Stable Model Predictive Control Based on TS Fuzzy Model with Application to Boiler-turbine Coordinated System

Transcription

1 5th IEEE Confeence on Decson and Contol and Euopean Contol Confeence (CDC-ECC) Olando, FL, USA, Decembe -5, Stable Model Pedctve Contol Based on S Fuy Model wth Applcaton to Bole-tubne Coodnated System Xao Wu, Jong Shen, Yguo L, Kwang Y. Lee, Fellow, IEEE Abstact In ths pape, we popose a stable fuy model pedctve contolle based on extended-fuy Lyapunov functon. he man dea of the poposed appoach s to desgn a fee contol vaable and a non-paallel dstbuted compensaton contol law n such a way that an extended-fuy Lyapunov functon s constucted wth mnmng the uppe bound of the nfnte hoon objectve functon n the fuy model pedctve contol. heefoe, the pedctve contolle can guaantee both the stablty of the closed-loop fuy model pedctve contol system and nput constants whle obtanng the optmal tansent pefomance. It s shown that the contolle s obtaned by solvng a set of lnea matx nequaltes. he extended-fuy Lyapunov functon educes the consevatsm of common Lyapunov functon and fuy Lyapunov functon, and t also enlages the feasble aea of the pedctve contolle. Moeove, appopate slac and collecton matces ae used n all lnea matx nequaltes, whch can futhe educe the consevatsm. he smulatons on a numecal example and a nonlnea bole-tubne coodnated system demonstate the advantage and effectveness of the poposed appoach. Index ems S fuy model, model pedctve contol, extended-fuy Lyapunov functon, slac and collecton matces. M I. INRODUCION ODEL pedctve contol (MPC) uses an explct model to pedct the futue behavo of the plant and solves a constaned optmaton poblem on-lne to obtan the optmal contol sequence [],[]. Snce MPC can guaantee the stablty of the closed-loop system and deal wth the nput constant n an optmal way, MPC has become a popula stategy n pocess contol [3],[4]. Geneally, thee ae two poblems wth conventonal MPC; the fst s that t s dffcult to analye the stablty of the closed-loop system whle consdeng the constant, the second s that the exact hs wo was suppoted by the Natonal Natual Scence Foundaton of Chna (NSFC) unde Gant 5767 and Gant 536, the Specaled Reseach Fund fo the Doctoal Pogam of Hghe Educaton unde Gant 995, the Key Poject of Chnese Mnsty of Educaton unde Gant 86, and the U.S. Natonal Scence Foundaton unde gant ECCS 844. Xao Wu, Jong Shen and Yguo L ae wth the Depatment of Enegy Infomaton and Automaton, Southeast Unvesty, Nanjng, 96, Chna (e-mal: wux@seu.edu.cn; shenj@seu.edu.cn; lyg@seu.edu.cn ). Kwang Y. Lee s wth the Depatment of Electcal and Compute Engneeng, Baylo Unvesty, One Bea Place #97356, Waco, X , USA (e-mal: Kwang_Y_Lee@baylo.edu). model of the nonlnea plant s dffcult to obtan and the computatonal buden of nonlnea optmaton s heavy. One way to deal wth the above poblems s usng the combnaton of seveal lnea models to appoxmate the behavo of nonlnea plant and adoptng the nfnte hoon objectve functon n MPC. Based on lnea paamete-vayng (LPV) systems, Kothae et al. [5] have used a feedbac contol law to mnme an uppe bound of the wost-case nfnte hoon objectve functon. he stablty and obustness of the closed-loop system can be guaanteed though solvng a convex optmaton poblem n the fom of lnea matx nequaltes (LMIs). he appoaches poposed n [6], [7] have used a quas-nfnte hoon objectve functon to mpove the method n [5]. By sepaatng the fst o seveal contol moves fom the est of the contol moves govened by the feedbac law and settng them as fee vaables, the desgned contolle can be less consevatve. Howeve, these model pedctve