Comparative Studies on Decentralized Multiloop PID Controller Design Using Evolutionary Algorithms

Transcription

1 Comparative Studie on Decentralized Multiloop PID Controller Deign Uing Evolutionary Algorithm Sayan Saha, Saptarhi Da, Anindya Pakhira, Sumit Mukherjee, Indranil Pan Abtract Decentralized PID controller have been deigned in thi paper for imultaneou tracking of individual proce variable in multivariable ytem under tep reference input. The controller deign framework take into account the minimization of a weighted um of Integral of Time multiplied Squared Error (ITSE) Integral of Squared Controller Output (ISCO) o a to balance the overall tracking error for the proce variable required variation in the correponding manipulated variable. Decentralized PID gain are tuned uing three popular Evolutionary Algorithm (EA) viz. Genetic Algorithm (GA), Evolutionary Strategy (ES) Cultural Algorithm (CA). Credible imulation comparion have been reported for four benchmark 2 2 multivariable procee. Index Term-Cultural Algorithm; Evolutionary Strategy; Genetic Algorithm; multivariable proce control; PID controller I I. INTRODUCTION N mot of the indutrial proce control ytem, generally few manipulated variable are adjuted to control a number of controlled variable. Such control ytem are known a multi-variable procee more commonly termed a multiple-input multiple-output (MIMO) ytem [1]. In multivariable proce control, unlike ingle-input, ingleoutput (SISO) ytem, change of any ingle manipulated variable affect more than one controlled variable, giving rie to the loop interaction [2]. The implet MIMO procee which are often referred in multivariable proce control literature conit of two input two output (TITO) [3]. Conventional method of deigning PID type controller for uch MIMO procee require correct pairing of one manipulated variable to one controlled variable, to avoid poor controller performance reduced tability margin, which can be achieved by mean of Relative Gain Array (RGA) approach. Other improved meaure of loop interaction, neceary ufficient condition for pairing, control tructure election etc. have been thoroughly reviewed in [2], though the RGA baed loop interaction analyi till dominate the proce control indutrie. Only teady-tate information about the proce gain matrix i required to develop a RGA, which provide a meaure of proce S. Saha A. Pakhira are with the Department of Intrumentation Electronic Engineering, Jadavpur Univerity, Salt-Lake Campu, LB-8, Sector 3, Kolkata , India. S. Da, S. Mukherjee, I. Pan are with the Department of Power Engineering, Jadavpur Univerity, Salt-Lake Campu, LB-8, Sector 3, Kolkata , India. ( aptarhi@pe.jul.ac.in). interaction between the manipulated variable controlled variable. The mot effective pairing can be achieved if a manipulated variable i ued to monitor a controlled variable with which it meaure of interaction i highet, preferably, cloe to unity. Thi allow pairing of a ingle controlled variable with a ingle manipulated variable via a feedback controller for two uch loop of TITO proce a in Fig. 1. Thu in decentralized PID control of 2 2 MIMO ytem, the control ytem conit of two uch controller. Each of them take care of a ingle loop only the interaction between the two loop i greatly reduced unlike the centralized PID control where imilar 2 2 controller tructure i aumed to tabilize a TITO proce [4]-[5]. However, the decentralized controller deign can be eaily applied if the loop do not heavily interact with each other i.e. the correponding RGA hould have dominating principal diagonal. If the loop interaction change the proce gain of the individual loop coniderably, then well-tuned controller for the individual loop fail to keep the controlled variable at their repective et-point. In uch cae, pairing of any manipulated variable with any controlled variable reult in poor controller performance. One method to overcome thi problem, a attempted in thi paper, i to tune both the loop imultaneouly intead of tuning decoupled loop individually. Thi i becaue in the later cae when one loop i being tuned, the controlled variable of the other loop i not traced. Thu it i likely to deviate from the et-point in cae of large loop interaction. One of the conventional method of tuning SISO control loop in frequency domain i uer-pecified gain phaemargin (GPM) method. However, in cae of MIMO ytem, the number of control loop i two or more, reulting in a number of pair of gain-margin phae-margin i.e. one pair for each loop, calculated by opening that individual loop only. Thu, the preence of multiple loop make it impoible to correctly aign a particular gain-margin phae-margin to a MIMO ytem hence tuning of MIMO ytem in frequency domain become very difficult. Huang et al. [6] have propoed a methodology to decompoe MIMO procee a everal effective open loop procee applied the GPM method for thoe individual loop. Thi method may be ued to monitor the peed of repone for each loop of the MIMO ytem. However thi doe not take into conideration the required variation in the manipulated variable or control ignal which i required in order to limit the actuator ize variation of the manipulated variable. The time domain integral performance index baed PID controller tuning for

