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1、 ELSEVIER Journal of Systems Engineering and Electronics Vol. 19, No. 3, 2008, pp.578 583 Available online at ScienceDirect Design of PID controller with incomplete derivation based on differential evolution algorithm* Wu Lianghong1,2, Wang Yaonan2, Zhou Shaowu1 Sz Tan Wen1 1. School of Informatio
2、n and Electric Engineering, Hunan Univ. of Science and Technology, Xiangtan 411201, P. R. China; 2. Coll, of Electric and Information Engineering, Hunan Univ., Changsha 410082, P. R. China (Received November 5, 2006) Abstract： To determine the optimal or near optimal parameters of PID controller wit
3、h incomplete derivation, a novel design method based on differential evolution (DE) algorithm is presented. The controller is called DE-PID controller. To overcome the disadvantages of the integral performance criteria in the frequency domain such as IAE, ISE, and ITSE, a new performance criterion i
4、n the time domain is proposed. The optimization procedures employing the DE algorithm to search the optimal or near optimal PID controller parameters of a control system are demonstrated in detail. Three typical control systems are chosen to test and evaluate the adaptation and robustness of the pro
5、posed DE-PID controller. The simulation results show that the proposed approach has superior features of easy implementation, stable convergence characteristic, and good computational efficiency. Compared with the ZN, GA, and ASA, the proposed design method is indeed more efficient and robust in imp
6、roving the step response of a control system. Keywords： PID controller, incomplete derivation, differential evolution, parameter tuning. that the transient response of system has a greater overshoot. In general, it is often difficult to determine optimal or near optimal PID parameters with the Ziegl
7、er-Nichols method in many industrial plants6. Evolutionary algorithms (EAs) have been received much interests recently and have been applied success- fully to solve the problem of optimal PID controller parameters. P. Wang in Ref. 4 used an advanced ge- netic algorithm to auto-tune classical PID con
8、trollers. L. Wang proposed a GAS A hybrid strategy for design- ing a class of PID controller for non-minimum phase systems in Ref. 5. Particle swarm optimization al- gorithm was used to optimize PID controller parame- ters in Refs. 6 7. In Ref. 8, ant system algorithm was applied to design PID contr
9、oller with incomplete derivation. Chaotic ant swarm was used to tune the PID parameter in Ref. 9. Differential evolution (DE), first introduced by R. Storn and K. Price in 1995, is one of the modern * This project was supported by the National Natural Science Foundation of China (60375001) and the S
10、cientific Research Foundation of Hunan Provincial Education Department (05B016). 1. Introduction During the past decades, the control techniques in the industry have made great advances. Numerous control methods such as fuzzy control, neural network control, expert system, and adaptive control have
11、been studied deeplyt1. Among them, the best known is the proportional-integral-derivative (PID) controller, which has been widely used in the industry because of its simple structure and robust performance in a wide range of operating conditions2. Unfortunately, it has been quite difficult to tune p
12、roperly the gains of PID controllers because many industrial plants are often burdened with problems such as high order, time delays, and nonlinearities3 6. Over the years, some methods have been proposed for the tuning of PID controllers. The first method that used the classical tuning rules was pr
13、oposed by Ziegler and Nichols3. But the drawback of this method is Design of PID controller with incomplete derivation based on differential evolution algorithm 579 heuristic algorithms10. The DE algorithm has gradually become more popular and has been used in many practical cases, mainly because it
14、 has demonstrated good convergence properties and is principally easy to understands11. Because the DE algorithm is an excellent optimization methodology, it may be a promising approach for solving the optimal PID controller parameters problem. In this article, the DE is employed to design the PID c
15、ontroller with incomplete derivation. The controller is called DE-PID controller. The integral performance criteria in frequency domain were often used to evaluate the performance of the controller, but these criteria have their own advantages and disadvantages. In Ref. 4, a simple performance crite
16、rion in time domain was proposed. However, the performance criterion will be invalidated when the system step response has not overshoot. In this article, a new simple performance criterion in time domain is introduced for evaluating the performance of a PID controller. 2. PID controller with incomp
17、lete derivation In general, derivation control can improve the dynamic behavior of a control system, but a pure derivative cannot and should not be implemented, because it will give a very large amplification of measurement noise. To overcome this drawback, a low-pass filter is often added to deriva
18、tion term. The derivation control with a low-pass filter is called the incomplete derivation control. PID controller with incomplete derivation has better control performances compared with general PID controller, and this PID controller is adopted in this article. The transfer function of the PID c
19、ontroller with incomplete derivation is expressed as KpTds + T f sJ E(s) = U + U + U a i s ) where Kp is the proportional gain, Ti is the integral time constant, Td is the derivative time constant. In the discrete-time domain, the controller can be described as follows uk) upk) + Ui(k) + Udk)= k Kpe
20、(k) Ki2 eU) + ud(k) (2) j=i where Ki = KpT/Ti and T is the sampling period. can be derived as follows. From Eq. (1) we can see that Ud(s) KpTds E ( s ) 1 + T/S The differential equation of Eq. (3) is as follows ud(t) +T/ dud(t) dt KpTd de(t) dt (3) In the discrete-time domain, Eq. (4) can be describ
21、ed as udk) Kd(l - A)e(A;) - e(k - 1)+ Xud(k - 1) (5) where A = T f / ( T f + T) CR and j randn(i) (14) where j = 1, 2 , . . . , rand(j) G 0,1 is the jth evaluation of a uniform random generator number. CR G 0,1 is the crossover probability constant, which has to be determined previously by the user.