contolles ae developed based on common Lyapunov functon (CLF) whch need to fnd a common postve defnte matx fo all submodels. hs may lead to consevatsm snce t s dffcult to fnd such a matx, especally fo complex systems. Zhang et al. [8] have poposed a aag-sugeno (S) fuy model pedctve contolle based on pecewse Lyapunov functon (PLF) whch can educe the consevatsm n CLF. Howeve, the stablty esult of ths method depends on the pattons of state space. An altenatve way to educe the consevatsm n CLF s to adopt fuy Lyapunov functon (FLF) whch only needs to fnd an ndependent postve defnte matx fo each submodel. he FLF has been used successfully n fuy model pedctve contolles [9]. Moe ecently an extended-fuy Lyapunov functon has been poposed to mpove the pevous esults [], and t can also be used to obtan bette esults of MPC. Besdes mpovng Lyapunov functons, ecently, the technque of slac and collecton matces has been contnuously developed n fuy contol to educe the consevatsm n stablty analyss o stablaton esults []-[3]. Yu et al. [4] have used slac and collecton matces n MPC fo the LPV systems; howeve, they do not use ths technque to constan the uppe bound of the nfnte hoon objectve functon. Moeove, wth the mpoved Lyapunov functons moe slac matces and collecton matces can be used to futhe educe the consevatsm [3]. Based on dscete S fuy model, a stable model pedctve contolle usng extended-fuy Lyapunov functon s poposed n ths pape. he man dea of the //$6. IEEE 98

2 poposed appoach s to desgn a fee contol vaable and a non-paallel dstbuted compensaton (non-pdc) contol law n such a way that an extended-fuy Lyapunov functon s constucted wth mnmng the uppe bound of the nfnte hoon objectve functon n the fuy model pedctve contol. heefoe, the pedctve contolle can guaantee both the stablty of the closed-loop fuy model pedctve contol system and nput constants whle obtanng the optmal tansent pefomance. Combned wth some new esults n fuy contol, we use appopate slac and collecton matces wth extended-fuy Lyapunov functon to futhe educe the consevatsm and enlage the feasble aea of the pedctve contolle. he smulaton on a numecal example demonstates the advantage and effectveness of the poposed appoach. hen, the poposed stable model pedctve contolle wll be appled to a nonlnea bole-tubne coodnated system of a powe unt to change the load n a wde ange. II. PRELIMINARIES A. S Fuy Model Suppose a nonlnea dscete system can be epesented as the followng S fuy model: R : IF ( ) s M, ( ) s M,..., n( ) s Mn HEN x( ) Ax( ) Bu( ),,... () whee R denotes the -th fuy nfeence ule, s the numbe of nfeence ules, M j ( j,,..., n) ae fuy sets, x ( ) R n, u ( ) R m ae espectvely the system state and nput vaables, ( ( ), ( ),..., n( )) ae pemse vaables such as nown state vaables, and ( A, B) s the -th local model of the fuy system. By usng fuy blendng, the dynamc fuy model () can be expessed by the followng global model: x( ) w( )( Ax ( ) Bu ( )) () Ax ( ) Bu ( ) whee A w ( ) A, B w ( ) B n M( ) j j M( ) j w ( ), M ( ) M ( ( )). B. Extended-Fuy Lyapunov Functon As shown n [], an extended-fuy Lyapunov functon s defned by: ( ( ), ) ( ) ( ( ) j( ) j) ( ) j V x x w w S x (3) Fo notatonal convenence, we denote the followng notatons fo a matx X : X w ( )X w ( )X w w w wj j w wj j j j w wj j w wj j j j X ( )X ( )X X ( ) ( )X ( ) ( )X X ( ) ( )X ( ) ( )X thus the extended-fuy Lyapunov functon can be smply descbed as: V x x S x (4) ( ( ), ) ( ) ( ) hen we have: Δ V x x S x x S x (5) ( ( ), ) ( ) ( ) ( ) ( ) If the Lyapunov functon (4) satsfes Δ V < fo all tme, the system can be guaanteed to be