Fractional Order PID Controller with an Improved Differential Evolution Algorithm

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1 2016 International Conference on Micro-Electronics and Telecommunication Engineering Fractional Order PID Controller with an Improved Differential Evolution Algorithm Rinki Maurya a, Manisha Bhandari b a,b Rajasthan Technical University, Kota, India a rinki.maurya.21@gmail.com b mbhandari@rtu.ac.in Abstract Differential Evolution algorithm has recently emerged as a simple yet very powerful technique for real parameter optimization. This article describes an application of DE for the design of fractional order proportional Integral Derivative controller. FOPID controller parameter are composed of the proportional constant, integral constant, derivative constant, derivative order and integer order, and its design is more complex than that of conventional integer order PID controller. Here the controller synthesis is based on minimising the integral square error of given plant by which a single objective optimization problem achieved. This article proposes a tuning method of fractional order PID controller for a given plant by nature inspired algorithms which is Differential Evolutionary for better dynamic and static performance.. The mutation of DE algorithm is modified by which the optimal response is obtained and also compared with Genetic Algorithm and PSO algorithm. The comparison is done on the basis of step response with characteristics like maximum overshoot (M p%), rise time(t r) and steady state error. Simulated results are represented on matlab 2012(a). Index Terms Fractional Order System, Differential Evolutionary Algorithm, Particle Swarm Optimization, FOMCON toolbox,genetic Algorithm. I. INTRODUCTION In recent years ordinary PID controllers are principally runs in industries,which gives gratifying responses, provide effective control. In the absence of the complete knowledge of the process these types of controllers are the most efficient of choices. Their are three main parameters namely Proportional part (P), Integral part (I) and Derivative part (D)[14]. The proportional part is used to give the desired set-point, the integral part and derivative part are used for the counting of past errors and the rate of change of error in the process respectively. PID tuning by conventional method is not very efficient because of the presence of non-linearities in the plant. Conventional PID system gives a quite high maximum overshoot with large settling time. With the upgrade of fractional calculus, the fractional order PID controllers are preferred because they have five parameters to tune in which two parameters are fractional in nature as compared to conventional PID controller and it converts their resultant graph from two-dimensional (2D) or point to three-dimensional (3D) or plane, hence there are lots of opportunity to blueprint the controllers. Tuning of a PID controller refers to the tuning of its various parameters (P, I and D) to achieve an optimized value of the desired response. The output parameters which are responsible for controlling the system are stability, rise time, peak time and maximum overshoot and steady state error. The tuning of PID controller are done with respect to desire of the plant specification. If control system taken from offline,the step input response are used for tuning of PID controller or FOPID controller. But mostly the large industrial applications have control system must be online where tuning is done manually with very experienced personnel.but there is a possibility of human error. The main focus of this paper is to apply two specific soft-computing techniques viz. Differential Evolutionary and PSO algorithm to design and tuning of FOPID controller to get an output with better dynamic and static performance. The application of Evolutionary algorithm to the PID or FOPID controller makes it give an optimum output by searching for the best set of solutions for the FOPID parameters. The paper also covers the benefits and problems of both methods. The simulation outputs are the MATLAB results obtained for a step input to a DC Motor[12]. In Evolutionary Algorithm (EA) optimum solution is obtained by evolving a population of candidate solution over a number of generations. Differential Evolution Algorithm (DE) is a population based and stochastic search technique, relatively a uncomplicated approach to find an optimum solution. The fractional order PID controllers have modified response in terms of maximum overshoot, rise time and steady state error. But computation of FOPID controller is tough in evaluation as compared to normal PID controller. Fractional order PID controller is a controller in which there are five parameters that are variable in nature. First one is integer order and second one is a differential order known as lambda (λ) and mue (μ). II. MATHEMATICAL BACKGROUND OF FRACTIONAL CALCULUS Differentiation and Integration are usually performed on integer values in which differentiate or integrate any function with many times is done. But after 30th September 1965 with L Hospital rule, Leibniz generates the possibility by developing the method of differentiation and integration to /16 $ IEEE DOI /ICMETE

