Decentralized robust control design using LMI

Transcription

1 Acta Montanstca Slovaca Ročník 3 (28) číslo -4 Decentralzed robust control desgn usng LMI Anna Flasová and Dušan Krokavec ávrh robustného decentralzovaného radena pomocou LMI he paper deals wth applcaton of decentralzed controllers for large-scale systems wth subsystems nteracton and system matrces uncertantes. he desred stablty of the whole system s guaranteed whle at the same tme the tolerable bounds n the uncertantes due to structural changes are maxmzed. he desgn approach s based on the lnear matrx nequaltes (LMI) technques adaptaton for stablzng controller desgn. Key words: Decentralzed control state feedback parameter uncertantes lnear matrx nequaltes Introducton Recently a number of efforts have been made to extend the applcaton of robust control technques to decentralzed systems usng convex optmzaton nvolvng LMI (Lnear Matrx Inequaltes). he paper presents some extensons and modfcatons of the problems concernng the system robust stablty n the presence of nterconnecton among subsystems as well as parameter uncertantes n the subsystem state transton matrces and was motvated by the technque presented n (Befekadu & Erlch 25) whch s generalzed here for uncertan large-scale systems. System descrpton Generally a large-scale nterconected system composed of subsystems can be modelled by dfferental equatons of the form q& () t = ( A + A ()) t q () t + B u () t + ( A + A ()) t q () t () h h h h= h y() t = Cq () t (2) wth gven subsystem matrx A subsystem nput matrx B and subsystem output matrx C of approprate dmenson where = 2... q (t) s the subsystem state vector u (t) s the subsystem control nput vector and y (t) s the subsystem measurement vector. It s assumed the matrces B C to be of full rank all the subsystem states can be observed or measured pars (A B ) and (A C ) are stablzable and observable respectvely and the subsystem () (2) s controlled by local state feedback control law u( q( t )) = Kq ( t) (3).e. each subsystem s controlled by local control law where K s a m x n constant matrx. It s supposed that the nterconnecton uncertanty terms n () can be wrtten as ( A + A ()) t q () t = G g ( q()) t G ε H q () t G ε H q () t (4) h h h h h h h= h h= h where h h 2 H = [ H LH ] H = ε = max ε q( t) = ( t) ( t) ( t) q q L q (5) g q g q q H H q (6) 2 ( ( t)) ( ( t)) ε ( t) ( t) ε > s a parameter related to nterconnecton uncertantes n the subsystem and H and G are constant matrces of approprate dmensons. It s assumed that consdered uncertanty matrces A (t) = 2... are norm bounded and can be descrbed as (Krokavec & Flasová 22) Anna Flasová Dušan Krokavec Department of Cybernetcs and Artfcal Intellgence echncal Unversty of Košce Košce Slovak Republc. anna.flasova@tuke.sk dusan.krokavec@tuke.sk (Recenzovaná a revdovaná verza dodaná )

