Output Feedback Robust Stabilization of the Decoupled Multiple Model

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1 Output Feedback Robust Stablzaton of the Decoupled Multple Model Ahmed ahar, Mohamed Naceur Abdelkrm Abstract hs paper ams to desgn a controller to robustly stablze uncertan nonlnear systems wth norm bounded uncertantes and unmeasured state varables va decoupled mult-model. he stablzaton condtons are gven n the form of lnear matrx neualtes. Suffcent condtons are derved for robust stablzaton n the sense of Lyapunov asymptotc stablty and are formulated n the format of lnear matrx neualtes (LMIs. he effectveness of the proposed decoupled mult-model controller and mult-observer desgn methodology s fnally demonstrated through numercal smulatons. Index erms Decoupled multple model, LMI, Mult-observer, robust control. I. INRODUCION he control of dynamc systems n the presence of severe nonlneartes and uncertantes s of great attenton for several researchers. In order to overcome ths knd of problems, t s necessary to develop an ntellgent modellng and control approaches. Indeed, multple model and mult-control approaches are consdered to be useful for ndustral processes whch are, often, complex, uncertan, ll-defne, and have avalable ualtatve knowledge from doman experts for ther controller desgn. here have been many successful applcatons n the ndustry to date ([] [3], [9], [], [3]. wo basc structures of multple model can be llustrated for aggregatng the local models between them ([4], [5], []. In the frst structure, the local models have a common state vector (akag-sugeno multple model; n the second one, the local models are decoupled and ther state vectors are dfferent (decoupled multple model. he decoupled state multple model has been unfortunately poorly studed n the lterature. However, t represents an ncreasng relevance to akag-sugeno multple model. Indeed, the usefulness of ths multple model has been clearly employed for the control and the modellng of nonlnear systems ([4], [6], [7], []. More recently, the state estmaton problem has been consdered n [6] and [7]. he man aspect of the decoupled multple model s that submodels of dfferent order (e.g. number of states can be used. hs fact ntroduces some adaptablty degrees n the modellng step because the dmensons of the submodels can Manuscrpt Receved on February 5. Ahmed ahar, Department of Electrcal Engneerng, Natonal School of Engneers of Gabes, Unversty of Gabes, St Omar Ibn El-Khattab, 69, Gabes, unsa. Mohamed Naceur Abdelkrm, Department of Electrcal Engneerng, Natonal School of Engneers of Gabes, Unversty of Gabes, St Omar Ibn El-Khattab, 69, Gabes, unsa. be well adapted to the complexty of the system nsde each operatng zone and conseuently the total number of parameters necessary for descrbng the system can be reduced. As well as stablty, robustness s another performance to be examned n the study of uncertan nonlnear control systems. he parametrc uncertanty s a prncpal cause responsble for the degraded stablty and the performance of an uncertan nonlnear control system. In fact, n many cases t s very dffcult, f not mpossble, to obtan the correct values of some system parameters. hs s due to the mprecse measurement, unmeasurable system parameters, or on-lne varaton of the parameters. herefore, ths has mproved some actve research n the last few years ([8], [9]. Robustness n multple model-based control has been largely studed n the past, such as the stablty robustness aganst modellng errors ([], [], some control technues for -S multple models ([], [3]. Motvated by the aforementoned concerns, ths paper deals wth the parametrc uncertantes ssue n a nonlnear system wth decoupled multple model. It s well recognzed that the observer desgn s a very mportant complcaton n control systems. Snce n many practcal nonlnear control systems, state varables are often naccessble, output feedback or observer-based control s reured and has caused some nterest. anaka et al. [4], Ma et al. [5] and Jun Yoneyama et al. [6] studed fuzzy observer desgns for -S fuzzy control systems, and they proved that a state feedback controller and an observer always yelds a stablzng output feedback controller provded that the stablzng property of the control and asymptotc convergence of the observer are guaranteed by the Lyapunov method. However, n the above output feedback fuzzy controllers, the parametrc uncertantes for -S fuzzy control system were not consdered. So the robustness of the closed-loop system cannot be guaranteed. ong et al. [] studed the robust fuzzy control problem for nonlnear systems n the presence of parametrc uncertantes and the state varables unavalable for measurement. he akag-sugeno (-S fuzzy model system wth parametrc uncertantes s adopted for modellng the nonlnear system and establshng fuzzy state observer. hs paper s dedcated to the desgn of an output feedback robust control strategy for nonlnear systems descrbed by decoupled mult-model. Some suffcent condtons n the LMI format and systematc desgn procedures for both controller and observer desgns for general nonlnear systems wth parametrc uncertantes are proposed. he stablty condtons for nonlnear systems wth parametrc uncertantes gven n [] are extended to nonlnear systems represented by decoupled mult-model, whch are formulated n the LMI format. he paper s organzed as follows: Secton revews 94

