Research Article The Design of Frequency Filters of Iterative Feedback Tuning Using Particle Swarm Optimization

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1 Advances n Electrcal Engneerng, Artcle ID 46527, 5 pages Research Artcle The Desgn of Frequency Flters of Iteratve Feedback Tunng Usng Partcle Swarm Optmzaton Arman Sharf Department of Electrcal Engneerng, Islamc Azad Unversty, Kermanshah Branch, Kermanshah , Iran Correspondence should be addressed to Arman Sharf; arman85sh@gmal.com Receved 9 May 24; Revsed 7 August 24; Accepted August 24; Publshed 25 August 24 Academc Edtor: Ana Alejos Copyrght 24 Arman Sharf. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. Iteratve feedback tunng (IFT) s a data-based tunng approach that mnmzes a quadratc performance ndex usng some closedloop expermental data. A control weghtng coeffcent, known as lambda, and two frequency flters are the most mportant parameters whch can sgnfcantly mprove the performance of the method. One of the major problems n IFT s tunng these parameters. Ths paper presents a new approach to tune frequency flters usng partcle swarm optmzaton (PSO). At the end, the performance of the proposed method s evaluated by two case study smulatons.. Introducton Iteratve feedback tunng (IFT) s a data-based method for the tunng of controllers wth restrcted complexty whch was proposed by Hjalmarsson et al. [] n 994. In ths method, the problem of model bas could be avoded by replacng the nformaton carred by the model wth nformaton acheved drectly from the system tself. Ths leads to an teratve method where the controller parameters are successvely updated usng nformaton from closed-loop experments wth the most recent controller n the loop. IFT has proved to be very effectve n practce and s now greatly used n process control [2]. Snce 994 many experences have been acheved by IFT algorthm and lots of mprovements have been observed [3 3]. In order to fully take advantage of IFT, a new scheme needs to be ntroduced to tune the effcent parameters. In earler studes, the mportance of IFT parameters was confrmed and a few tunng key ponts were presented [8], but the tunng of these parameters has not been treated n much detal. However, some approaches have been proposed to tune smlar parameters. Shrdhar and Cooper [4] derved an analytcal expresson n whch the suppresson coeffcents arecalculatedasafunctonoftheplantmodelparameters. Ka et al. [5]proposedamn-maxalgorthmtodesgntunng parameters. Al-Ghazzaw et al. [6]presentedan approachto tunng MPC on-lne based on senstvty equatons derved from a step response model wth lnear constrants. The IFT s parameters are manly a scalar, named lambda, and two frequency flters. Ths paper provdes a method to tune the frequency flters whch can emphasze or suppress specfc frequency bands of the output and control sgnals. These flets affect the IFT s performance greatly. Therefore, convenent desgn of these flters seems crtcal. In order to tune these flters, the partcle swarm optmzaton (PSO) s proposed, whch tunes frequency flters usng the nputoutput data acheved from IFT algorthm. Ths method s proposed for the frst order flters wth two parameters. The paper s organzed as follows: a bref explanaton of IFT algorthm s presented n Secton 2. InSecton 3, the partcle swarm optmzaton method s brefly descrbed and mplemented to tune the frequency flters. Consequently, we shed some lght on the performance of the proposed method for tunng the frequency flters, by applyng the mproved IFT to some case study systems n Secton 4. Fnally,thepaper s concluded n Secton 5 wth a summary of key ponts and results.

