Automatic Tuning of Model Predictive Control Using Particle Swarm Optimization

Transcription

1 Automatc Tunng of Model Predctve Control Usng Partcle Swarm Otmzaton Rohe Suzuk Fukko Kawa Hdeuk Ito Chkash Nakazawa Yoshkazu Fukuama Etaro Aosh Keo Unverst 3-- Hosh Kohoku-ku Yokohama 3-5 Jaan Fuj Electrc Advanced Technolog Co. Ltd. Fujmach Hno Toko 9-5 Jaan (kawa-fukko to-hdeuk nakazawa-chkash Abstract Ths aer resents an automatc tunng method of model redctve control (MPC usng artcle swarm otmzaton (PSO. Although conventonal PID s dffcult to treat constrants and future lant dnamcs MPC can treat ths ssues and ractcal control can be realzed n varous ndustral roblems. One of the challenges n MPC s how control arameters can be tuned for varous target lants and usage of PSO for automatc tunng s one of the solutons. The numercal results show the effectveness of the roosed PSO-based automatc tunng method. I. INTRODUCTION Varous control methods have been utlzed n ndustres. Among these control methods MPC becomes one of the major control strateges because of ts ntutve control concet. Recentl MPC has man successful alcatons ncludng chemcals food rocessng automotve and aerosace alcatons []. Tunng of the MPC arameter s consdered to be eas for skllful MPC controller desgners. However the relaton between the tunng arameters and hscal meanngs s not alwas clear. Therefore the arameter tunng s dffcult for unsklled controller desgners. Tunng technques for the MPC have been roosed n man lteratures. Ino et al. roosed a arameter tunng method consderng robust stablt based on frequenc resonse analss []. Rowe et al. roosed the H-nfnt looshang method [3]. Droges et al. roosed a heurstc tunng method based on exert rules []. Ino s and Rowe s methods requre comlcated comutng rocedure and t s dffcult to use t b unsklled ordnar controller desgners. Practcall Droges exert rules have to be modfed for the target roblem. Namel the revous tunng methods are stll dffcult for unsklled controller desgners and smle MPC tunng method s eagerl requred n ndustres. Tunng of MPC can be formulated as an nverse roblem and otmzaton technques can be aled. Snce MPC should be treated as a blackbox n the nverse roblem conventonal mathematcal rogramg te otmzaton technques cannot be aled and evolutonar comutaton te otmzaton technques should be aled. In the formulaton state varables are contnuous and PSO [5] s one of the best canddates for the roblem. Recentl PSO has been aled n ndustres ractcall [-] and a PSObased tunng method must be acceted. Ths aer resents an automatc tunng of the model redctve control usng artcle swarm otmzaton. The roosed PSO-based Automatc Tunng for MPC (PSO- MPC comutes the otmal tunng usng tme-doman erformance crteron. In the roosed PSO-MPC as descrbed above MPC s treated as a blackbox and unsklled ordnar controller desgners do not have to care MPC tself. Moreover t s verfed through smulatons that the roosed PSO-MPC can tune MPC arameters aroratel usng the same PSO arameters. Namel t s suggested that unsklled ordnar controller desgners do not have to tune PSO and consequentl the can tune MPC easl. The numercal results show the effectveness of the roosed PSO-MPC. II. MODEL PREDICTIVE CONTROL In ths secton bref summar of model redctve control and target arameters s descrbed. Model redctve control has been frst roosed b J.Rchalet and C.Cutler ndeendentl at the end of 7s [9][]. MPC estmates future behavor of the control target wthn a certan erod usng a model of the control target nsde the controller. Then t deteres manulated sgnals so that an objectve functon s mzed [][-3]. Illustrated comarson between MPC and conventonal PID s shown n fg.. hle a PID controller deteres a manulated sgnal usng nstantaneous observed and target values drectl MPC estmates behavor of the control target wthn a certan erod n the future. Therefore t consder not also nstantaneous control roert but also control roert wthn a certan erod n the future. hle PID can onl treat lmters as constrants MPC can treat PID controller resent PID Controller Control target observe onl nstantaneous behavor Detere a manulated sgnal sgnal usng usng onl onl current current (nstantaneous observed observed values values t MPC controller Estmate future behavor of the control target nsde the controller MPC Control target Estmates Estmates future future behavor behavor of of the the control control target targetusng a model model of of the the control control target target and and deteres manulated sgnals sgnals so so that that an an objectve functon s s mzed Present Fg.. Comarson between conventonal PID controllers and MPC. Future t --7-7/7/$. 7 IEEE

