HYBRID APPROACH COMPARED TO GRADIENT METHOD FOR OPTIMAL CONTROL PROBLEM OF HYDRAULIC SYSTEM

Transcription

1 04.pdf HYBRID APPROACH COMPARED TO GRADIENT METHOD FOR OPTIMAL CONTROL PROBLEM OF HYDRAULIC SYSTEM Nrin Majdoub,*, Ani Saly & Mohamd Bnrjb 3,3 Unité d Rchrché LARA Automatiqu, Ecol National d Ingéniur d Tuni, Univrité d Tuni El Manar, BP 37, 00, Tuni Blvédèr, Tuniia Ecol National d Ingéniur d Monatir, Univrité d Monatir, Ibn Aljazzar, 509, Monatir, Tuniia ABSTRACT In thi papr, hybrid approach includ mta-huritic and convntional mthod, for optimal control problm of witchd ytm i propod. Th maximum principal i ud to find an optimal continuou input and Particl Swarm Optimization (PSO) to find optimal witching intant minimizing a prformanc function dpnd on th intant, in a finit horizon. W dmontrat via hydraulic application th ffctivn of propod approach whr th rult ar compard to tho obtaind by gradint mthod. Kyword: Nonlinar witchd ytm, optimal control, witching intant, Particl Swarm Optimization (PSO), gradint mthod.. INTRODUCTION A witchd ytm i a ytm that conit of vral ubytm and a witching law indicating th activ ubytm at ach tim intant []. For an optimal control problm of witchd ytm, w nd to find not only an optimal continuou input but alo an optimal witching qunc inc th ytm dynamic vary bfor and aftr vry witching intant. Svral rarchr hav tudid thi framwor uch a Piccoli [], Lincoln and Brnhardon [3] Bmporad t al. [4]. In particular, Xu and Antali [, 5-7], formulat th problm in two tag optimization problm. Stag (a), dal a convntional optimal control problm that find th optimal cot giving th qunc of activ ubytm and th witching intant. Stag (b), dal a nonlinar optimization problm that find th local optimal witching intant bad on gradint mthod. Th concptual algorithm applid for thi optimization mthodology, allow at ach itration to find a valu J J of cot J by olving an optimal control problm (tag (a)), find and with rpct to th witching t t intant and u th gradint projction mthod to updat witching intant. In fact, for vral numbr of witching intant, th calculation of th drivativ bcom mor difficult. Alo, gradint mthod rquir much rgularity about function to optimiz and don't ncarily guarant global optimum olution. In ordr to ovrcom th diadvantag, w changd, only in tag (b), th gradint mthod by an mta-huritic algorithm uch a Particl Swarm Optimization (PSO) algorithm. So, our approach includ convntional mthod in tag (a) and mta-huritic in tag (b) which w call hybrid approach. In th qul, an optimal control problm for witchd ytm i formulatd in ction. Sction 3 dcrib th hybrid approach and prnt it implmntation for optimal control problm. In th following, w dmontrat via hydraulic application, th ffctivn of th propod approach compard to gradint mthod.. PROBLEM FORMULATION.. Hybrid Sytm Conidring a witchd ytm dcribd by th following ubytm [7]: i,,,,..., x f x u i I M () with f i an lmnt of a t of indxd fild vctor and I i a t of finit dicrt variabl that indicat that th i ytm will b in M configuration... Switching Squnc For a witchd ytm (), th input of th ytm conit of both a continuou input u, witching qunc. W dfin a witching qunc a follow [7]: t t, t 0 f, and a 00

2 t, i, t, i 0 0,..., t, i,..., t, i K K () with 0 K, t t... t t, and i I for 0,,..., K. A witching qunc a dfind abov 0 K f indicat that, if t t, thn ubytm i i activ in t, t ( t, t K f if K ); th ytm witch from ubytm i to ubytm i..3. Optimal Control Problm Conidr a witchd ytm a dcribd by (), givn an initial continuou tat x, a fixd tim intrval, 0 t t 0 f and a prpcifid qunc of activ ubytm, find a continuou input u and witching intant t,..., t K minimizing th cot functional givn a follow: = J t f, f J x t L x t u t dt final J t0 continuou whr J i a componnt charactriz th contribution of trminal condition and J i componnt final continuou charactriz th ffct of tmporal accumulation du to continuou volution of ytm. 3. HYBRID APPROACH FOR OPTIMAL CONTROL PROBLEM Two tag can b propod to olv th optimal control problm of witchd ytm rfrring to wor of Xu and Antali in 000, 00, 003 and 004 [, 5-7]. 3.. Stag (a) Stag (a) dal with convntional optimal control problm which an optimal continuou control input u. W apply th maximum principal, whr an vctor function pt p t p t condition hold: (i) for almot any with,..., t t, t 0 f, th following tat and cotat quation hold: T dx t H x t p t u t f x t u t dt p,,, dp t H f L xt, pt, u t p dt x x x,,, T, n T T T T (3) xit uch that th following H x p u L x u p f x u (6) (ii) for almot any t t, t 0 f, th tationary condition hold: T T T 0 H f L xt, pt, u t p u u u (iii) at t th function p atifi th boundary condition: f f xt f T p t x (iv) at any t,,,..., K, w hav continuity condition: pt p t (9) Howvr, th diffrntial algbraic quation can b olvd fficintly by uing th function bvp4c of Matlab. (4) (5) (7) (8) 0