2 MIMO procee, a attempted in thi paper wa firt introduced by Zhuang Atherton [3] wherea the preent paper improve thi technique by taking both loop error index (ITSE) required controller effort (ISCO) imultaneouly into conideration. In cae of optimal tuning of PID controller to hle MIMO ytem, if each of the controlled variable i enforced to track a et-point uing ome optimization technique, while ignoring the variation in the other controlled variable, it become almot impoible to track both the controlled variable to deired et-point with the tuned controller parameter. In thi paper, multivariable PID controller are deigned by imultaneou optimal tuning of all the control loop for 2 2 multivariable benchmark procee with high loop interaction viz. Berry (WB), Vinante Luyben (VL), Wardle (WW), Ogunnaike Ray (OR) [6]. Optimum tuning i attempted uing three well known evolutionary algorithm viz. Genetic Algorithm (GA), Evolutionary Strategy (ES) Cultural Algorithm (CA) for imultaneou tracking of all the controlled variable around deired et-point, implemented a unit tep reference input. Figure 1. Schematic of the decentralized PID control tructure for benchmark TITO procee. Similar computational intelligent optimization baed attempt have been made in few contemporary literature for the tuning of multi-loop PID controller e.g. Chang et al. [7] propoed an on-line cheme for PID controller deign for multivariable procee, uing auto-tuning neuron employing hyperbolic tangent activation function. Iruthayarajan Bakar [8] Chang [9] ued evolutionary algorithm multi-croover genetic algorithm to minimize the ummed integrated abolute error (IAE) for each loop while tuning the PID controller parameter. Rajabioun et al. [10] deigned a decentralized PID controller by minimizing total IAE for all loop uing colonial competitive algorithm. Han et al. [11] tuned PID controller baed on a cloed loop particle warm optimizer (PSO) algorithm. Zhao et al. [12] minimized integral quare error (ISE) employing two-lbet baed PSO for deigning robut PID controller for MIMO ytem. Thi paper put forward a new methodology of tuning MIMO control loop taking et-point tracking controller effort both into conideration comparion i made between three different EA baed decentralized PID controller tuning. Ret of the paper i organized a follow. Section II decribe the baic of time domain tuning of PID controller to hle MIMO procee. Brief decription of three EA, ued for controller tuning i preented in Section III. Comparion of control performance for the benchmark TITO procee are outlined in Section IV. The paper end with the concluion a Section V, followed by the reference. II. TIME-DOMAIN OPTIMUM PID CONTROLLER DESIGN FOR MIMO PROCESSES A. Propoed Approach of PID Controller Tuning for Multivariable Procee The controller for each loop ha been conidered a the PID type in parallel tructure ince PID controller till dominate proce indutrie due to their implicity, robutne eae of implementation. The PID controller parameter K, K, K for each loop of the multi-variable proce i { p i d } tuned in an optimal fahion o a to keep each of the controlled variable at their et-point, irrepective of any change in et-point of other controlled variable. In many proce control application, large variation of manipulated variable are not allowed to keep the phyical train of the final control element within limit. The control action i hence limited to minimize the deviation of manipulated variable. Evolutionary algorithm are ued to minimize the objective function which take into account both the deviation of the controller output that of the controlled variable, given by: 2 2 J = w1 t e ( t) + w2 u ( t) dt 0 The firt term in the above expreion correpond to the ITSE which minimize the overhoot ettling time, wherea the econd term denote the ISCO. The two weight w, w balance the impact of control loop error (ocillation { } 1 2 /or luggihne) control ignal (larger actuator ize chance of integral wind-up) both have been choen to be unity in the preent imulation tudy, indicating ame penalty for large magnitude ITSE ISCO. Evolutionary Algorithm (EA) have now been employed to obtain optimum PID controller parameter within the range of K, K, K 6, 6 to minimize the objective function { p i d } [ ] (1). In the propoed approach the tranfer function matrix of any proce need not necearily be a quare one, a criterion that mut be met if de-coupler are to be ued for controlling the multivariable proce. However, ince mot of the claical tet-bench problem have 2 2 tranfer function matrice, the imulation tudy