22、 randn(i) G ( 1 , 2 , . . . , D) is a randomly chosen index which ensures that u1 gets at least one element from . Otherwise, no new parent vector would be produced and the population would not be altered. 4.3 Selection DE adapts greedy selection strategy. If and only if, the trial vector uf5 1 * yi
23、elds a better fitness function value than then is set to . Otherwise, the old value xf is retained. In this article the minimization optimization is considered. The selection operator is as follows. G+l ,G+1 /(f+1) f ? ) (15) 5. Optimization procedures for PID controller parameters 5.1 Parameters se
24、arching space The parameters searching space of DE is extended on the base of results obtained by Ziegler-Nichols (ZN) method, which not only take use of the reasonable kernel of ZN method but also reduce the parameter searching space. If the optimization result is close to the boundary of the searc
25、hing space, the searching space should be further extended based on the result. The range of the controller parameters is determined by the following strategies. (l- a ) T v ； (l + a)Kp(16) (l- a ) T / T ; ( l + a ) (17) (l- a ) T T d ( l + a ) Z (18) where the Krp T/, Trd are the tuning result usin
26、g ZN method, a is set in the rang of 0 to 1. 5.2 Optimization procedures of DE-PID The searching procedures of the DE-PID controller with incomplete derivation are shown as below. Step 1 Specify the number of population NP, difference vector scale factor F, crossover probability constant CR, and the
27、 maximum number of generations. Initialize randomly the individuals of the population and the trial vector in the given searching space. Step 2 Use each individual as the PID controller parameters and calculate the values of the four performance criteria of the system unit step response in the time
28、domain, namely Ess, tr and ts. Step 3 Calculate the fitness value of each individual in the population using the performance criterion function given by (11) and (12) for non-minimum phase system. Step 4 Compare the fitness value of each individual and get the best fitness and best individual. Step
29、5 Generate a mutant vector according to (13) for each individual. Step 6 According to (14), do the crossover operation and yield a trial vector. Step 7 Do the selection operation in terms of (15) and generate a new population. Step 8 G G + l , return to Step 2 until to the maximum number of generati
30、ons. 6. Simulation research Three typical control systems are chosen to evaluate the performance of the proposed DE-PID controller. The transfer functions of the plants are given as follows8. Case 1 (Three-order system) Gi(s) = (19) s(s2 +1105 + 6 068) v Case 2 (Time-delay system) G2(s) = rS (20) Ca
31、se 3 (Non-minimum phase system) G养器 （ 21) To verify the performance of theproposed DE- PID controller with incomplete derivation, three existing PID controllers, including ZN-PID, GA-PID, and ASA-PID, are compared with the controller. The three PID controllers are introduced in Ref. 8. 582 Wu Liangh
32、ong, Wang Yaonan, Zhou Shaowu 8z Tan Wen The following simulation parameters are used in this article. The member of each individual is Kp Ti: and Population size NP = 30. Scale factor F = 0.5, crossover probability con- stant Ci?二 0 丄 The maximum number of generations Maxitera- tion=400. The sampli
33、ng time T 0.01 s for Case 1, and T = 0.1 s for Case 2 and Case 3. The statistical results of 20 independent runs for each case are shown in Tables 1 3. Figs. 1 3 are the unit step response of Case 1 to Case 3. Table 1 Statistical results for Case 1 PID controller ZN-PID GA-PID ASA-PID DE-PID Kp 33.1