3 3.. Stag (b) Stag (b) dal with non linar optimization problm givn by th quation (0) which an optimal witching intant. Jˆ tˆ T tˆ T tˆ t,..., t t t... t t with min J (0) tˆ K 0 K f For optimal witching intant tˆ,..., t ˆ K, w u th PSO algorithm. Thi mta-huritic, i an volutionary optimization tchniqu [8], invntd by Rul Ebrhart and Jam Knndy [9, 0]. Th baic PSO i dvlopd from rarch on warm uch a fih chooling and bird flocing []. Th population i initializd randomly and th potntial olution, namd particl, frly fly acro th multidimnional arch pac []. Each particl i charactrizd by a poition p and vlocity v. During flight, ach particl updat it own vlocity and poition by taing bnfit from it bt xprinc and th bt xprinc of th ntir population. Lt b th itration indx, th nw particl vlocity and poition ar updatd according to th mov quation [3, 4] givn a follow: v w v c r pbt p c r pgbt p () p p v () with: p : poition of ach particl at itration, v : vlocity of ach particl at itration, pbt : bt poition dicovrd by th particl until th itration, pgbt : global bt particl poition of th ntir population. c and c ar trngth of attraction contant and r and r ar uniformly ditributd random numbr in 0,. Inrtia wight, w, i a paramtr ud to control th impact of th prviou vlociti on th currnt vlocity. It influnc th trad-off btwn th global and local xploitation abiliti of th particl. For initial tag of th arch proc, larg inrtia wight to nhanc th global xploitation i rcommndd whil for lat tag, th inrtia wight i rducd for bttr local xploration. Wight i updatd a: w w max max min w w max whr w, w and ar minimum, maximum valu of w and pr-pcifid maximum numbr of itration min max max cycl, rpctivly. A chmatic rprntation of particl flight within th arch pac i givn in figur. (3) 0

4 pbt v p p wv c b pgbt p pgbt v c b pbt p Figur. Schmatic rprntation of particl flight. In ordr to hav optimal witching intant, w conidr that poition of particl p rprnt a witching qunc. By tting th arch pac t t, 0 f, th witching numbr of ytm c n and th warm iz Sw, th optimization algorithm calculat th numrical valu of th cot function at ach qunc and lct th on with th minimum cot. In th nxt, w prnt th optimization algorithm. 3.. Propod Optimization Algorithm Stp : dfin th paramtr of th algorithm: t, t 0 f, Sw, n, c, c, c gnrat randomly p Sw n c and v, Sw n c w, w and, min max max Stp : by olving an optimal control problm (tag(a)), valuat ach particl by th cot function J, idntify th witching intant corrponding to th bt fitn J ˆ t, affct th bt qunc at affct pbt at pgbt, pbt, initializ th itration indx, Stp 3: updat v and p by th mov quation () and (), valuat ach particl by th cot function J by olving tag (a), find th witching intant corrponding to th bt fitn and affct it to compar initial pgbt to pbt, pbt, if pbt pgbt thn pgbt pbt, Stp 4: incrmnt, tt th fluctuation of th cot valu if it tnd to ( 0 ) thn rturn to tp 3, 03

5 l go to tp 5, Stp 5: giv th bt witching intant of th minimum cot. 4. ILLUSTRATIVE EXEMPLES Conidr a hydraulic ytm coniting of two tan [5] rprntd in figur. q h V h V Figur. Hydraulic ytm. Th tudid framwor conit in obtaining fluid lvl h 5m and h m f f 0,30 by in th tim intrval controlling th input flow q and th witching law applid to th valv V and V. Th valv witchd btwn two configuration: clod or fully-opn. Th modl quation of th hydraulic ytm ar givn by th following quation: dh S q V h (4) dt dh S V h V h (5) dt whr S i th tan ction, and ar rpctivly th hydraulic ritanc. Thir corrponding numrical valu ar givn a follow: 5 / 5 / S m 0.0 m / 0.03 m / (6) To how th ffctivn of th propod approach, w prnt two opration ca of hydraulic application. Th paramtr of th propod algorithm for th ca ar fixd to: c c 0.75 w w 0.4 (7) min max 0.9 Sw 40 5 max 4.. Ca Initially, w aum that th valv V i clod and V i opn and th initial lvl for two tan ar h 3m and 0 h 0m. Th valv V opn at t t, thu th hydraulic ytm ha two dynamic dcribd by th following 0 quation: t 0, t for 04