2 non-integer orders. Analysis of dynamic behaviour of control system is obtained normally by transfer function, which is obtained by taking Laplace transform of fractional function which is same as in integer order[7]. The relationship between input x(t) and output y(t) for general single input single output system is a p d ap y(t) dt ap + a p 1 d ap 1y(t) dt ap 1 d βqx(t) d βq 1x(t) = b q + b q 1 dt βq dt βq a 0 d a0y(t) dt a b 0 d β0x(t) dt β0 (1) By taking Laplace transform of above equation, transfer function of the system is given as follows described by fractional order differential equations.[6][4]. The expansion of PID controller is FOPID controller by using fractional calculus. The orders of integration and differentiation are respectively λ and μ (both positive real numbers, not necessarily integers)[9]. Taking λ =1and μ =1, we will have an integer order PID controller. So we see that the integer order PID controller has three parameters, while the fractional order PID controller has five parameters. A fractional PID controller is given as C(S) =K p + K i s λ + K ds μ (6) Y (s) G(s) = X(s) = b qs βq + b q 1s βq b 0 s β0 a p s αp + a p 1s αp 1 (2) a 0 s α0 The frequency domain analysis of the fractional order continuous system (FOCS)gives an expression for (jω) γ = ω γ (cos π 2 γ + j sin π γ) (3) 2 With the help of mathematics of fractional calculus [8]a Differintegral operator is developed which is a combination of differentiation and integration operation which is useful for any function (f). There are several definitions are given to describe differintegral operator by some researchers like by Riemann-Liouville (Their definition are not used for periodic functions)[4], by Grunwald-Letnikov (Their definition are used when binomial expressions is needed), by Caputo(Thier definiton are used to reveal on initial conditions of individual function), by Hadamard etc. With advancement of fractional terms in ordinary PID controller, fractional order PID has more degree of freedom, good flexible, which gives better dynamical properties. 1. By Grunwald-Letnikov definition of fractional order derivative is given below :- [t a/h] adt u f(t) = lim h u h 0 v=0 where ( 1) v( u v) are binomial coefficient Cv u, where (v =0, 1, 2, 3, 4,...) ( 1) v ( u v ) f(t vh) (4) 2. Fractional order Caputo s definition is given as adt u 1 t f (x) f(t) = dτ (5) Γ(u x) a (t τ) u x+1 III. FRACTIONAL ORDER PROPORTIONAL INTEGRAL DERIVATIVE CONTROLLER The concept of the fractional order PID controllers was proposed by Podlubny in 1997[13]. He also certififed the better response of FOPID controllers, in comparison with classical PID controllers, when it is used for the controlling of fractional order plants. Fractional Order control Systems(FOCS) are Fig. 1: FOPID controller A. Proposed method of tuning of FOPID controller Differential Evolution algorithm is relatively uncomplicated, speedy and population based stochastic search techniques, presented by Price and Storn[11]. Mutation, selection and crossover are three fundamental operators used in Differential Evolution algorithm. At the initial phase, uniformly distributed population is generated randomly. A trial vector is produced by mutation that is used within the crossover to generate offspring and then selection is used to select the best from the offspring and the parent, for the next generation. The initial population X 1,i,0 =(x 1,i,0,x 2,i,0,x 3,i,0..., x D,i,0 ) i = 1, 2, 3..., NP. is randomly generated according to a normal or uniform distribution x low j x j,i,0 x up j for j =1, 2, 3, 4, 5,..D Where, NP= Population size D = Dimension of the problem x low j = Lower limit of j th vector component = Upper limit of jth vector component x up j After initialization, DE enters a loop of evolutionary operations: mutation, crossover and selection. B. Mutation At each generation g, this operation creates mutation vectors v i,g based on the current parent population X 1,i,0 =(x 1,i,0,x 2,i,0,x 3,i,0..., x D,i,0 ) i =1, 2, 3..., NP. the following are different mutation strategies frequently used

3 basically, DE/rand/1 v i,g = X r0,g + F i (X r1,g X r2,g) DE/current-to-rest/1 v i,g = X i,g + F i (X best,g X i,g )+ F i (X r1,g X r2,g) DE/best/1 v i,g = X best,g + F i (X r1,g X r2,g) Where, r 0,r 1,r 2 =,distinct integers uniformly chosen from the set 1,2,...NPı (X r1,g X r2,g) = difference vector to mutate the parent, X best,g = best vector at the current generation F i = the mutation factor which usually ranges on the interval (0,1+). In classical DE algorithms, F i = F is a single parameter which is used to generate all mutation vectors, whereas in others adaptive DE algorithms, each individual vector i is linked with its own mutation factor F i.the above mutation strategies can be generalized by implementing multiple difference vectors other than (X r1,g X r2,g). C. Cross-over After mutation, a binomial crossover operation forms the final trial vector u i,g =(u 1,i,g,u 2,i,g,...u D,i,g ) { uj,i,g = v x i,g+1 = j,i,g...ifrand j (0, 1) CR i orj = j rand x j,i,g,...otherwise. (7) Where, rand j (a, b) = uniform random number on the interval (a, b) and newly generated for each j, j rand = j randiant (1,D)=integer randomly chosen from 1 to D and newly generated for each i, the crossover probability, CR i [0, 1] roughly corresponds to the average fraction of vector components that are inherited from the mutation vector. In classic DE,CR i = CR is a single parameter that is used to generate all trial vectors, while in many adaptive DE algorithms, each individual I is associated with its own crossover probability CR i. D. Selection The selection operation selects the better one from the parent vector x i,g and the trial vector u i,g according to their fitness values f(.).for example, if we have a minimization problem, the selected vector is given by { ui,g,...iff(u x i,g+1 = i,g ) <X i,g (8) x i,g,...otherwise. and used as a parent vector in the next generation. E. Modification in Differential Evolutionary Differential Evolution (DE)[3] is a vector population based and stochastic search optimization algorithm. DE converges faster, finds the global minimum independent to initial parameters, and uses few control parameters. In order to maintain the proper balance between exploration and exploitation in the population a novel strategy algorithm is proposed. In which,a new phases of scaling factor on mutation is introduced into classical DE, which exploits the methodology of search space for a new population without increasing the function evaluation. With the help of experiments over 2 well known optimization problems, it has been shown that Modified DE outperform as compared with classical DE. All the N parameter vectors processes from mutation, recombination then selection. Mutation is used to expand the local search space. we know the the difference vector is weighted by a multiplying factor known F i in basic DE. Therefore, by taking inspiration from reference paper [1], a combined scale factor is developed for to improve the convergence speed. Hence the donor vector is given by v i,g = X i1,g +F (X i1,g X i,g )/2+K(X r2,g X r3,g )/2 (9) where K is the combination coefficient, which has been proven to be effective when it is chosen with a uniform random distribution from [0.2, 0.5] and F = K.F i is a new constant here with F i is a random distribution from [1.5,2]. For find the optimal solution of given plants, a objective function is needed with the control system output integral error square goes minimum.here the Matlab 2012a used for simulating the results of fractional order system and fractional order controllers by adding the FOMCON toolbox in matlab, which is used to simulate fractional order systems. IV. VECTOR REPRESENTATION IN DE The solution space of the given plants is 5-dimensional, the five dimensions beingkp, Ki,Kd,λ,μ. So each parameter vector in DE has 5 components i.e. the j-th population member at g-th generation may be given as: X j,g =(K p,k i,k d,λ,μ) T From the practical consideration of the FOPID controller design, we fixed the following numerical ranges for each parameter: 1 K p 1000, 1 K i 5000, 5000 K d 500 A. Case-study 1 G p1 (s) = 0 λ, μ s s (10)

