Evolutionary Algorithms Based Fixed Order Robust Controller Design and Robustness Performance Analysis

Transcription

1 Proceeding of 01 4th International Conference on Machine Learning and Computing IPCSIT vol. 5 (01) (01) IACSIT Pre, Singapore Evolutionary Algorithm Baed Fixed Order Robut Controller Deign and Robutne Performance Analyi Faizullah Mahar 1, Syed Saad Azhar and Zuhaibuddin Bhutto 3+ 1, Faculty of Engineering Science &Technology, Iqra Univerity, Karachi, Pakitan 3 Balochitan Univerity of Engineering and Technology, Khuzdar, Pakitan Abtract. Evolutionary Algorithm (EA) are tochatic earch technique that direct a population of olution toward bet poible reult by uing the natural principle. In recent year, thee algorithm have grown to be an accepted optimization tool for many area of cientific and engineering reearch, together with control ytem engineering deign. Significant reearch exit concerning evolutionary algorithm to control ytem deign and robutne performance analyi of controller. But, little work ha been done with evolutionary optimization algorithm control becaue of the problem related with robutne performance in early period of the evolution of controller. Moreover, until recently the robutne performance of controller baed on evolutionary algorithm ha not been reearched in tipulate. Keyword: Evolutionary algorithm, optimization, robutne, performance and controller 1. Introduction A uccefully deigned control ytem hould be alway able to maintain tability and performance level in pite of uncertaintie in the ytem. Due to it importance, however, the reearch on robut deign ha been going on all the time. A breakthrough came with the pioneering work by [1] on the theory, now known a the H control theory. A controller that perform uitably in the preence of plant parameter variation i aid to be a robut. The aim in control ytem deign i to deign a controller, which connected to a ytem, provide a deired behaviour of the control ytem. Fixed order robut controller have become intereting area of reearcher becaue of implicity and acceptable controller order []. Regardle of the exiting elegant method of robut control, engineer complain about the gap between theorie and practice in control ytem the deign technique cannot incorporate realitic contraint uch i fixed tructure [3]. Intereting way olving the problem are heuritic algorithm. Motly, the controller ued in indutrial proce are PI or PID controller. Unfortunately, tuning of control parameter of thee controller for achieving both robutne and performance pecification i difficult. To overcome thi problem, the approache to deign a robut control for tructure pecified controller were propoed in [4-5]. EA have emerged a a contetant due to it flexibility effectivene in variety of optimization application. PSO i general purpoe optimizer that olve the wide range of optimization problem thu the PSO can be adapted to variou categorie of optimization. In IA, antigen repreent the problem to be olved and an antibody et i generated where each number repreent a candidate olution [6]. In IA n number of antibodie generated randomly [7]. In thi paper, the performance and robut tability condition of the ytem atifying the H loop haping are formulated a the cot function. EA i.e., Particle Swarm Optimization (PSO) and Immune Algorithm (IA) are adopted to olve thi problem and to achieve the performance of the deigned controller. + Correponding author. Tel.: addre: zuhaib_bhutto@hotmail.com. 137