34、04 25.765 15.380 15.794 Ti 0.301 8 1.759 5 2.033 6 2.854 7 Td 0.072 4 0.027 2 0.013 5 0.017 3 Mp/% 17.21 5.23 2.04 2.0 0.06 0.07 0.13 0.11 ts/s 0.305 2 0.602 5 0.180 2 0.13 Ess 0 0 0 0 Table 2 Statistical results for Case 2 PID controller ZN-PID GA-PID ASA-PID DE-PID Kp 2.121 5 1.856 1 1.222 5 1.428
35、 6 Ti 2.624 5 7.511 5 6.018 6 4.835 9 Td 0.629 9 0.525 9 0.319 9 0.304 0 Mp/% 52.69 11.98 5.65 2.06 2.3 2.5 3.6 2.9 ts/s 8.368 4 7.726 7 7.509 1 3.4 Ess 0 0 0 0 Table 3 Statistical results for Case 3 PID controller ZN-PID GA-PID ASA-PID DE-PID Kp 1.550 5 1.244 5 1.234 4 1.185 9 Ti 3.343 0 2.703 7 2.
36、703 5 2.284 3 Td 0.802 3 0.995 8 0.990 0 0.862 Mp/% 18.11 6.48 6.19 2.10 Mu/% 10.16 9.16 8.85 8.16 tr/s 2.8 3.2 3.3 3.7 ts/s 11.315 5.727 7 5.759 4 4.4 Ess 0 0 0 0 Fig. 2 The unit step response of Case 2 Fig. 3 The unit step response of Case 3 From Tables 1 3, we can see that for each of the three s
37、tudy cases, the performance of the proposed DE-PID controller is much better than that of the ZN-PID controller and GA-PID controller and better than that of ASA-PID controller. As can be seen from the tables and figures, for Case 1 and Case 2, the overshoot Mp and the setting time ts of the DE- PID
38、 controller are better than those of the other three PID controllers, for Case 3, the ts and the undershoot Mu of the DE-PID controller are also better than those of the other three PID controllers. Design of PID controller with incomplete derivation based on differential evolution algorithm 583 7.
39、Conclusions This article presents a novel and intelligent design method for tuning the parameters of the PID controller with incomplete derivation using differential evolution algorithm. The proposed method integrates the DE algorithm with the new time-domain performance criterion into a DE-PID cont
40、roller. Three typical systems are used to evaluate the performance of DE-PID controller. The simulation results show that the DE algorithm can perform an efficient search for the optimal PID controller parameters. Compared with ZN-PID controller, GA-PID controller, and ASA-PID controller, the propos
41、ed DE-PID controller has better dynamic performances and robust stability of unit step response. The DE algorithm is easy to be understood and realized and it is very efficient and robust for complex function optimization; therefore, the DE-PID controller can be used in practice engineering widely.
42、Different DE control parameters are required for solving different practice problems, such as difference vector scale factor F and crossover probability constant CR. Hence, how to select proper parameters for the target problem is an important focus of our future studies. References 1 Wang Y N. Inte
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47、rm. Chinese Journal of Scientific Instrument, 2006, 27(9): 1104 1106. 10 Storn R, Price K. Differential evolution a simple and efficient adaptive scheme for global optimization over continuous spaces. International Computer Science Institute, Berkley, 1995. 11 Wu L H, Wang Y N, Yuan X F, et al. Diff
48、erential evolution algorithm with adaptive second mutation. Control and Decision, 2006, 21(8): 898 902. Wu Lianghong was born in 1978. He is currently a lecturer at the School of Information and Electric Engineering, Hunan University of Science and Technology, and a Ph. D. candidate of the College o
49、f Electric and Information Engineering, Hunan University. His research interests are intelligent control and computation intelligent. E-mail: Wang Yaonan was born in 1957. He is currently a professor and Ph. D. supervisor at Hunan University. His research interests are intelligent control, computation intelligent, and image processing. Zhou