ha been limited to 2 2 multi-variable procee only in thi paper. The ame principle can be ued to tune higher dimenional MIMO procee. However, in uch cae the time required for uch an optimization with large number of deciion variable, repreenting the controller parameter will alo increae. Since the propoed multi-loop controller tuning methodology i offline, thu computational complexity required time for the EA to converge i not a major concern. B. Multi-loop Tet-bench Procee In order to tet the effectivene of the propoed tuning methodology, four different 2 2 multi-variable procee, normally encountered in proce control application, have been conidered a a tet-bench [6]. The tranfer function matrice for Berry, Vinante Luyben, Wardle, Ogunnaike Ray MIMO procee are given by equation (2)-(5) repectively: (1)

3 3 12.8e 18.9e (2) GWB ( ) = e 19.4e e 1.3e 7 7 (3) GVL ( ) = e 4.3e e 0.101e 60 1 ( 48 1)( 45 1) GWW ( ) (4) = e 0.12e e 11.64e (5) GOR ( ) = e 5.8e A previouly mentioned, proper pairing of manipulated variable with the controlled variable i required to minimize the effect of loop interaction a much a poible for the deign of decentralized controller for multivariable procee. The common criterion, ued to obtain the knowledge of correct pairing i the relative gain array or RGA, derived from G 0 of the teady tate proce tranfer matrix the dc gain ( ) a: p ( ) ( ) 1 RGA = G 0 ( G 0 ) T p p (6) where, denote the element-by-element multiplication of the matrice. The RGA for the above four tet-bench procee are: RGA WB = (7) RGA VL = (8) RGA WW = (9) RGA OR = (10) It i oberved from the RGA value of the tet-bench TITO procee that the procee uffer from high loop interaction. Hence deigning decentralized controller by pairing of any manipulated variable with any controlled variable will not lead to a atifactory performance. In thi paper, we utilized imultaneou tuning of both the PID controller at a time for conidering the effect of loop interaction in the tuning phae while minimizing the objective function (1) for all loop, intead of tuning one controller a in a SISO loop. III. BRIEF DESCRIPTION OF THE OPTIMIZATION ALGORITHMS USED IN THE PRESENT CONTROLLER TUNING PROBLEM In MIMO procee, it i often oberved that one loop i table while the other one i untable when all other loop in the ytem except the concerned one are opened. However, the MIMO ytem a a whole may be table when all the loop are cloed. It i alo een that while one of the individual loop involve a poitive proce gain, ome other loop ha negative proce gain. Alo, ome loop exhibit a leading nature while ret of the loop may have lagging characteritic. To tabilize a MIMO proce, where co-exitence combination of uch different dynamical behavior are inevitable, it i important to avoid local minima while earching for optimum controller parameter. To reolve thi problem evolutionary algorithm are ued in thi tudy, ince they are derivative free global optimizer, capable of avoiding local minima. We have developed in-houe MATLAB code to implement Genetic Algorithm, Evolutionary Strategie Cultural Algorithm a three repreentative cae of evolutionary algorithm [13]. A. Genetic Algorithm Genetic Algorithm i a tochatic optimization technique baed on Darwin theory of natural election. In GA, an initial et of probable olution, viewed a a pool of chromoome, i evolved over ucceive generation uing the procee of election, croover mutation, to arrive at an optimum. A fitne function i defined which meaure the cloene of a chromoome to the required optimum. In each generation the fitne of each individual chromoome in the population i meaured a et of mot fit chromoome i elected, baed on ome well-defined criteria. Thi i termed a election. The elected chromoome are then treated a parent for the reproduction of the next generation. A fraction of the parent undergo croover, where genetic information between two chromoome i blended. The remaining fraction undergoe rom mutation. The croover repreent traveral of the earch pace for optimum olution. Mutation i ued to prevent pre-mature convergence trapping into local minima/maxima. In thi way, a new generation i formed, the proce i repeated. In certain GA variant, a mall et of bet fit individual i directly put into the population of the next generation. Thi i termed elitim. Real GA, where each chromoome i repreented a a vector of real value, poe a a very viable variant of GA i pecifically uited for olving optimization problem employing large continuou earch pace ha been ued in the preent multivariable PID controller tuning problem. B. Evolutionary Strategie Among everal variant of ES, elf-adapting ES i ued in thi tudy. Evolutionary trategy begin with an initial et of population coniting of real valued vector termed a individual, each one repreenting the genetic characteritic of the deciion vector to be optimized. Evolution of the initial population formed by romly generated individual, called parent, conit of evolving of the genetic characteritic controlled by the trategy parameter (σ ) which i alo evolved dynamically, depending upon the performance. Strategy parameter of each individual i et to a common value of 0.5 initially. Multi-membered ES, denoted by ( µ ρ + λ ) i ued for optimization. For the purpoe of determining the optimized value of the deciion vector, number of parent ( µ ) number of off-pring ( λ ) generated in one generation are fixed to 30. In each generation two parent ( ρ = 2 ) were elected at rom recombined by a romized real-valued croover operator to generate an