2 Acta Montanstca Slovaca Ročník 3 (28) číslo -4 A() t = ML() t (7) L() t L() t < I (8) where M s a n x l constant matrx s a h x n constant matrx and I s h x h denttty matrx. Known matrces M defne the structure of the state transton matrx uncertantes of the th subsystem and the parameter uncertanty matrces L (t) belong to the set l h L= { L( t) R : LL < I} = 2 K (9) Usng the overall system state varable vector q(t) defned n (6) the nterconnected system model can be compactly wrtten as follows q& () t = ( A+ A()) t q() t + Bu() t + Gg( q()) t () y() t = Cq () t () 2 2 L u() t = u () t u () t L u () t y() t = y () t y () t y () t (2) A = dag A L t L A A( t) = dag A( t) A( ) (3) B= dag BLB C= dag CL C G = dagglg (4) wth the uncertantes upper-bounds (9) and nterconectons upper-bounds g 2 ( q ( t)) g ( q ( t)) = g ( q ( t)) g ( q ( t)) q ( t) ε ( t) H H q. (5) hus the control law for the overall system wll be u() t = Kq() t K = dag K K2 L K. (6) Lyapunov functon Snce overall system () () s lnear n q(t) the Lyapunov functon canddate v(q(t)) can be of the form v( q( t)) = q ( t) P( t) q( t ). (7) Here v(q(t)) s a quadratc postve defnte functon wth symmetrc postve defnte weghtng matrx P(t). If a steady-state soluton P of P(t) under the boudary condton P() = exsts the evaluatng dervatve of (7) for steady-state soluton P gves dv( q( t )) = q ()( t A P+ A () t P+ PA+ P A() t + K B P+ PBK+ g ( q()) t G P+ PGg( q())) t q() t < d t (8) It now follows for (8) usng dentty ( e.g. Krokavec & Flasová 26a) that UV + VU UU + VV (9) q ()( t A P + A () t P + PA + P A() t + K B P + PBK + g ( q()) t g( q()) t + PGG P) q() t dv( q( t)) dt ()( t () t () t 2 ε ) () t q A P+ A P+ PA+ P A + K B P+ PBK+ PGG P+ H H q < t t 2 ε A P + A () P + PA + P A() + K B P + PBK + PGG P + H H < (2) respectvely where (2) s the matrx Lyapunov equaton. Usng (7) (8) the sum P A(t) + A (t)p n (2) can be rewrtten as (2)

3 Anna Flasová and Dušan Krokavec: Decentralzed robust control desgn usng LMI P A() t + A () t P = Pdag ML() t L ML() t + dag L() t M L L() t MP (22) where υ > = 2... are desgn parameters. Also usng (9) an upper bound of (22) s 2 2 P A() t + A () t P Pdag M M L MM P+ dag 2 L() t L() t L () () 2 L t L t 2 2 Pdag M M L MM P+ dag 2 L. 2 (23) Denotng M = L M L = L L (24) nequalty (23) takes on the form as follows P 2 A () t + A P () t + PM M P 2 (25) and n adton the matrx Lyapunov equaton (2) has an expresson of the form 2 2 ε 2 A P + PA + K B P + PBK + PGG P + H H + PM M P + <. (26) hat ensures the negatve defnteness of the quadratc functon dervatve under the constrants (9) and (5) for all trajectores of the closed-loop nterconnected system () () under the control. Let the lnear matrx nequalty be gven as Schur complement Q S < Q = Q R = R S R hen Gauss elmnaton yelds (Boyd at al. 994). (27) I SR Q S I Q+ SR S I SR = det = (28) I S R R S I R I and (28) mples Q S Q+ SR S < < + < Q SR S S R R R >. (29) LMI form of desgn condtons Inequalty (26) can be converted nto LMI form usng a technque based on Schur complements (29). ote that matrx nequalty (26) s not convex n P and K but wth any lnear fractonal transformaton can be transformed to those form. hus pre-multplyng (26) from left and rght hand sde by matrx P - (matrx P s a postve defnte matrx and then matrx P - s postve defnte too) gves 2 2 ε 2 P A + AP + P K B + BKP + GG + P H H P + M M + P P <. (3) Usng substtutons 2 2 X= X = P > µ = ε > η = > = 2 K Y = KP = KX (3) nequalty (3) takes on the followng form ( µ η η ) XA + AX + Y B + BY + GG + XH H X + M M + X X < (32) 2