2 Simulation reult validate that the robutne and performance of the propoed robut controller deign approach. Paper i organized a: ection, preent brief overview of evolutionary algorithm, Loop haping deign i given in ection 3, deign procedure of the propoed cheme i dicued in ection 4, Section 5 preent imulation reult, robutne performance analyi i preented in ection 6 and concluion i placed in ection 7.. Evolutionary Algorithm Optimization ha been eential component of many engineering technology field, a number of approache exit to get an optimal behaviour in a proce of plant. EA have emerged a a contetant due to it flexibility effectivene in variety of optimization application [, 4, and 9]. PSO i population baed optimization technique that ha many advantage over other claical optimization procedure [6, 9]. A population of particle i initialized with random poition and velocitie and a function i evaluated, uing the particle poitional coordinate a input value. Poition and velocitie are adjuted and the function evaluated with the new coordinate at each time tep. When a particle dicover a pattern that i better than any it ha found previouly, it tore the coordinate in a vector. The difference between (the bet point found by o far) and the individual current poition are tochatically added to the current velocity, cauing the trajectory to ocillate around that point [8]. In IA, antigen repreent the problem to be olved and an antibody et i generated where each number repreent a candidate olution. Alo an affinity i the fit of an antibody to the antigen. The role of antibody i to eliminate the antigen. Alo an affinity i the fit of an antibody to the antigen. The role of antibody i to eliminate the antigen [9]. While affinity of all antibodie i known new population i generated through three tep: replacement, cloning and hyper mutation. In replacement tep low antibodie are replaced thoe with highet affinity are elected by cloning and hyper mutation i applied where the mutation rate i inverely proportional to it affinity [10]. 3. Loop Shaping Two weighting function W 1 and W are pecified to hape the original plant. The ingular value of the haped plant atify the cloed-loop performance requirement. Thi organized procedure ha it foundation in [11]. G W 1 G o W The weighting function are choen a W, W I where I, i the identity matrix, the controller K i yntheized (Mat lab code) and final controller K() i contructed by multiplying K with weighting function a hown in Eq.(3) K( ) W K W (3) final 1 4. Fixed Order Robut Controller Deign In the propoed technique, PSO and IA are ued to minimize the cot function. The tranfer function of the identified plant model i given in Eq. (4) e G( ) (4) The tructure K (p) of controller i pecified before tarting the optimization proce. A et of controller parameter p i evaluated to minimize CF. In thi paper, EA (PSO and IA) are adopted to find the optimal value of controller parameter p* in tabilizing controller K (p) uch that T i minimized. wz (1) () 138

3 By uing eq. (3) controller K (p) can be written a: K( p) W K W (5) 1 It i aumed that W 1 and W are invertible, therefore, K W 1K( p) W 1 (6) 1 A controller K which tabilize the original cloed loop ytem and minimize gamma. I inf 1 1 γ tab ( I + GK ) M (7) k K By ubtituting Eq. (6) in Eq. (7), the H -norm of the tranfer matrix from diturbance to tate, which ha to be, minimized i.e. cot function i written a: I T ( I + G W 1K( p)( I G ) (8) zw 1 1 ( ) W1 K p The pecified controller tructure i expreed in Eq. (9) k K ( p ) i k + p (9) 4.1. Propoed cheme uing PSO Step-1 initialize everal et of population parameter p a population of particle, where p i conidered a a vector of controller parameter. Step- Specify the controller tructure, evaluate CF of each particle uing Eq. (8) Step-3 at each generation the velocity of each particle and poition of the next i calculated. Step-4 if current iteration i le maximum iteration then top, go to tep Propoed Scheme by Uing IA Step-1 Generate initial et of parameter p a population of antibodie Step- Specify the controller tructure K (p) where p i conidered for each tring of antibodie, evaluate CF of each antibody uing Eq. (8). Step-3 Bet antibody in the preent problem i choen a an antigen, which ha minimum CF. 5. Simulation Reult The fixed order robut controller deign by uing EA ha been imulated to predict performance of the propoed approach. By uing the LSDP and Mat lab coding the K controller i obtained a: K ( ) (10) By uing the H LSDP the final controller i obtained a: K ( ) (17) The controller obtained by conventional technique Eq. (17) i of 4th order and it tructure i complex and difficult to implement practically. The pecific controller tructure i expreed in Eq. (18). k K ( p ) i k + p (18) 139

4 The imulation wa carried out uing repreentation of particle. The ize of initial population i 10 particle. Algorithm converged in 4th iteration, and gave optimal CF value of Fig. how the plot of convergence of cot function veru iteration of PSO. The optimal olution of controller parameter wa obtained which ha atified tability margin of The computed optimal value of controller parameter are hown in Eq. (19) K( p) The tep repone of the control ytem with optimized controller parameter uing PSO i hown in Fig.1; the tep repone preent rie time 1.5 ec., about % overhoot and the ettling time i about 1.3 ec. (19) Fig. 1: Step repone obtained with PSO Fig. : Convergence of CF V.iteration of PSO The imulation wa carried out uing IA with repreentation of antibodie. The ize of initial population wa 10 antibodie, colonial affinity wa calculated and ingle bit mutation wa ued. IA converged on the 3rd iteration and gave the optimal CF of Fig. 3 how the plot of convergence of cot function veru iteration of IA, which ha atified tability margin of Obtained optimal value of controller parameter are hown in Eq. (0) K ( p) (0) The tep repone of the control ytem with optimized controller parameter by uing IA i hown in fig.4; the tep repone preent rie time 1.06 ec., % overhoot and the ettling time i about ec. the reult obtained clearly how the effectivene of propoed cheme. Fig.3: Convergence of CF V. iteration of IA 140