4 offpring. Evolution of trategy parameter i done following the modified 1/5th update rule, reported in Greenwood Zhu [14] according to which, σ i updated after every n = 9 trial depending on the number of ucceful mutation. Mutation of each offpring i then done by adding the zeromean Gauian variable with tard deviation (σ ). Selection to form the new population i done by chooing the bet ( µ ) individual out of the pool of the total ( µ + λ ) individual compriing of all the parent offpring depending upon their fitne value. Thi whole proce i repeated until the required value of fitne i reached or the maximum number of generation i exceeded. C. Cultural Algorithm Cultural Algorithm ha been developed by modeling how human culture work. Culture i viewed a a vehicle for toring relevant information gathered ince the tart of the culture, i available to all the ubequent generation of the ociety. Thi information can be ueful for the generation to guide their problem olving activitie, at the ame time being dynamically modified by new information gathered by each new generation. The CA i modeled uing two eparate information pace, viz. the population pace the belief pace. The population pace contain the et of poible olution to the problem available in the preent generation. The belief pace model the actual cultural apect. It tore information related to the problem olution that ha been found till the preent generation in turn influence the evolution of the population pace in ubequent generation. Communication between the two pace i hled by a protocol coniting of two function: an acceptance function, which elect the et of individual that will influence the belief pace an influence function which influence the creation of the next generation. In addition the belief pace require an update function which i baically reponible for updating the belief pace when required. The belief pace i compoed of a few knowledge ource viz. normative knowledge, topographical knowledge, ituational knowledge hitory knowledge. In the preent cae, a variation of the CA ha been ued, where the evolutionary apect i hled by a Genetic Algorithm [15]. In thi Genetic Algorithm baed Cultural Algorithm (GACA), the acceptance function accept the bet 25% of the population uing tochatic uniform ampling, to be reponible for updating the belief pace. Belief pace i compoed of normative ituational component. The normative knowledge component i compoed of the upper lower bound of each of the variable among the individual accepted. The ituational knowledge i a et of the bet or elite individual. At the end of each generation the normative knowledge i updated with the bound of the accepted individual the ituational component i updated if neceary. Mutation in the Genetic Algorithm part i influenced by the belief pace. The direction of mutation i determined by the poition of current individual with repect to individual preent in the ituational knowledge pace. Mutation i directed toward the bet individual. Alo, mutation range i.e. the maximum range through which the individual can be mutated, i determined by the width of the normative knowledge component for each of the variable of an individual olution. Finally, if an individual after mutation or croover violate the normative knowledge, it i forced back into the earch pace dictated by the normative knowledge. IV. SIMULATION AND RESULTS The above dicued three EA have now been applied to tune decentralized PID controller parameter (Fig. 1) for each of the tet-bench TITO procee (2)-(5). The et-point tracking performance required variation in manipulated variable of each loop have been hown in Fig. 2-9 repectively. Table I-II report the bet found optimum controller parameter for the two loop of each TITO proce uing GA, ES CA. It can be eaily inferred from figure preented that all the three algorithm viz. GA, ES CA perform atifactorily in imultaneou tuning of both the PID controller for the multivariable procee with high loop interaction. From the reult preented in Table III it i evident that though performance of all the three algorithm are almot ame, CA outperform the other two by ome margin in all the cae. Figure 2. Variation in proce variable for WB MIMO proce. Figure 3. Variation in manipulated variable for WB MIMO proce. It i to be noted that variation in the reult obtained from five independent run of each EA i minimum for CA. Some of the obtained controller gain are negative which i due to the fact that the MIMO proce ha negative tranfer function in ome loop. The tuned repone with the controller, uing the propoed methodology, i omewhat proce dependent. From Fig. 2-3, it can be een that the WB proce ha light overhoot in tracking of proce variable ocillation in manipulated variable. From Fig. 4-5 it i oberved that the

