Optimal Guaranteed Cost Control of Linear Uncertain Systems with Input Constraints

Transcription

1 Internatona Journa Optma of Contro, Guaranteed Automaton, Cost Contro and Systems, of Lnear vo Uncertan 3, no Systems 3, pp 397-4, wth Input September Constrants Optma Guaranteed Cost Contro of Lnear Uncertan Systems wth Input Constrants L Yu, Qng-Long Han, and Mng-Xuan Sun Abstract: he guaranteed cost contro probem for a cass of near systems wth normbounded tme-varyng parameter uncertantes and nput constrants s consdered A suffcent condton for the exstence of guaranteed cost state feedback controers s derved va the near matrx nequaty (LMI) approach, and a desgn procedure to guaranteed cost controers s gven Furthermore, a convex optmzaton probem s formuated to determne the optma guaranteed cost controer An exampe s gven to ustrate the effectveness of the proposed resuts Keywords: Guaranteed cost contro, nput constrants, LMI, uncertan systems INRODUCION he probem of desgnng robust controers for systems wth mode uncertanty has drawn consderabe attenton n recent contro system terature Much effort has been drected towards fndng a controer n order to guarantee robust stabty [-3] However, when controng a rea pant, t s aso desrabe to desgn a controer whch not ony makes the cosedoop system asymptotcay stabe but aso guarantees an adequate eve of performance One approach to ths probem s the so-caed guaranteed cost contro approach gven by Chang and Peng [4] hs approach has the advantage of provdng an upper bound on a gven performance ndex and thus the system performance degradaton ncurred by the mode parameter uncertantes s guaranteed to be ess than ths bound Based on ths dea, many sgnfcant resuts have been proposed [5-] In partcuar, Petersen and McFarane [6] ntroduced a noton of quadratc guaranteed cost contro whch extends the noton of quadratc stabzabty to aow for a quadratc performance ndex and presented a Rccat equaton approach for desgnng quadratc guaranteed cost controers Yu et a [9] presented a near matrx nequaty (LMI) approach for the desgn of guaranteed Manuscrpt receved February 6, 5; accepted June 4, 5 Recommended by Edtora Board member Jae Weon Cho under the drecton of Edtor-n-Chef Myung Jn Chung hs work was supported by the Natona Natura Scence Foundaton of PRChna under grant L Yu and Mng-Xuan Sun are wth the Department of Automaton, Zhejang Unversty of echnoogy, Hangzhou 33, PRChna (e-mas: {yu, mxsun}@zjuteducn) Qng-Long Han s wth the Facuty of Informatcs and Communcaton, Centra Queensand Unversty, Rockhampton, Qd 47, Austraa (e-ma: qhan@cqueduau) cost controer, and the desgn probem of optma guaranteed cost controer, whch mnmzes the assocated guaranteed cost, was formuated as a convex optmzaton probem wth LMI constrants On the other hand, a physca contro systems have to operate under constrants on the magntude of the contro nput due to the physca mtatons of actuators hese mtatons n terms of nput constrants must be consdered n the controer desgn Otherwse the desred cosed-oop system performance cannot be guaranteed and even the cosed-oop system w become unstabe herefore, t s necessary to consder nput constrants n the desgn of the guaranteed cost controers However, at the knowedge of the authors, the guaranteed cost contro probem for uncertan system subject to nput constrants has been receved very tte attenton n terature hs paper s concerned wth the guaranteed cost contro probem for a cass of uncertan systems subject to nput constrants he mode parameter uncertantes are assumed to be tme-varyng and norm-bounded Condtons for the exstence of state feedback guaranteed cost controers satsfyng the gven constrants are derved va the LMI approach Furthermore, a convex optmzaton probem wth LMI constrants s presented to desgn the optma guaranteed cost controer of uncertan systems wth nput constrants Fnay, an exampe s gven to ustrate the proposed resuts, and the comparson wth the exstng resuts s made PROBLEM AND PRELIMINARIES Consder the foowng near uncertan systems: x () t = [ A+ A] ) t + [ B+ B] u(), t () ) = x,