6 for t t dh q dt S dh dt S h,30 dh q h dt S (9) dh h h dt S Our goal i to find th optimal control input q and th witching intant ˆt minimizing th cot function givn a follow: 30 J h 30 5 h 30 h t 5 h t u t dt (0) 0 For optimal input, by applying th ncary optimality condition, it com: t 0, t, th hamiltonian quation i: for H x, p, q h 5 h u q p p h S () and th diffrntial quation ar: with: for t t,30 dh q dt S dh h dt S dp h 5 dt dp p h dt S h q p (3) S, th hamiltonian quation i: H x, p, q h 5 h u q h p h h p S S (4) and th diffrntial quation ar: (8) () dh q h dt S dh h h dt S dp dt S h dp p h dt S h p p h 5 (5) with: 05

7 q p (6) S th boundary condition ar: p 30 h 30 5 p 30 h 30 (7) For optimal witching intant, th propod algorithm i ud whn w obtaind, a oon th firt itration (figur 3), th optimal witching intant t ˆ.54 corrponding to th optimal cot J ˆ Th optimal input q and th fluid lvl h and h ar hown rpctivly in figur 4, figur 5 and figur 6. It can b obrvd in figur 7 that th cot function i nonconvx. Howvr, whn w applid th gradint mthod for a variou initial intant, tabl, w not that it nd up with th firt olution foundd and convrg to a global olution only if w hav th uitabl initial intant. 4.. Ca Initially, w aum that th two valv ar clod and th initial lvl for two tan ar h 3m and h 0.5m. 0 0 At t t, th valv V opn and V at t t. Each of th dicrt witch onc at a dfinit momnt, th ytm thn prnt thr dynamic dcribd, at ach intrval, givn by th following quation: t 0, t for dh q dt S dh 0 dt (8) J() Figur 3. Fitn function for ca. 06

8 .5 q (m 3 /min) t() Figur 4.Optimal control input u h (m) t() Figur 5. Fluid lvl h. 07

9 h (m) t() Figur 6. Fluid lvl h 0 9 J(t ) t () Figur 7. Cot function J t. 08

10 Tabl. Obtaind rult by gradint mthod for ca. Intant initial t Réultat optimaux ˆt Ĵ W conclud that gradint mthod don't guarant an global olution, for that, w p th propod approach. t t, t for for t t dh q h dt S dh h dt S,30 dh q h 3 dt S (30) dh h h dt S W to find th optimal control input q and th witching intant ˆt and ˆt minimizing th am cot function givn a follow: J h h h t h t u t dt Applying th ncary optimality condition, w obtain: t 0, t, th hamiltonian quation i: for (9) (3) 0,, 5 H x p q h h u q p (3) S and th diffrntial quation ar: dh q dt S dh 0 dt dp dt dp dt h h 5 (33) with: q p (34) S for t t, t, th hamiltonian quation i: H x, p, q h 5 h u q h p h p S S (35) and th diffrntial quation ar: 09

11 dh q h dt S dh h dt S dp dt S h dp h dt p p h 5 (36) with: for t t,30 q p (37) S, th hamiltonian quation i:,, H x p q h h u q h p h h p 3 S S (38) and th diffrntial quation ar: dh q h 3 dt S dh h dt S dp dt S h dp p h dt S h p p h 5 (39) with: q p (40) 3 S th boundary condition ar: p 30 h 30 5 p 30 h 30 (4) By applying th our algorithm and aftr 5 itration, figur 8, w obtain th optimal witching intant t ˆ.560 and tˆ corrponding to th optimal cot J ˆ Gradint mthod convrg to th optimal olution obtaind by th propod approach, a hown in tabl, unl w tart by th initial intant blong to th olution zon. Tabl. Obtaind rult by gradint mthod for ca. Intant initial Réultat optimaux ˆt ˆt Ĵ t t 0

