems like first order delay system, second order unstable and delay system and higher order systems are considered for the current study. Form the simulation, it has been observed that the proposed algorithm improves the margin (gain and phase margin) for the delay system; which is usually difficult to obtain due the non-minimum phase behavior of the system. Furthermore, the robustness analysis for the considered systems reveals that the proposed method tolerates more gain variation and yields better robustness characteristics. It has been observed from the simulation study that the proposed method ensures superior transient performance (less overshoot and quick settling time) along with good robustness characteristics. In addition, it maintains the desired phase margin and gain margin. The ALO method performs promisingly better for the tuning of the controller parameters as compared with other methods. The study focuses on the prospective of ALO as a substitute method to look for the feasible better solution for designing the FOPID controllers.

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Please cite this article as: R. Pradhan, S. K. Majhi, J. K. Pradhan et al., Optimal fractional order PID controller design using Ant Lion Optimizer, Ain Shams Engineering Journal, https://doi.org/10.1016/j.asej.2019.10.005

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Dr. Rosy Pradhan is currently working as Assistant Professor in Electrical Engineering, Veer Surendra Sai University of Technology, Burla, Smbalpur, India. She received her BTech from College of Engineering and Technology, Bhubaneswar in 2010 and MTech in Control and Automation from National Institute of Technology, Rourkela, India and PhD in Control System Engineering from VSSUT, Burla, India in the year 2019. Her research area includes fractional order control system and Computational Intelligence, AI etc.

Dr. Jatin Kumar Pradhan is currently working as Assistant Professor in Electrical Engineering, Veer Surendra Sai University of Technology, Burla, Smbalpur, India. He received her BTech from Veer Surendra Sai University of Technology, Burla, Smbalpur, India in 2008 and MTech in Control and Automation from National Institute of Technology, Rourkela, India and PhD from Indian Institute of Technology, Bhubaneswar, India. His research area includes robust control system and multi variable control.

Dr. Santosh Kumar Majhi is currently working as Assistant Professor in Computer Science and Engineering, Veer Surendra Sai University of Technology, Burla, Smbalpur, India. He received her BTech from Veer Surendra Sai University of Technology, Burla, Smbalpur, India, MTech in CSE from Utkal University, Bhubaneswar and PhD in Computer and Information Systems from Sri Sri University, Cuttack, India. He is the recipient of young scintist award in 2016 for his contribution to Cloud Computing Security. His research area includes Computational Intelligence, Cloud Computing, Security Analytics, Data Mining, Soft Computing and AI.

Prof. (Dr.) Bibhuti Bhusan Pati is a Senior Professor in Electrical Engineering at Veer Surendra Sai University of Technology, Burla, Smbalpur, India. He has done his BSc (Engg) from UCE (VSSUT), Burla, India, MSc (Engg) from Indian Institute of Science (IISc) Bangalore, India and PhD from Utkal University, Bhubaneswar, India. He has 28 years of teaching and industrial experience. He is the fellow of Institute of Engineers, India. He has published more than 45 international journal in repute. He has executed six research and consultancy projects. He has guided 7 PhD scholars. His research areas includes Control System, Power System control, AUV Control.

Please cite this article as: R. Pradhan, S. K. Majhi, J. K. Pradhan et al., Optimal fractional order PID controller design using Ant Lion Optimizer, Ain Shams Engineering Journal, https://doi.org/10.1016/j.asej.2019.10.005