2 varous constrants of the control target when t mze an objectve functon. Ths s another advantage comared wth conventonal PID controllers. MPC can be formulated as follows. Concet of MPC s shown n Fg.. MPC utlzes redctve oututs ŷ whch s estmated wthn H stes (redcton horzon n the future usng a nternal model. It mze the followng objectve functon whch conssts of weghted sum of trackng error of redctve oututs ŷ to set onts r manulated sgnals u and varances of manulated sgnals u. A model redctve control roblem can be formulated as an otmzaton roblem whch detere nut sgnals u(k... u(k+h c - wthn H c stes (control horzon n the future so that the objectve functon s mzed consderng the followng constrants []. ( Objectve functon of MPC H J { ( r( k ˆ( k ( ( uˆ( k u u ( uˆ( k u ˆ( k uˆ( k uˆ( k where ( s weght for each term r( s set ont ŷ s redctve outut u ˆ( s redctve control nut. ( Constrants of MPC ˆ ( k H ( where s lower bound of outut s uer bound of outut. u uˆ ( k u H (3 where u s lower bound of control nut u s uer bound of control nut. u uˆ ( k u H ( where u s lower bound of control nut dfference u s uer bound of control nut dfference. It s assumed that nut sgnals after control horzon are equal (u(k+h c -=u(k+ H c =...=u(k+h -. The onl ast u future ŷ r û k k+h c - k+h P - Tme H P H c Predcton Horzon Control Horzon resent P P P } nut sgnals at the frst ste are utlzed as manulated sgnals. After one samlng erod actual oututs are nut and the same rocedure (an otmzaton rocedure of MPC s aled usng the redcton horzon whch s shfted one ste to the future drecton. III. FORMULATION OF AN AUTOMATIC TUNING PROBLEM OF PARAMETERS IN MPC Ths secton shows a formulaton of an automatc tunng roblem of arameters n MPC. In the tunng roblem lant oututs from t=... t calculated b numbers of MPC are evaluated. 3. Target tunng Parameters of MPC Generall snce redcton horzon H control horzon H c and constrants are fxed weghted coeffcents u and u are tuned n ths aer. In ths aer for the sake of smlct ( s fxed to constant values. Namel n the tunng roblem t s assumed that objectve functon can be a functon of weghts of MPC. 3. Problem Formulaton The target tunng roblem can be formulated as follows: ( Objectve functon The basc (sngle objectve functon can be exressed as follows: f / G G ( e M ( E M ss t M e ( t s t where s weghted coeffcent for resonse and varance roert E ss s stead state error t s s settlng tme t r s rsng tme M s mum value of t s at the end tme. If we have number of oututs the objectve functon can be exressed as a - te functon. { f f f m} ( x where m s number of outut. A general weghted sum te objectve functon can be utlzed as well. However accordng to our exerence - te functon has better erformance and we decded to utlze the - te objectve functon. ( Constrants Uer and lower bounds of weghts should be consdered. r (5 Fg.. Concet of MPC.

3 F F... m (7 u u u... m ( u u u... m (9 where s lower bound of outut s uer bound of outut u s lower bound of nut u u s uer bound of nut u u s lower bound of nut varance u u s uer bound of nut varance u. The concet of the tunng roblem s shown n Fg. 3. Parameters (weghts of the objectve functons of MPC control roblem are searched n the uer tunng roblem b PSO and the are sent to the lower level MPC roblem. Next the oututs of the rocess are calculated b MPC and the are sent to the uer level PSO. Then objectve functon values n the uer level tunng roblem are calculated. Ths rocedure s reeated untl the redetered teraton number. As shown n the fgure usng ths tunng concet MPC arameters are automatcall tuned b PSO and unsklled control desgners can utlze t easl. hen state varables are udated at the one teraton ste of PSO lower otmzaton roblem (MPC s calculated as shown n Fg.3. Therefore total otmzaton rocedure (ntegraton of uer and lower level s exected to be tme-consug. However because of recent rogress of comuter erformance ths knd of brute force rocedure can be realzed. 3.3 Characterstcs of the Problem In ths secton characterstcs of the tunng roblem esecall the shae of the objectve functon s exaed. Fg. and 5 show shaes of the objectve functons of PID controllers and MPC. In PID roortonal dervatve and Integral arameters (k k d k are tuned. The two controllers are aled to the same rocess and have the same manulator. Conventonal Controller tunng lteratures b PSO have alread ublshed [5][]. As shown n these fgures a MPC tunng roblem has to treat a more comlcated and multmodal objectve functon than a PID controller tunng roblem. Parameters (eghts of the objectve functons of MPC Uer level tunng roblem (PSO Oututs of the rocess untl stead states IV. AUTOMATIC TUNING OF MPC USING PSO Ths secton shows the wa how automatc tunng of MPC s realzed usng PSO. The target tunng roblem descrbed n cha.iii can be rewrtten as the followng otmzaton roblem wth uer and lower bound constrants: F( u u ( F { f f f n} ( 3m. Equaton (( can be exressed as an otmzaton roblem (3 wthout constrants usng the followng transformaton functon (: h( f ( k ( ( k f ( k f ( k. F ( h( u u (3. kd.. k kd k Fg.. An examle of the shae of an objectve functon of a PID tunng roblem (two PIDs u u Fg. 5. An examle of the shae of an objectve functon of a MPC tunng roblem. Lower level control roblem (MPC Fg.3 Concet of the tunng roblem. 3