2 2 Advances n Electrcal Engneerng 2. Iteratve Feedback Tunng Approach Consder the SISO closed-loop system as presented n Fgure.Fromthefgure,tcanbefollowedthat r C r u G y y (t) =Gu(t) + V (t), u (t) =C r (ρ) r t C y (ρ) y t, where G s an unknown sngle nput-sngle output (SISO) operator whch s lnear tme nvarant (LTI), y(t) s the process output, u(t) s the correspondng process nput, and V(t) s an mmeasurable dsturbance whch s assumed to be stochastc. r t s an external determnstc reference sgnal whch s ndependent of {V(t)}. C={C r (ρ), C y (ρ)} are lnear tme nvarant transfer functons parameterzed by parameter vector ρ. Assumng y d as a desred output response to the reference sgnal r, for closed-loop system, the error between the acheved and desred response s () y (ρ) y(ρ) y d. (2) Thecontroldesgnobjectvecanbeformulatedas J= N 2N E ( (L y y t (ρ)) 2 N +λ (L u u t (ρ)) 2 ), (3) t= where E[ ] denotes expectaton wth respect to the dsturbance V. Scalar λ s the weghtng coeffcent expressng the relatve mportance of the penalty on the control effort, and the symbols L y and L u are some frequency flters whch can emphasze or suppress specfc frequency bands of the outputandcontrolsgnals.theseflterscanbeusedforsome purposes such as () emphaszng or suppressng specfc frequency bands of the outputs and control sgnals, for nstance, to prevent unwanted oscllatons n these sgnals: () usng as notch flters n the frequency bands where the measurement nose domnates; () meetng specfc frequency doman performance specfcatons, such as constrants on the senstvtes. For mnmzng J, IFTestmatesthegradent J/ ρ only usng some specal closed-loop experments. The detaled procedureofestmatngthegradentcanbeseenn[8]. After estmatng the gradent J/ ρ, the parameters of the controller are updated by the followng teratve algorthm: t= ρ + =ρ γ R J (ρ). (4) ρ Here, R s an approprate postve defnte matrx whch s typcally a Gauss-Newton approxmaton of the Hessan of J whle γ s a real postve scalar whch defnes the step sze of the algorthm. C y Fgure : SISO closed loop system. 3. Partcle Swarm Optmzaton Partcle swarm optmzaton (PSO) s a populaton-based optmzaton approach frst proposed n 995 [7, 8]. Ths method s urged by the observaton of socal nteracton and anmal behavours such as fsh schoolng and brd flockng [9 2]. Smlar to most optmzaton technques, PSO requres a cost functon evaluaton functon relevant to the partcle s poston. As L y and L u affect the output sgnal and the nput sgnal, the proposed cost functon conssts of the ntegrated absolute error (IAE) defned as follows [22]: IAE = r (t) y(t) dt. (5) And the second part of the cost functon s total varaton (TV) of the manpulated nput. Ths crteron s used to evaluate the requred control effort whch s a convenent measure of smoothness of control nput and should be as small as possble [22]: TV = k= Thus, the cost functon can be defned as u (k+) u(k). (6) CF = IAE + TV. (7) All solutons n PSO can be represented as partcles n a swarm. Each partcle has a poston and velocty vector and each poston coordnate represents a parameter value. The current poston of th partcle of the swarm s denoted by x. x pbest and x gbest are the personal best (Pbest) poston and global best (Gbest) poston of the th partcle. Each partcle s ntalzed wth a random poston and velocty. The velocty of each partcle s accelerated toward the global best and ts own personal best based on the followng equaton: V (t+) = wv (t) +c r (x pbest (t) x (t)) +c 2 r 2 (x gbest (t) x (t)). Here r and r 2 are two random numbers generated from a unform dstrbuton n the range [, ]; c and c 2 are the acceleraton constants and w s the nerta weght factor. Sutable selecton of these parameters provdes a balance between global and local exploratons and helps the partcles (8)

3 Advances n Electrcal Engneerng 3 Table : Controller settng for Example (evaluatng the proposed frequency flters effect). Tunng method K T K 2 T 2 S. T. Os% IAE TV Tuned flters Random flters Random flters Random flters S. T.: Settlng tme..5 System output 2 Control sgnal Tme Tme IFT-tuned flters IFT-random flters IFT-random flters2 IFT-random flters3 IFT-tuned flters IFT-random flters IFT-random flters2 IFT-random flters3 (a) (b) Fgure 2: Smulaton results of Example for evaluatng the effect of proposed flters. converge to Gbest. In ths paper, the constrcted verson of PSO [23] s used whch sets the parameters as follows: w= 2, φ 2+ φ 2 4φ c =wφ, c 2 =wφ 2, where φ =φ 2 = 2.5, φ=φ +φ 2. The postons are updated based on ther movement over a dscrete tme nterval (Δt) as follows: (9) x (t+) =x (t) +V (t+) Δt, () where Δt usually s set to. Then the cost functon at each poston s reevaluated to update Gbest and Pbest. In ths paper, the frequency flters are consdered as the frst order flter as follows: L y = L u = K T s+, K 2 T 2 s+. () Therefore, the proposed PSO s employed to tune four parameters K, T, K 2,andT 2. In each teraton of PSO, the IFT algorthm obtans the optmal control parameter wth the current flters parameters. Usng the updated controller parameters, cost functon (7) s changed and t leads to dfferent parameters of frequency flters acheved n PSO algorthm. 4. Smulaton Study Example. A gas boler system s consdered where the gas valve poston V 2 s used to control the temperature T of the water. The followng transfer functon s used to represent the relatonshp between the varatons n V 2 and T when the flow of water s constant [24]: e 5s P (s) = (+3s)(+s). (2) Ths system s controlled by a PI controller tuned by IFT algorthm. The frequency flters tuned by the proposed PSO method are as follows: L y =.57.26s +, (3).94 L u = 3.7s +. The smulaton results n Fgure 2 and Table ndcate that the IFT algorthm tuned by the proposed method gves an acceptable response whch can substtute the tral-and-error tunng of the flters. As mentoned before, the frequency flters play a sgnfcant role n IFT performance. By many smulatons t was observed that IFT s very senstve to flters parameters. By a random selecton of the flters parameters, t was observed that poorly tuned flters can lead to unstable responses. To show the flters effect on the responses, the flters parameters are obtaned randomly around the optmal values by a 3% tolerance. From Fgure 2 and Table,tcan be seen that small varatons of flters parameters can lead to large changes n output response.