stable n the sense of Lyapunov. III. SABLE MPC BASED ON S FUZZY MODEL he closed-loop system stablty as well as a satsfactoy contol pefomance s moe desable fo ndustal pocesses such as fossl fed powe plant, and MPC s an excellent stategy to acheve these goals. he dynamc fuy model () can be ewtten n a pedctve fom as: x( s ) A x( s ) B u( s ) (6) Consdeng the nfnte hoon objectve functon: s J ( ) [ x( s ) Q x( s ) u ( s ) Ru ( s )] whee Q Q >, R R > ae espectvely symmetc weghtng matces of states and contol moves, we pesent the followng man esult. heoem : Fo the dscete S fuy system () unde nput constant: u ( s ) u, s, p,,..., m, p p,max f thee exst contol vaable u ( ), matces Y, G, Q ( Q ) ( j > ), Q l Q l ( Q l ) j j j j j l ( Qj ) ( j >, l > ), Θ l ( Θ l ) ( l > ), m ( ) j mj (7) 98

3 ( j > ), cj ( cj ) ( j > ) and symmetc matces S j S l l j ( j > ), Q Q ( l ), Θ, m, c such that the followng convex optmaton poblem s feasble: mn l γ, u ( ), YGQ,,, Θ, m, c, S st..(9) () j l j j j γ then, the contol acton u ( ) and non-pdc law u( s ) YG x( s ), s > mnme the uppe bound of the objectve functon (7) whle guaanteeng the stablty of the closed-loop system. Q,,,..., (9) j j j j Q Q j > ;, j,,,..., () l l l l l l l l l l l l j j j j j j j j Q Q l > ;,, l,,..., () Q Q Q Q j >, l > ;, j,, l,,..., () Θ Θ Θ Θ Θ Ψ > Θ ( ) Θ Θ( ) Θ (3) Φ,,..., (4) l l Φ Φ l > ;, l,,..., (5) n m,... (6) nj nj mj mj, j,,..., (7) m m m m m Μ > m ( ) m m( ) m (8) u ( ) u p,,..., m (9) p p,max o c,... () oj oj cj cj, j,,..., () c c c c c Γ > () c ( ) c c( ) c whee G Gj S * * * j l ( AG j BY j) S l j Gj γ Q (3) Yj γ R, j,, l,,..., (8) Q Θ Q Θ Q Θ l l l l l l l l l Q Θl Q Θl Φ l Q( ) Θl l l l Q Θl Q( ) Θl Q Θl l,,,..., (4) * * * A( ) x( ) B( ) u( ) S j nj / Q x( ) γ I / R u( ) γ I, j,... (5) U Yj oj, j,,..., Yj G Gj S j U,,,..., pp up,max p m (6) whee U pp s the dagonal elements of the matxu. * n a matx stands fo the coespondng tems of a symmetc matx. Poof: Pat (mnmng the uppe bound of nfnte hoon objectve functon): Fo the nfnte hoon objectve functon (7), dvde t nto two pats [6]: J ( ) x ( ) Qx ( ) u ( ) Ru ( ) J ( ) Suppose the extended FLF satsfes: V( x ( s )) V( x ( s )) [ x ( s ) Q x( s ) u( s ) R u( s )] (7) (8) Summng (8) fom s to s, and wth x( ) and V( x( )), we get: J ( ) V( x( )) x( ) S x( ) (9) hus we can get the uppe bound of J ( ) : J ( ) x ( ) Qx ( ) u ( ) Ru ( ) x ( ) S x ( ) (3) We also dvde the contol nput nto two pats: U [ u( ), U ], whee the fst computed move u ( ) s defned as a fee nput vaable and mplemented on the plant, whleu ae gven though the non-pdc law [5],[]: u s Y G x s s > (3) ( ) ( ), At samplng tme, snce the cuent pemse vaables ae assumed to be avalable, the cuent 983

4 model{ A( ), B( )} can be obtaned and x( ) can be pedcted exactly as: x( ) A ( ) x( ) B ( ) u( ) (3) Substtutng(3) nto(3), we can obtan J ( ) x ( ) Qx ( ) u ( ) Ru ( ) [ A( ) x( ) B( ) u( )] S [ A ( ) x( ) B ( ) u( )] Defne a scala γ and suppose (33) x( ) Qx( ) u( ) Ru( ) [ A ( ) x( ) B( ) u( )] S[ A( ) x( ) B( ) u( )] γ (34) hen mnmng the uppe bound of J ( ) s equvalent to the mnmaton of γ, subject to (34). By defnng S S (35) γ and usng Schu complements [5],(34) can be expessed as: * * * A( ) x( ) B( ) u( ) S / Q x( ) γ I / R u( ) γ I whch s equvalent to(37) accodng to (5), j (36) w( ) wj( ) nj, j,... (37) Now we use slac matces and collecton matces to elax the suffcent condtons fo (37). By applyng (6), (7) and (8), the left sde of (37) w n w wj nj nj j> w m w wj mj mj j> ( ) ( ) ( )( ) ( ) ( ) ( )( ) [ w( ) I w( ) I w ( ) I] M[ w I w I w I] ( ) ( ) ( ) > heefoe, (37) holds. Pat (stablty constant): Wth the non-pdc law: ( ) ( ), (38) u s Y G x s s > (39) the pedctve closed-loop fuy system can be descbed as: x s A x s B Y G x s ( ) ( ) ( ) ( A BYG ) x( s ) (4) Substtutng (35), (39), (4) nto (8) and notcng that ( G ) ( ) S S G S > G S G G G S, the stablty constant (8) s satsfed f: G G S ( AG BY ) S ( AG BY ) Q R (4) G G Y Y > γ γ (4) can be expessed by the LMI below: G G S * * * ( AG BY ) S > G γ Q Y γ R whch s equvalent to (43) accodng to (3): l w wl w wj j l j ( ) ( ) ( ) ( ) > (4), j,, l,,..., (43) Next, combned wth the appoach n [3], we use slac matces and collecton matces to obtan less consevatve suffcent condton fo (43). Applyng (9)-(), yelds: he left sde of (43) w w Q w wj Qj Qj j> l l w( ) wl( )[ w ( )( Q Q ) l> l l l l w ( ) wj ( )( Qj Qj Qj Qj )] j> ( )[ ( ) ( ) ( )( )] Let (44) w ( ) Q w( ) wj( )( Qj Qj ) Θ j> (45) l l w ( )( Q Q ) j> l l l l w ( ) w ( )( Q Q Q Q ) Θ Θ j j j j j l l and apply (3), we have: he left sde of (43) l l l w w w l> [ w ( ) I w ( ) I w ( ) I] ( ) Θ ( ) ( )[ Θ Θ ] (46) 984

5 [ w I w I w I] Ψ ( ) ( ) ( ) > (47) heefoe, (43) holds. Smlaly, by applyng (4), (5) and (4), we can show that (45) and (46) hold. Pat 3(nput constant): Snce we splt the nput nto fee vaables and futue contol moves detemned by non-pdc law, we must constan them accodngly. Fo the fee vaables, we dectly constan them by the pea bound (9). Fo the futue nput moves, by usng the appoach n [5] and consdeng ( G S G ), the constant s satsfed f thee exsts a ( G G S ) matx U, such that the followng LMIs ae feasble: U Y, Upp up,max, p,... m Y G G S (48) Smlaly usng the slac and collecton matces, we can show that ()-() guaantee (48). heefoe, the nput constant s acheved. Rema : We apply an extended-flf to mpove the stablty condton of MPC. Note that f we mpose S S... S S, the extended-flf educes to FLF; and f all Lyapunov matces S j ae mposed to be the same, the extended-flf educes to CLF. heefoe the extended-flf leads to less consevatsm n stablty analyss and stablaton desgn. Rema : he technque of slac and collecton matces s used n all LMIs, ncludng mnmaton of the uppe bound of nfnte hoon objectve functon; stablty constant and nput constant. hs technque can educe the consevatsm n that t collects the nteactons among subsystems. Notce that, combned wth the extended-flf, we use the slac and collecton matces twce n stablty constant, thus less consevatve esult can be obtaned. If all slac matces ae chosen to be eos, the conventonal esult wll be acheved. And snce we use ths technque n all LMIs, we can educe the numbe of matces to educe the computatonal buden fo some matces appeang n the fom of summatons. ae S j and S j, ( j > ) fo example, snce they always appea n the fom of S j S j,we can set S j S j ( j > ) to educe the numbe of matces. Rema 3: Compaed wth the conventonal fuy contol, the poposed appoach has the advantage that t can acheve stablty n an optmal way; moeove, (8) guaantees the Lyapunov functon of stable MPC to decease faste than conventonal fuy contol ( ΔV ). IV. ILLUSRAIVE EXAMPLE In ths secton, an example s pesented to show the advantage of the poposed stable MPC. Consde the followng dscete nonlnea model: x( ) x( ) x( ) x( ) (5 x( )) u( ) (49) x( ) x( ).5 x( ) x( ) u( ) Defne x as the pemse vaable, then (49) can be descbed by the S fuy model: R : IF x () s M HEN x( ) A x( ) B u( ), wth membeshp functons:.65 x( ).65 x( ) M( ) ; M( ) he system matces ae gven by: A, B A, B We apply the appoach n heoem and appoaches n [6], [9] and [6] to ths model wth the ntal state.8 x() [.65 5] ; contolle paametes Q.95, R.9 