2 398 L Yu, Qng-Long Han, and Mng-Xuan Sun n where R s the state vector, ut ( ) R s the contro nput vector, A and B are known constant rea matrces of approprate dmensons, A and B are rea vaued matrx functons representng tme-varyng parameter uncertantes n the system mode he contro nput u n system () s subjected to the foowng constrants: u u u, =,,, m, () where u s the th eement n the contro nput u, u, =,,, m are known constants he parameter uncertantes under consderaton here are assumed to be norm-bounded and of the form [ A B] = DF( [ E E ], (3) where D, E, E are known constant rea matrces of approprate dmensons, whch represent the structure j of uncertantes, and F( R s an unknown matrx functon wth Lebesgue measurabe eements and satsfes F ( F( I, (4) n whch I denotes the dentty matrx of approprate dmenson he uncertantes s sad to be admssbe f they satsfy the reatons (3) and (4) Assocated wth the system () s the cost functon J = [ x ( Q + u ( Ru( ] dt, (5) where Q and R are gven postve-defnte symmetrc matrces Defnton: A memoryess state feedback contro aw u ( = K s sad to be a quadratcay guaranteed cost controer of system () wth cost functon (5) f there exsts symmetrc postve defnte n n matrx P R such that Q + K RK + P[ A + BK + DF( E + E K)] + [ A+ BK + DF( E + E K)] P < m (6) for a admssbe uncertantes Lemma [6]: If u( = K s a quadratcay guaranteed cost controer of system () wth cost functon (5), then the cosed-oop uncertan system x = [ A + BK + DF( E + E K)] ) (7) ( t s quadratcay stabe, and the cost functon vaue of the cosed-oop system s no more than J = x Px, whch s sad to be a guaranteed cost of system () From the proof of Lemma, t foows that the matrx P s a Lyapunov matrx of the cosed-oop system wth the controer u( = K Furthermore, a guaranteed cost of system () can be determned n terms of the matrx P and the nta state It s cear that such a guaranteed cost depends on the choce of guaranteed cost controers In partcuar, the guaranteed cost controer to mnmze the correspondng guaranteed cost s more nterestng, such a controer s sad to be the optma guaranteed cost controer he objectve of ths paper s to deveop a procedure to desgnng the optma guaranteed cost controer for the system () subject to nput constrants 3 MAIN RESULS We frst present the foowng resut: heorem : If there exst a postve scaar α, a matrx K and symmetrc postve defnte matrces P and Z such that the matrx nequaty (6) hods for a admssbe uncertantes and x Px α, (8) Z K, K α P (9) Z, =,,, m () ( ) u hen u( = K s a quadratcay guaranteed cost controer satsfyng the constrant () of the system () Proof: It foows from the condton of ths theorem and Lemma that u( = K s a quadratcay guaranteed cost controer of the system () and the matrx P s a Lyapunov matrx of the assocated cosed-oop system herefore, the nequaty (8) mpes that the cosed-oop state trajectory x ( satsfes x ( P α By the Schur compement, t foows that the matrx nequaty (9) s equvaent to α KP K Z Denote the row of the matrx K by K, then () = () = () KP P xt () = KP K x () tpxt () KP K α ( Z) u t K x t K P P x t From the nequaty () we can concude that the contro aw u( = K satsfes the constrant () hs competes the proof of the theorem he foowng theorem s the man resuts of ths paper heorem : If there exst postve scaars α and ε, a matrx Y and symmetrc postve defnte matrces X and Z such that the foowng matrx nequates hod:

3 Optma Guaranteed Cost Contro of Lnear Uncertan Systems wth Input Constrants 399 Ω X Y ( E X + EY ) X αq <, Y αr E X + EY εi () x, () x X Z Y, (3) Y X Z, =,,, m, (4) ( ) u where Ω = AX + BY + ( AX + BY ) + εdd, hen u(=yx - s a quadratcay guaranteed cost controer satsfyng the constrant () of the system (), and the cost functon of the correspondng cosedoop system satsfes J α Proof: Pre- and post-mutpyng the eft-hand sde of the matrx nequaty (9) by matrx dag{ I, α P } mpy that the matrx nequaty (9) s equvaent to Z αp K αkp αp (5) By denotng X = α P and Y = KX, the matrx nequaty (3) s mmedatey obtaned from the nequaty (5) Pre- and post-mutpyng the efthand sde of the matrx nequaty (6) by matrx α P, t foows that the matrx nequaty (6) s equvaent to αp + E QP + αp K RKP + α[ A+ BK + DF( E K)] P + αp [ A+ BK + DF( E + E K)] <, whch can be further wrtten as α XQX + α Y RY + AX + BY + ( AX + BY ) + DFEX ( + EY) + [ DFEX ( + EY)] < By the Schur compement, the above matrx nequaty s equvaent to AX + BY + ( AX + BY ) X Y X α Q Y α R D + F[ EX + EY ] D + [ EX + EY ] F < It foows from Lemma n [] that the above matrx nequaty s true for a F satsfyng F F I f and ony f there exsts a postve scaar ε such that AX + BY + ( AX + BY) X Y D D + ε + ε [ E X + E Y ] [ E X + E Y ] < αq X Y αr Quotng the Schur compement agan, the above matrx nequaty s equvaent to the matrx nequaty () Fnay, from the Schur compement and X = α P, t foows that the nequaty (8) s equvaent to the matrx nequaty () herefore, the resuts of heorem can be obtaned from heorem, whch competes the proof of the theorem ()-(4) s a near matrx nequaty system n ε, α, X,Y,Z and defnes a convex set of ( ε, α, X,Y,Z) Hence, the exstng convex optmzaton technques such as nteror-pont agorthms can be used to test whether ths set s nonempty and to generate partcuar soutons f the LMI system s feasbe Moreover, ts soutons parametrze the set of guaranteed cost controers hs parametrzed representaton of guaranteed cost controers can be expoted to desgn the guaranteed cost controers wth some addtona requrements In partcuar, we sha use ths representaton to present a desgn procedure for the optma guaranteed cost controer that mnmzes the guaranteed cost of the cosed-oop uncertan system Accordng to the heorem, the desgn probem of the optma guaranteed cost controer can be formuated as the foowng optmzaton probem: mn α (6) st (), (), (3), (4) If the probem (6) has an optma souton ε, α, X, Y,Z, then u( = YX s the optma guaranteed cost controer satsfyng the constrant () It s cear that the probem (6) s a convex optmzaton probem wth LMI constrants herefore, the goba mnmum of the probem can be reached f t s feasbe, and t can be easy soved by usng the sover mncx n the LMI oobox of MALAB 4 BLOCK-DIAGONAL PARAMEER UNCERAINY In the secton we use the above resuts to sove the guaranteed cost contro probem for systems wth

4 4 L Yu, Qng-Long Han, and Mng-Xuan Sun bock-dagona tme-varyng parameter uncertantes and nput constrants Consder the uncertan system () wth the parameter uncertanty descrbed by (3)-(4) Suppose that the uncertan matrx F ( s of the foowng bock-dagona form: where { F (, F (,, F ( )} F( = dag t, (7) k k jk F ( R and satsfes k k jk jk F () t F () t I, k =,,,, n whch I j k j k denotes j k jk dentty matrx hen, for any constant vector [ ] ε = ε ε ε, εk >, k =,,, Defne { ε ε ε } { ε j j ε j j ε j j } M = dag I, I,, I, N = dag I, I,, I Obvousy, we have ~ ~ ~ [ E E ] DMF( [ NE N ] DF ( = E he foowng theorem provdes a souton to the desgn probem of the optma guaranteed cost controer for systems wth bock-dagona tmevaryng parameter uncertantes and nput constrants heorem 3: If the foowng convex optmzaton probem mn α (8) ε, α, X, Y, Z Ω X Y Ω DM X αq st () Y αr < Ω N MD M () (), (3), (4) has a souton ε, α, X, Y, Z, where Ω = AX + BY + ( AX + BY ), Ω = E X + EY hen u( = YX s the optma guaranteed cost controer satsfyng the constrant () of the system () wth bock-dagona tme-varyng parameter uncertantes (7) Where [ ε ε ε ], ε = M = dag{ εi,,, }, ε I ε I N = dag{ ε I, ε I,, ε I } j j j j j j Athouth the probem (6) can be aso used to sove the guaranteed cost contro probem for systems wth bock-dagona parameter uncertantes, heorem 3 w gve a ess conservatve resuts due to the ntroducton of free parameters ε, ε,, ε 5 ILLUSRAIVE EXAMPLES Consder the same exampe as n [] hs exampe represents an uncertan mode of the dynamcs of a hecopter n a vertca pane he uncertan dynamca mode s as foows: x = ( A + r = x, (9) where A + r A ) x + ( B + sb ) u, ) A =, B =, x =, A =, A =, B =, 3 r, r, s he contro nput u n system (9) s subjected to the foowng constrants: u, =, the assocated performance ndex s J = ( x Qx + u Ru) dt, where Q =, R = Defne F = dag{ r, r, s}

5 Optma Guaranteed Cost Contro of Lnear Uncertan Systems wth Input Constrants 4 D =, 9 E = 3, E = 673 hen (9) can be rewrtten as x ( = ( A + DFE) + ( B + DFE ) u( hs s a system wth bock-dagona tme-varyng parameter uncertantes By appyng heorem 3 and sovng the correspondng optmzaton probem (8), we obtan the optma guaranteed cost controer u( = () and the guaranteed cost of the uncertan cosed-oop system s J * = 64 If we do not consder the nput constrants, Yu et a (999) gave the optma guaranteed cost controer u( = () and the guaranteed cost of the uncertan cosed-oop system s J * = 534 o compare the effect of the controers () and () by smuaton, we assume that s = sn t, r = sn t, r = sn 3t he contro aw () (sod ne) and () (dot ne) are shown n Fg he state varabes of correspondng cosed-oop systems are shown n Fg Where the sod nes stand for the x x u x u x Fg Contro aw Fg he cosed-oop state varabes