5 6. Robutne Performance Analyi Fig. 4: Step repone obtained by IA In order to validate the uitability and robutne performance of deigned controller, ome parameter of the nominal plant in Eq. (4) were varied a follow: G Δ ( ) ( e ) The deigned optimal controller Eq. (19) i teted on perturbed plant Eq. (0). The reult hown in fig. 5, demontrate that the deigned controller uing IA have reaonably good performance and robutne. (1) Fig.5: Show robut check of IA controller Tab.1: Comparion between optimized parameter Parameter PSO IA H K p K i CF The CF value and the parameter of the controller optimized uing PSO and IA were compared. Reult hown in Tab.1, indicate that EA gave much better olution than conventional H. Moreover, the bet CF value i obtained from IA a compared to PSO Tab.: Comparion between PSO and IA Parameter PSO IA S t in ec R t in ec %Overhoot.5% % 141

6 Comparion hown in Tab., PSO ha higher ettling time than IA. So tuning PI controller for plant uing IA i more optimal than PSO. The controller optimized with IA ha provided much better repone than controller optimized with PSO. 7. Concluion An appropriated performance that atifying the time domain pecification and robutne i evaluated by PSO and IA, the repone from the deigned controller from propoed EA cheme were compared. Thi propoed technique i an alternative method which directly conider the performance pecification and robutne in the deign and in which the tructure of controller i not retricted to PID. The controller K (p) can be replaced by any fixed-tructure controller and the propoed algorithm can till be applied functionally. 8. Reference [1] G. Zame. Feedback and optimal enitivity: Model reference tranformation multiplicative emi norm and approximate invere, IEEE Tranaction on Automatic Control, Vol.6, pp , 1981 [] F. Mahar and S. Saad Azhar Ali. Evolutionary Algorithm Baed Fixed Order Robut Controller Deign and Robutne Performance Analyi, American journal of cientific reearch, iue 51, February, 01.pp.16-6 [3] F. Mahar and S. Saad Azhar Ali. Deign of Fixed Order Controller by Uing Particle Swarm Optimization and Sytem Simulation. International journal for computer information ytem and indutrial management application, ISSN , Vol. 4 (01) pp [4] F. Mahar, S. Saad Azhar Ali and A. Karim. Deign of fixed order robut controller by uing evolutionary optimization technique: Comparion and Performance Analyi, journal of engineering and applied cience, vol.1, 010, pp [5] A. Smith, and A. Mont, Robut and optimal Control uing polynomial Chao theory, PhD diertation department of electrical engineering, univerity of outh California, 007 [6] F. Mahar and S. Saad Azhar Ali. PSO Baed Fixed Order Controller Deign and Sytem Simulation, (010) International Conference on Soft Computing and Pattern Recognition, vol.1, France, 010, pp [7] F. Mahar and S. Azhar Ali. Immune Algorithm Baed Fixed Order Controller Deign and Sytem Simulation, IEEE International Sympoium on Signal, Sytem and Electronic, Nanjing, China, vol.1, 010, pp.18-5 [8] M. Clerc and J. Kennedt The particle warm exploion, and Convergence in a multi-dimenional Complex pace, IEEE tranaction on Evolutionary Computation, vol.1, 00, pp.58-73, [9] M. Mori and M. Tukiyma. Immune algorithm with earching diverity and IA application to reource allocation Problem, Tranaction on Intrumentation Electronic Engineering, Japan. 1993, pp [10] S. Kyanzadeh, M. Farangi, N. Kwang and Y. Lee. Deign of power ytem tabilizer uing immune algorithm, the 14th International Conference on Intelligent Sytem to Power Sytem application, Kaohiunng Taiwan, vol. 3, 007, pp [11] A. Chritianon and B. Lennarton. Weight election for H control uing Genetic Algorithm, Proceeding of the Triennial World Congre Beijing China, 1999, pp