12 J() Figur 8. Fitn function for ca. Th corrponding optimal control input q and th fluid lvl h and h ar hown rpctivly in figur 9, figur 0 and figur. Figur how th optimal cot for diffrnt witching intant t and t. Th optimal olution chcd in thi plot i a am olution foundd by our approach. PSO algorithm can aily b adaptd vn for vral witching intant. In fact, w chang th variabl n and c chc th activ ubytm in hi tim intrval. Contrary for gradint mthod, thi ca it' mor difficult whn w hav to find th drivativ of th cot with rpct to ach witching intant. 5. CONCLUSION In thi papr, w propod hybrid approach includ a convntional mthod for optimal continuou input and mtahuritic uch a Particl Swarm Optimization algorithm for witching intant optimization. To how th ffctivn of our approach, w tudy a hydraulic application with nonconvx cot function. In th firt ca w conidr a impl configuration with on witching intant whr w compard th obtaind rult by tho obtaind by gradint mthod. In fact, gradint mthod don't guarant th optimal olution for that w conrv our approach. For vral witching intant, Hybrid approach can aily b adaptd whr it conidrd a gnral olution for optimal problm of witching intant.

13 q (m 3 /min) t() Figur 9. Optimal control input u h (m) t() Figur 0. Fluid lvl h.

14 h (m) t() Figur. Fluid lvl h J(t,t ) t () t () Figur 7. Cot function, J t t. 3

15 6. REFERENCES []. X. Xu, P. J. Antali, Optimal Control of Switchd Sytm: Nw Rult and Opn Problm, Procding of th Amrican Control Confrnc,, Illinoi, Chiacago, , Jun (000). []. B. Piccoli, Hybrid Sytm and Optimal Control, Procding of th 37 th IEEE Confrnc on Dciion and Control, Tmpa, Florida, 3 8, Dcmbr (998). [3]. B. Lincoln, B.M. Brnhardon, Efficint Pruning of Sarch Tr in LQR Control of Switchd Linar Sytm, Procding of th 39th IEEE Confrnc on Dciion and Control, Sydny, vol., , Dcmbr (000). [4]. A. Bmporad, F. Borrlli, M. Morari, On th optimal control law for linar dicrt tim hybrid ytm, Procding of th 5th Intrnational Worhop on Hybrid Sytm: Computation and Control, vol. 89, 05 9, (00). [5]. X. Xu, P.J. Antali, Optimal Control of Switchd Sytm Via Nonlinar Optimization Bad on Dirct Diffrntiation of Valu Function, Intrnational Journal of Control, vol. 75, no. 6/7, , (00). [6]. X. Xu, P. J. Antali, Rult and Prpctiv on Computational Mthod for Optimal Control of Switchd Sytm, Hybrid Sytm: Computation and Control, vol. 63, , (003). [7]. X. Xu, P. J. Antali, Optimal Control of Switching Sytm Bad on Paramtrization of th Switching Intant, IEEE Tranaction on Automatic Control, vol. 49, no., 6, (004). [8]. T.L. L, M. Nqnvity, M. Piutowi, Ntwor Equivalnt and Exprt Sytm Application for Voltag and VAR Control in Larg Scal Powr Sytm, IEEE Tranaction on Powr Sytm, vol., no. 4, , (997). [9]. R.C. Ebrhart, J. Knndy, A Nw Optimizr Uing Particl Swarm Thory, Procding of th Sixth Intrnational Sympoium on Micro Machin and Human Scinc, Picataway, NJ: IEEE Srvic Cntr, 39 43, Nagoya, (995). [0]. J. Knndy, R. C. Ebrhart, Particl Swarm Optimization, Procding IEEE Intrnational Confrnc on Nural Ntwor, Picataway, NJ: IEEE Srvic Cntr, vol. 4, , Prth, (995). []. O. Chao, L. Wining, Comparion btwn PSO and GA for paramtr optimization of PID controllr, Procding of th IEEE, Intrnational Confrnc on Mchatronic and Automation, 5 6, Luoyang, (006). []. O.K. Jon, A. Boufft, Comparaion of b algorithm ant colony optimiation and particl warm optimiation for PID controllr tuning, Intrnational Confrnc on Computr Sytm and Tchnologi, Gabrovo, 008. [3]. H. Yohida, K. Kawata, Y. Fuuyama, Sh. Taayama, Y. Naanihi, A Particl Swarm Optimization for Ractiv Powr and Voltag Control Conidring Voltag Scurity Amnt, IEEE Tranaction on Powr Sytm, vol. 5, no. 4, 3 39, (00). [4]. J. Wang, Y. Zhou, X. Chn, Elctricity Load Forcating Bad on Support Vctor Machin and Simulatd Annaling Particl Swarm Optimization Algorithm, Intrnational Confrnc on Automation and Logitic, , Jinan, (007). [5]. N. Majdoub, A. Saly, M. Bnrjb, Hybrid Appraoch for Optimal Control Problm of Switchd Sytm, IEEE Intrnational Confrnc on Sytm, Man, and Cybrntic, Itanbul, (00). 4