4 Acta Montanstca Slovaca Ročník 3 (28) číslo -4 whch s convex n X and Y. If ths LMI problem n X and Y has a soluton then the Lyapunov functon v(q(t)) = q (t)x - q(t) proves the quadratc stablty of the closed-loop system wth state feedback u(t) = = YX - q(t) = K q(t). Usng LMI varables (3) the constrant (32) can be represented as the LMI where R G XH L XH X L X G I L L HX µ I L L M M M O M M M M < µ (33) H X L I L X L η I L M M M L M M O M X L L η I = η R XA AX Y B BY M M. (34) ote that t can also be consdered an LMI n X and µ η = 2 defned n (3). he soluton of ths problem can be expressed as a state feedback (3) wth state feedback gan matrx K = YX = YP (35) X Y s the unque soluton of (33) X s a symmetrc block-dagonal postve defnte matrx and Y K are block dagonal matrces respectvely generally nonsymmetrc. Illustratve example o demonstrate the algorthm propertes the multarea model of a power system was used (Veselý et al. 22 Krokavec & Flasová 26b). Gven model structures satsfy () - () where A B A = A = B = B2 = A = = B A2 B2.55 G G = G2 = G = [.55 ] 2 [.55 ] H = H = G2.5 A( t) = A2( t) =.75 A( t) M 2 = M = = 2 =.75 a32 ( t) a34 ( t) 2.5 M M = M2 = = 2 = 2 M2 he goals were to desgn equal feedback gan matrces for both subsystems ndependent on dfferent subsystems parameter uncertantes. he problem was solved usng the Self-Dual-Mnmzaton (SeDuM) package for MALAB (Peaucelle et al. 22). hs package s constructed based upon the prncples that one can descrbe the system to be analyzed usng a MALAB standard functon and LMI varables nequalty constrants as well as LMI soluton to be specfed wthn the SeDuM nterface add-on for MALAB. he LMI problem was feasble n X and Y and results n 3

5 Anna Flasová and Dušan Krokavec: Decentralzed robust control desgn usng LMI X = Y = µ = µ = η =.274 η = Accordng to (35) the gan matrces were obtaned as follows [ ] K = K 2 = One can easly verfy that closed loop s stable wth the same egenvalues spectrum of the subsystem transton matrx ρ(a ) = ρ(a 2 ) ={-.692 ± j ± j }. Concludng remarks he decentralzed robust controller desgn s formulated as an optmzaton problem nvolvng lnear matrx nequaltes and solved by LMI programmng method where the controller structures take nto account the nteractons among subsystems and the subsystem state transton matrces uncertantes. he most mportant practcal applcaton s that there are effectve and powerful algorthms for these by LMI reformulated problems that s algorthms that compute the global optmum wth non-heurstc stoppng crtera (the global optmum s obtaned to wthn some pre-specfed accuracy). Presented decomposton prncple gves enough flexblty to allow the ncluson of more general nteracton structures and parametrc uncertantes wthn all matrces of lnear systems gven by the state-space descrpton. Acknowledgement: he work presented n ths paper was supported by Grant Agency of the Mnstry of Educaton and the Academy of Scences of the Slovak Republc VEGA under Grant o. /273/5. References Bekefadu G. K. Erlch I.: Robust decentralzed controller desgn for power systems usng convex optmzaton nvolvng LMIs. In Preprnts of the 6th IFAC Word Congress 25. Prag Czech Republc [CD-ROM]. Boyd S El Ghaou L. Peron E. Balakrshnan V.: Lnear Matrx Inequaltes n System and Control heory. SIAM Socety for Industral and Appled Mathematcs Phladelpha 994. Krokavec D. Flasová A.: Optmal Stochastc Systems (n Slovak). Elfa Košce 22. Krokavec D. Flasová A.: Dscrete-me Systems (n Slovak). Elfa Košce 26. Krokavec D. Flasová A.: Decentralzed control of power systems. In Proceedngs of the 7th Conference Control of Power & Heatng Systems 26 Zlín Czech Republc. Academa Centrum Zlín 26 pp. P3-5. Peaucelle D. Henron D. Labt Y. atz K.: User s Gude for SeDuM Interface.4. LAAS-CRS oulouse 22. Veselý V. Rosnová D. Kozáková A.: Decentralzed controller desgn. A subsystem approach. In Proceedngs of the Conference Process Control 22 Kouty nad Desnou Czech Republc. Unversty of Pardubce 22 [CD-ROM]. 4