4 u u m u ( u m 3m m The basc Gbest model s utlzed n ths aer. An udate equaton of agents can be exressed as follows: best v ( k v ( k c ( ( k (5 gbest c ( ( k ( k ( k v ( k ( The udate equatons of gbest and bests can be exressed as follows: best arg { F( ( l l k} ( P l (7 gbest arg { F( best P} ( where P s total number of agents. A calculaton rocedure of PSO-MPC can be exressed as follows: Ste [Intalzaton rocedure] Intalze ( of agent (=...3m usng unform random numbers wthn uer and lower bounds of. eghts ( calculated at ste are sent to MPC and a control model smulates the target lant behavor untl t. The evaluaton values are calculated b (5. Pbests and gbest are calculated usng (7(. The rocedure s reeated for =...P. Ste [Udate of bests and gbest] eghts (k calculated at ste are sent to MPC and a control model smulates the target lant behavor untl t. The evaluaton values are calculated b (5. Pbests and gbest are calculated usng (7(. The rocedure s reeated for =...P. Consderng uer bounds of v v can be exressed as follows: v ( k sgn( v ( k{ v ( k.5( } (9 bests and gbest are udated usng(7(. Ste3 [Sto crteron and udate of the weghtng functon] The rocedure s stoed when k reaches k (mum teraton number. Otherwse k=k+ and of ( s udated usng the followng equaton (nerta weghts aroach and go to Ste: Outut Outut r ( r r u ( u u ( MPC lant Fg.. Block dagram of the lant wth a MPC controller k (9 k Tme(s Tme(s Fg. 7. Ste resonse of the lant wthout the PSO-MPC controller. V. NUMERICAL EXAMPLES For the urose of llustratng the effectveness of the roosed method two examles are studed numercall. 5. Numercal Examle No. ( Smulaton Condtons The followng lant s tested n examle No.. s 3s Y.e.9e U.7s s 7s 3s (.e 9.e Y U.9s.s The block dagram of PSO-MPC controller and the lant s shown n Fg.. Fg. 7 shows the orgnal ste resonses of the lant wthout the PSO-MPC controller. As shown n these fgures stead state errors are aeared n the orgnal resonses. In ths smulaton the followng arameter settngs are utlzed through re-smulatons: P k 5 c c..9.;

5 These arameters are fxed for all smulatons. The followng MPC controller arameters are also utlzed n the smulaton: w u wu w u H H c. u As shown n the Fg.3 MPC arameters are automatcall tuned b PSO and unsklled control desgners can utlze t easl. Therefore ractcal alcablt of the roosed method deends on whether arorate arameters can be obtaned wthn arorate calculaton tme. In addton Outut Outut Tme(s PSO-MPC Tunng PSO-PID Tunng Tme(s PSO-MPC Tunng PSO-PID Tunng Fg.. Ste resonse of the lant wth dfferent controllers. Table The otmal arameters. PSO- PID tunng PSO-MPC tunng k. u.997 k d.75 u 9.9 k d 55.5 k.37 u.39 k.35 u 9.99 k d Table PIDC & MPC tunng results. Objectve functon PID tunng MPC tunng value Mean value Best value orst value Standard devaton.3e-5.39 n order to verf the advantage of the roosed PSO-MPC we also al a PSO based PID tunng method (PSO-PID to the same lant wth two PID controllers. In ths smulaton the same fxed arameters P k c c are utlzed for PSO-PID. Uer and lower bounds of PID arameters are set to and.. PSO-MPC and PSO- PID are comared usng the same objectve functon (5. ( Smulaton Results The best values of the otmzed arameters are gven n Table. In order to obtan close arameters n Table sklled control desgners sent aroxmatel hours. On the contrar t onl takes 3 utes b PSO-MPC. Fg. shows ste resonses of the lant wth the PSO-MPC and the PSO-PID controllers. In order to comare ther convergence characterstcs best average worst and standard devaton of the objectve functon values are evaluated through 5 trals wth dfferent ntal condtons as shown n Table. Accordng to the results the PSO- MPC controller has better evaluaton values than the PSO- PID controller. Snce PSO-PID can generate the best control resonses of the conventonal method t s sad that arorate arameters can be obtaned wthn arorate calculaton tme b PSO-MPC. 5. Numercal Examle No. The roosed method s aled to a ractcal lant model wth nuts and oututs. Outut sgnals are mutuall couled (nterfered or have strong connectons n the lant. Smulaton for 5 cases (5 combnatons of arameters takes aroxmatel hours. On the contrar t onl takes aroxmatel utes b PSO- MPC. Best average worst and standard devaton of the objectve functon values are evaluated through 5 trals wth dfferent ntal condtons as shown n Table 3. Fg. 9 shows ste resonses of the lant wth and wthout the PSO-MPC (best results. Accordng to the results t s observed that arorate arameters are obtaned b PSO- MPC. Table 3. Objectve functon values through 5 trals. Mean value 5.77 Best value.799 orst value Standard devaton VI. CONCLUSIONS Ths aer resents an automatc tunng method of model redctve control (MPC usng artcle swarm otmzaton (PSO-MPC. Ths s a frst alcaton of PSO for automatc tunng of MPC. Usng PSO-MPC unsklled control desgner can tune MPC easl wthn arorate tunng tme. 5