4 TABLE I: Specification of system with step response for different controllers for case study 1 Step response specifications Modified Differential Evolutionary method By GA [1] By PSO [1] M p(%) % 0.12 % t r 0.307s 1.312s 0.982s Steady state error TABLE II: Specification of system with step response for different controllers for case study 2 Step response specifications Modified Differential Evolutionary method By GA [1] By PSO [1] M p(%) 4.7% 6.31% 3.91 % t r s 0.695s 0.822s Steady state error B. Case-study 2 G p2 (s) = K (Js + b)(ls + R)+K 2 (11) where J=0.01,k=0.01,b=0.1,R=1 and L=0.5 We have tested the proposed method on two specific instances of the design problem. The first one instance involve a fractional order plant. In some cases a real system is better described by such fractional order differential equations and from this consideration, it is important to investigate the controlling mechanism of such systems through FOPID type controllers. The second problem involves the speed control of a DC motor. Where the uncompensated motor rotates at 0.1 rad/sec with 1V input voltage, which is obtained by open-loop response. Since the basic requirement of dc motor is it should be rotate at the desired speed, with the steady-state error (ess) should be less than 1%. and other one is that its settling time is less. In this case, we want it to have a very less settling time. Since a speed faster than the reference may damage the equipment, we want to decrease an overshoot. 1) Parameters of Differential Evolutionary Algorithms: Maximum iteration =20 Population size= 25 Scaling Factor [1.5 2] Combination Coefficient [ ] crossover [ ] After tuning with the modified DE Algorithm the parameter of controllers for both plants is given below:- 1 For plant G p1 (s)= 0.9s s [1]the fractional order PID controller parameter is K p =556.84,K i =1,K d =-5000,λ= and μ = K For plant G p2 (s)= (Js+b)(Ls+R)+K 2 J=0.01,k=0.01,b=0.1,R=1, L=0.5 the parameter of FOPID controller is K p = ,K i =2048.2,K d = ,λ= and μ = The proposed design method has been compared with two state-of-the-art design methods for FOPID controllers based on the binary GA [5],[2] and PSO namely (Particle Swarm Optimizer). The Genetic Algorithm was proposed by Cao. [2] in which iteration is 50 bits binary string for encoding a 5 parameters of the FOPID controller. The fitness-functions in Caos method employ the integral of the squared error are Fig. 2: Step response of plant 1 with FOPID controller typically borrowed from the realm of optimal control [10]. There is no hand tuning of parameters are done in any phase. V. RESULT Figure 3 shows the dynamic response characteristics of the closed loop systems for case study I and II, as specified in Table 1 and 2. Table 4provides the FOPID Controller transfer functions for two test problems as found with modified DE. In above table the maximum overshoot (M p %), rise time (t r s) and steady state error (ess %) for the unit step response of controlled closed loop system with different type of tuning methods. It is noted that for the given common performance criteria on peak overshoot (M P ), rise time (tr) sec, and steady state error the fractional order controller achieves better results than its integer counterpart in general. VI. CONCLUSION An optimization method for designing fractional order PID (FOPID) controllers based on the DE is presented in this paper. Fractional calculus gives novel and higher performance extension for FOPID controllers. However, their is more difficulties in designing of FOPID controllers, because it accounts the derivative order and integral order in comparison with traditional PID controllers. To design the parameters of the FOPID controllers efficiently, the DE/rand/1/bin algorithm is modified by reducing the search space with respect to its scale factor F. The proposed method has been shown to outperform a state-of-the-art version of the PSO algorithm and a binary GA based method especially for the fractional order plants. The proposed scheme of fractional PID controller design will thus

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