4 4 Advances n Electrcal Engneerng Table 2: Controller settng for Example 2 (evaluatng the proposed frequency flters effect). Tunng method K T K 2 T 2 S. T. Os% IAE TV Tuned flters Random flters Random flters Random flters S. T.: Settlng tme. 2 System output Control sgnal Tme Tme IFT-tuned fltersdata IFT-random flters IFT-random flters2 IFT-random flters3 IFT-tuned fltersdata IFT-random flters IFT-random flters2 IFT-random flters3 (a) (b) Fgure 3: Smulaton results of Example 2 for evaluatng the effect of proposed flters. Example 2. Consder the control system of a heated tank. The dynamcs relatng the control acton and the temperature can be modelled wth an FOPDT model as follows [24]: P (s) = 3.9e 2.s.2s +. (4) The output response and control sgnal of the system are depcted n Fgure 3 n whch the IFT tuned by the proposed method has been compared to IFT wth the randomly tuned flters. Here, more freedom was assgned to select the random flters. Therefore, the effect of these flters has been shown better. Thesmulatonresultscanbeseen ntable Concluson The earler attempts to tune the frequency flters were based on tral-and-error. In ths study, these flters were desgned usng the PSO method. The proposed method was appled to a case study system. It was observed that the performance of IFTmodfedbytheproposedmethodswasgreatlymproved. Therefore, IFT has become more capable wth a better performance. The proposed method for desgnng frequency flters n ths study s not merely lmted to IFT algorthm; therefore, future studes are recommended to generalze the proposed approach to other methods wth smlar parameters. Also, a further study wth focus on hgh order flters s suggested. Conflct of Interests The author declares that there s no conflct of nterests regardng the publcaton of ths paper. References [] H. Hjalmarsson, S. Gunnarsson, and M. Gevers, Convergent teratve restrcted complexty control desgn scheme, n Proceedngs of the 33rd IEEE Conference on Decson and Control, pp , IEEE, Orlando, Fla, USA, December 994. [2] H. Hjalmarsson, Iteratve feedback tunng an overvew, Adaptve Control and Sgnal Processng, vol.6,no.5,pp ,22. [3] J.K.Huusom,N.K.Poulsen,andS.B.Jørgensen, Improvng convergence of teratve feedback tunng, Process Control,vol.9,no.4,pp ,29. [4] J. K. Huusom, H. Hjalmarsson, N. K. Poulsen, and S. B. Jorgensen, A desgn algorthm usng external perturbaton to mprove teratve feedback tunng convergence, Automatca, vol. 47, no. 2, pp , 2. [5] H. Hjalmarsson, Effcent tunng of lnear multvarable controllers usng teratve feedback tunng, Internatonal Journal of Adaptve Control and Sgnal Processng,vol.3,no.7,pp , 999. [6] W.K.Ho,Y.Hong,A.Hansson,H.Hjalmarsson,andJ.W.Deng, Relay auto-tunng of PID controllers usng teratve feedback tunng, Automatca,vol.39, no.,pp , 23. [7] K. Tsang, A. B. Rad, and W. Chan, Iteratve feedback tunng for postve feedback tme delay controller, Internatonal Journal of Control, Automaton and Systems, vol.3,no.4,pp , 25. [8] H. Hjalmarsson, M. Gevers, S. Gunnarsson, and O. Lequn, Iteratve feedback tunng: theory and applcatons, IEEE Control Systems Magazne,vol.8,no.4,pp.26 4,998. [9] A. E. Graham, A. J. Young, and S. Q. Xe, Rapd tunng of controllers by IFT for profle cuttng machnes, Mechatroncs, vol. 7, no. 2-3, pp. 2 28, 27.

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