6 Future works are as follows. In ths aer onl ste resonses are evaluated. e wll develo PSO-MPC evaluatng ste resonses dsturbance resonses and other tems. In ths aer weghts are assumed to be fxed all tme. Calculaton of ( (dfferent for each tme s one of the future works for mrovement of control roert. Moreover develoment of arorate man-machne nterface (MMI s one of the man ssues for unsklled control desgners. REFERENCES [] S.J.Qn and T.A.Badgwell "A surve of ndustral model redctve control technolog" Control Engneerng Practce No [] Y. Ino et al. A New Inut/Outut Constraned Model Predctve Control wth Frequenc Doman Tunng Technque and Its Alcaton to an Ethlene Plant Proceedngs of the IECON [3] C. Rowe et al. Tunng MPC usng H-nfnt Loo Shang Proceedngs of the Amercan Control Conference Vol.. [] S. Droges et al. Predctve Control: Proostons for the Desgn Methodolog Proceedngs of the 999 Amercan Control Conference Vol [5] J. Kenned and R. Eberhart Swarm Intellgence Morgan Kaufmann Publshers. [] Y. Fukuama "Foundaton of Partcle Swarm Otmzaton" n chater 5 of tutoral text on Modern Heurstc Otmzaton Technques wth Alcaton to Power Sstems organzed b IEEE Power Engneerng Socet IEEE nter Power Meetng Januar. [7] Y. Fukuama et al "A Partcle Swarm Otmzaton for Reactve Power and Voltage Control Consderng Voltage Securt Assessment" IEEE Trans. on Power Sstems Vol.5 No.. [] S. Ktagawa and Y. Fukuama "Comarson of Partcle Swarm Otmzatons for Otmal Oeratonal Plannng of Energ Plants" Proc. of Swarm Intellgence Smosum (SIS5 5. [9] J.Rchalet A.Rault J.L.Testud and J.Paon "Model redctve heurstc control: alcatons to ndustral rocesses" Automatca Vol. No [] C.R.Cutler and B.L. Ramaker "Dnamc matrx control A comuter control algorthm" In Proceedngs Jont Amercan Control Conferernce San Francsco U.S.A. 9. [] J.M.Macejowsk Predctve Control wth Constrants Prentce Hall. [] R.Soeterboek Predctve Control. A unfed aroach Prentce Hall 99. [3] C.A.Garca D.M.Prett and M.Morar "Model Predctve Control: Theor and Practce a Surve" Automatca Vol.5 No [] A.Bemorad N.L.Rcker and J.G.Owen "Model Predctve Control - New tools for desgn and evaluaton" In Proceedngs Amercan Control Conference Boston U.S.A.. [5] M. Abdo "Otmal Desgn of Power Sstem Stablzers usng Partcle Swarm Otmzaton" IEEE Trans. on Energ Converson Vol.7 No.3.-3 Se.. [] Z. L. Gang "A Partcle Swarm Otmzaton Aroach for Otmum Desgn of PID Controller n AVR sstem" IEEE Trans. on Energ Converson Vol.9 No June..... outut.. outut.... th the PSO-MPC controller thout the PSO-MPC controller Tme(s th the PSO-MPC controller thout the PSO-MPC controller Tme(s.... outut 3.. outut.... th the PSO-MPC controller thout the PSO-MPC controller Tme(s th the PSO-MPC controller thout the PSO-MPC controller Tme(s Fg. 9. Ste resonse of the nuts and oututs lant wth the PSO-MPC controller.