and nput constant u.3. x x u Fg.. Closed-loop contol pefomance fo heoem and appoach n [6]. (sold lne: heoem ; dotted lne: appoach n [6]) Smulaton esults shows that thee ae no feasble solutons fo the appoach n [6] and [9] whch ae MPC based on CLF and FLF. Whle the othe two appoaches can fnd feasble solutons and the contol esults ae shown n Fg.. hs clealy demonstated the advantage of the extended-flf. Appoach n [6] s anothe MPC based on extended-flf, howeve, snce we use appopate slac and collecton matces n all LMIs and tae full advantage of 985

6 extended-fuy Lyapunov functon, the appoach we popose has bette pefomance. V. APPLICAION O A BOILER-URBINE COORDINAED SYSEM OF POWER UNI In ths secton, the poposed contolle s appled to the typcal Bell-Åstöm bole tubne dynamc model [7]. he model s a 3d ode MIMO system whch has the chaactestcs of hghly nonlneaty. he state vaables n x [ x, x, x3] denote the dum pessue (g/cm ), the powe output (MW) and the densty of flud n the system (g/cm 3 ), espectvely. he nomaled nput vaables n u [ u, u, u3] denote the poston of fuel flow valve, the poston of steam contol valve and the poston of feedwate flow valve, espectvely. All valve poston vaables ae constaned to le n the nteval [, ]. he output vaables n y [ y, y, y3] denote the dum pessue (g/cm ), the powe output (MW) and the dum wate level (m), espectvely. w HIGH LOW Dum Pessue Fg.. Membeshp functons of the bole-tubne fuy model. dum pessue(g/cm) powe output(mw) dum wate level(m) tme(second) tme(second) tme(second) Fg. 3. Closed-loop contol pefomance of the bole-tubne coodnated system fo theoy (output). Lneang the nonlnea system aound heavy load opeatng pont (%) and low load opeatng pont (8%) usng aylo s sees appoxmaton, then choosng the membeshp functon as shown n Fg., we can easly obtan a two ules S fuy model n fom of () to epesent the nonlnea BA model. Owng to space lmtatons, we shall omt the system matces hee. fuel flow valve steam contol valve feedwate flow valve tme(second) tme(second) tme(second) Fg. 4. Closed-loop contol pefomance of the bole-tubne coodnated system fo theoy (nput). We now apply the poposed contolle to the bole-tubne coodnated system and consde the case of load change n a wde ange. he contol msson s tacng the expected opeatng ponts of dum pessue and output powe whle mantanng the dum wate level. We assume that at ts, the desed opeatng pont changes n step fom 7% to 3%, and the setpont of dum wate leve mantans at eo. Wth contolle paametes: 3 Q 5 ; R 4 the smulaton esults ae shown n Fg. 3 and Fg. 4. Fom the contol esults, we can see that when the load s nceased, the dum pessue and powe output espond apdly, and then appoaches to the expected opeatng ponts, whle the dum wate level jumps and then gadually etun to eo afte a peod of fluctuaton. he poposed fuy stable MPC can contol the bole-tubne coodnated system effectvely. VI. CONCLUSION he combnaton of model pedctve contol and S fuy model s an effectve way to solve the contol poblem of nonlnea system, whch motvates us to popose a stable fuy MPC based on extended-flf. By solvng a set of LMIs, both the stablty of the closed-loop system and the satsfacton of nput constant can be acheved n an optmal way. he extended-flf educes the consevatsm of CLF and FLF; and enlages the feasble aea of the pedctve contolle. Moeove, combned wth some new esults n fuy contol, we use appopate slac and collecton matces wth extended-flf to futhe educe the consevatsm. he smulatons on numecal example and nonlnea bole-tubne coodnated system of powe plant 986

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