5 VL proce give luggih output in one loop light overhoot in the other. For the WW proce the ettling time i very large hence the control ignal ha not ettled during the finite imulation time horizon (Fig. 6-7). From Fig. 8-9 it can be inferred that the OR proce give fat et-point tracking for both the loop. Figure 6. Variation in proce variable for WW MIMO proce. Figure 4. Variation in proce variable for VL MIMO proce. Figure 7. Variation in manipulated variable for WW MIMO proce. TABLE II. TUNED CONTROLLER PARAMETERS FOR THE SECOND LOOP Proce Algorithm K p2 K i2 K d2 Figure 5. Variation in manipulated variable for VL MIMO proce. TABLE I. TUNED CONTROLLER PARAMETERS FOR THE FIRST LOOP Proce Algorithm K p1 K i1 K d1 Berry Vinante Luyben Wardle Ogunnaik e Ray GA ES CA GA ES CA GA ES CA GA ES CA Berry Vinante Luyben Wardle Ogunnaik e Ray GA ES CA GA ES CA GA ES CA GA ES CA

6 Strategie Cultural Algorithm for each of the four tetbench TITO procee. It i oberved that for each cae, though all algorithm lead to table controller, CA wa the mot effective amongt the three EA. Thi i reflected by the tard deviation mean of the cot function, confirming the effectivene of CA. Simple GA i oberved to be the mot ineffective among the three EA. Figure 8. Variation in proce variable for OR MIMO proce. Figure 9. Variation in proce variable for OR MIMO proce. TABLE III. CONSISTENCY OF THE EAS FOR MULTI-LOOP PID CONTROLLER DESIGN Proce Algorithm J min (mean) J min (tard deviation) Berry Vinante Luyben Wardle Ogunnaike Ray GA ES CA GA ES CA GA ES CA GA ES CA REFERENCES [1] Dale E. Seborg, Thoma F. Edgar Duncan A. Mellichamp, Proce dynamic control, John Wiley & Son, New Delhi, [2] Qing-Guo Wang, Zhen Ye, Wen-Jian Cai, Chang-Chieh Hang, PID control for multivariable procee, Springer-Verlag, [3] M. Zhuang D.P. Atherton, PID controller deign for a TITO ytem, IEE Proceeding-Control Theory Application, vol. 141, no. 2, pp , March [4] Saeed Tavakoli, Ian Griffin, Peter J. Fleming, Tuning of decentralied PI (PID) controller for TITO procee, Control Engineering Practice, vol. 14, no. 9, pp , Sept [5] Ferno Morilla, Francico Vazquez, Juan Garrido, Centralied PID control by decoupling for TITO procee, IEEE International Conference on Emerging Technologie Factory Automation, ETFA 2008, pp , Sept. 2008, Hamburg. [6] Hiao-Ping Huang, Jyh-Cheng Jeng, Chih-Hung Chiang, Wen Pan, A direct method for multi-loop PI/PID controller deign, Journal of Proce Control, vol. 13, no. 8, pp , Dec [7] Wei-Der Chang, Rey-Chue Hwang, Jer-Guang Hieh, A multivariable on-line adaptive PID controller uing auto-tuning neuron, Engineering Application of Artificial Intelligence, vol. 16, no. 1, pp , Feb [8] M. Willjuice Iruthayarajan S. Bakar, Evolutionary algorithm baed deign of multivariable PID controller, Expert Sytem with Application, vol. 36, no. 5, pp , July [9] Wei-Der Chang, A multi-croover genetic approach to multivariable PID controller tuning, Expert Sytem with Application, vol. 33, no. 3, pp , Oct [10] Ramin Rajabioun, Farzad Hahemzadeh, Emaeil Atahpaz-Gargari, Bahman Megari, Farzad R. Salmai, Decentralized PID controller deign for a MIMO evaporator baed on colonial competitive algorithm, Proceeding of the 17 th World Congre, The International Federation of Automatic Control, July 2008, Seoul, Korea. [11] Kai Han, Jun Zhao, Zu-hua Xu, Ji-xin Qian, A cloed-loop particle warm optimizer for multivariable proce controller deign, Journal of Zhejiang Univerity-Science A, vol. 9, no. 8, pp , [12] S.-Z. Zhao, M. Willjuice Iruthayarajan, S. Bakar, P.N. Suganthan, Multi-objective robut PID controller tuning uing two lbet multiobjective particle warm optimization, Information Science, vol. 181, no. 16, pp , Augut [13] Andrie P. Engelbrecht, Computational intelligence: an introduction, John Wiley & Son, [14] Garrion W. Greenwood Qiji J. Zhu, Convergence in evolutionary program with elf-adaptation, Evolutionary Computation, vol. 9, no. 2, pp , [15] Zhengui Xue Yinan Guo, Improved cultural algorithm baed on genetic algorithm, IEEE International Conference on Integration Technology, ICIT 07, pp , March 2007, Shenzhen. V. CONCLUSION In thi paper, a new trategy wa propoed for the decentralized PID controller tuning for TITO procee. The optimization baed controller parameter election ue an objective function a a weighted um of ITSE ISCO. The minimization problem wa olved uing three popular global optimization technique viz. Genetic Algorithm, Evolutionary