6 4 L Yu, Qng-Long Han, and Mng-Xuan Sun state varabes of the cosed-oop system resuted from controer (), and the dot nes denote ones of the cosed-oop system resuted from the controer () It can be seen from Fg that the magntudes of the nput varabes u and u n the controer () are sgnfcanty reduced due to consderng the nput constrants () n the desgn 6 CONCLUSIONS In ths paper, we have presented an LMI based approach to the optma guaranteed cost contro probem va state feedback contro aws for a cass of uncertan systems Contrast to the Rccat equaton based approach, ths approach has the advantage that no tunng of parameters and/or matrces s nvoved, and some addtona requrements and constrants can be effectvey treated REFERENCES [] B R Barmsh, Stabzaton of uncertan systems va near contro, IEEE rans on Automatc Contro, vo 8, no 3, pp , March 983 [] P P Khargonekar, I R Petersen, and K Zhou, Robust stabzaton of uncertan near systems: Quadratc stabzabty and H contro theory, IEEE rans on Automatc Contro, vo 35, no 3, pp , March 99 [3] I R Petersen, A stabzaton agorthm for a cass of uncertan systems, Systems & Contro Letters, vo 8, no 3, pp 8-88, 987 [4] S S L Chang and K C Peng, Adaptve guaranteed cost contro of systems wth uncertan parameters, IEEE rans on Automatc Contro, vo 7, no 4, pp , Apr 97 [5] D S Bernsten and W M Haddad, Robust stabty and performance va fxed-order dynamc compensaton wth guaranteed cost bounds, Math Contr Sgnas and Systems, vo 3, no, pp 39-63, June 99 [6] I R Petersen and D C McFarane, Optma guaranteed cost contro and fterng for uncertan near systems, IEEE rans on Automatc Contro, vo 39, no 9, pp , September 994 [7] I R Petersen, D C McFarane, and M A Rotea, Optma guaranteed cost contro of dscrete-tme uncertan near systems, Internatona Journa of Robust & Nonnear Contro, vo 8, no 7, pp , Juy 998 [8] L Yu and J Chu, An LMI approach to guaranteed cost contro of near uncertan tmedeay systems, Automatca, vo 35, no 6, pp 55-59, June 999 [9] L Yu, G Chen, and M Yang, Optma guaranteed cost contro of near uncertan systems: LMI approach, Proc of the 4th IFAC Word Congress, vo G, pp , 999 [] L Yu and F Gao, Optma guaranteed cost contro of dscrete-tme uncertan systems wth both state and nput deays, Journa of the Frankn Insttute, vo 338, no, pp -, January [] A Fshman, J M Don, L Dugard, and A Neto, A near matrx nequaty approach for guaranteed cost contro, Proc of the 3th IFAC Word Congress, pp 97-, 996 L Yu receved the BS degree n Contro heory from Nanka Unversty n 98, and the MS and PhD degrees from Zhejang Unversty, Hangzhou, Chna He s currenty a Professor n the Coege of Informaton Engneerng, Zhejang Unversty of echnoogy, Chna Hs research nterests ncude robust contro, tme-deay systems, decentrazed contro Qng-Long Han receved the BS degree n Mathematcs from the Shandong Norma Unversty, Jnan, Chna, n 983, and the ME and PhD degrees n Informaton Scence from the East Chna Unversty of Scence and echnoogy, Shangha, Chna, n 99 and 997, respectvey From September 997 to December 998, he was a Post-Doctora Researcher Feow at LAII-ESIP, Unversté de Poters, France From January 999 to August, he was a Research Assstant Professor n the Department of Mechanca and Industra Engneerng, Southern Inos Unversty at Edwardsve, USA In September he joned the Facuty of Informatcs and Communcaton, Centra Queensand Unversty, Austraa, where he s currenty a Senor Lecturer Hs research nterests ncude tme-deay systems, robust contro, networked contro systems, compex systems and software deveopment processes Mng-Xuan Sun receved the PhD degree from Nanyang echnoogca Unversty, Sngapore, n He s currenty a Professor n the Coege of Informaton Engneerng, Zhejang Unversty of echnoogy, Chna Hs man research nterests ncude teratve earnng contro and optma contro