2 Output Feedback Robust Stablzaton of the Decoupled Multple Model decoupled mult-model and decoupled observers. he output feedback controller desgn for robust stablzaton of decoupled mult-model systems wth parametrc uncertantes are presented n Secton 3. Secton 4 shows a desgn example and smulaton results. Fnally, concluson s gven n Secton 5. II. DECOUPLED MULIPLE MODEL AND MULI-OBSERVER he multple model strategy s based on the basc dea that complex dynamc behavours can be accurately represented wth the help of an nterpolaton of smple submodels. In ths paper, heterogeneous multple model wll be employed [7]. he state space representaton of ths multple model s: xɺ t = A x t + B u t, ( ( ( ( = (, y t C x t L ( = µ ( ξ ( ( y t t y t, L s the number of the submodels, n x R and ( p y R are respectvely the state vector and the output of the th submodel; m u R s the nput and n n measured output. he matrces R, p n C R are known and approprately dmensoned. A p y R the n m B R, he complete partton of the operatng space of the system ξ t that s assumed to s performed usng a decson varable ( be known and real-tme avalable (e.g. the nputs and/or exogenous sgnals. Notce that the contrbuton of each submodel s uantfed by the weghtng functons µ ξ that satsfy the followng convex sum constrants: ( ( ( ( µ ξ t = and µ ξ t, =, t ( Let us notce that n ths multple model no blend between the parameters of the submodels s performed. Indeed, the submodel contrbuton s taken nto account va a weghted sum between the submodel outputs and conseuently the submodels do not share the same state space. hanks to ths fact, the dynamcs of the submodels are completely decoupled and conseuently the dmenson of the state vector x of each submodel can be dfferent (of course the output vectors dmensons must be dentcal. herefore, ths structure s well adapted for modellng strongly nonlnear systems whose structure vares wth the operatng regme, for example when the complexty of the dynamc behavour s not unform n the operatng range. y t of each submodel must be Remark : he outputs ( consdered as ntermedary modellng sgnals only used n order to provde a representaton of the nonlnear system. Hence, they cannot be employed for drvng an observer because they are not physcally avalable and conseuently no measurement s possble. Only the global output y ( t of the multple model can be used for ths purpose. In ths paper, an uncertan nonlnear system descrbed by a decoupled multple model s consdered. he state space representaton of ths multple model s gven by: xɺ ( t ( A A x ( t ( B B u ( t = y ( t = C x ( t (3 y ( t = µ ( ξ ( t y ( t he parametrc uncertantes n the system are represented by matrces A and B Assumpton. he parameter uncertantes consdered here are norm-bounded, n the form: A, B = D F t E, E [ ] ( [ ] F ( t F ( t I D,E and E are known real constant matrces of approprate dmenson, and F ( t s an unknown matrx functon wth Lebesgue-measurable elements, I s the dentty matrx of approprate dmenson. A. Augmented form of the decoupled multple model Consder the followng augmented state vector: n R, x ( t = x x x n = n he decoupled multple model (3 may be rewrtten n the followng compact form: xɺ ( t = ( A + A x( t + ( B + B u ( t (4 y t = C t x t ( ( ( A = dag A A A (5 { }, B = B B B, (6 ( µ ( µ ( µ ( C t t C t C t C (7 = L ( DF ( t E ( DF ( t E A t = (8 B t = (9 { } { } D = dag D D D ( E = dag E E E ( E = E E E ( { ( ( ( } F = dag F t F t F t (3 Remark. he matrx C ( t can be rewrtten as a weghted sum of matrces as follows: ( = µ ( C t t C, (4 C s a constant block matrx gven by: C = C (5 C [ ] Such that the term B. Decoupled mult-observer C s found on the th block column of he proposed observer based on the decoupled multple model has the followng form: 95

3 xˆɺ ( ˆ ( ( ( ( ˆ t = A x t + Bu t + K y t y ( t yˆ ( ˆ t = C x ( t (6 yˆ ( t = µ ( ( ˆ ξ t y ( t n p R s the gan assocated to the th observer. K he observer may be rewrtten n the followng compact form: xˆɺ ( t = Ax ˆ ( t + Bu ( t + K ( y ( t yˆ ( t (7 yˆ ( t = C ( t xˆ ( t = µ ( ( ˆ ξ t C x( t = xˆ ( t xˆ ( t xˆ ( t xˆ ( t s the state estmaton and ŷ ( t the output estmaton and K K K K s the augmented gan of the = observer to be determned such as the exponental ˆx t towards V x( t, e( t s guaranteed. convergence of ( ( III. OUPU FEEDBACK ROBUS SABILIZAION OF HE DECOUPLED MULIPLE MODEL Defne observaton error as e t = x t xˆ t (8 ( ( ( he objectve s to desgn a decoupled multmodel output feedback controller based on the observer for robust stablzaton of system (3 n the form u t = L xˆ t (9 ( ( ( µ ( ( ( ( ( ˆ ξ µ ξ ( u t = t u t = t L x t = L ˆ ( ( ˆ x t = L t x( t µ ξ ( µ ( µ ( µ ( = ( L t t L t L t L ( Remark 3. he tme-varyng matrx L ( t can be also rewrtten, usng the weghtng functons propertes, as the followng weghted sum of constants matrces: L ( t = µ ( t L ( L s a constant block matrx gven by: [ L ] L =, (3 From systems (3-(8 and (, we have ( A + A ( t ( B + B ( t L x( t xɺ ( t = µ ( ξ ( t (4 + ( B + B ( t L e( t ( µ ξ ( ( ( ˆ ( ( xˆɺ t = t A BL x t + KC e t (5 ( A + B ( t L KC e( t ( = µ ( ξ ( t + ( A ( t B ( t L x( t eɺ t (6 he man result for the global asymptotc stablty of decoupled multmodel, wth parametrc uncertantes and unavalable state varables, are summarzed n the followng theorem: heorem: If there exst symmetrc and postve defnte matrces P > and P >, some matrces M and N, and ε =, such that the followng LMIs are scalars, (,, satsfed, then the decoupled multmodel system (3 s asymptotcally stablzable va the decoupled multmodel-based output-feedback controller ( Ξ * * E Q E M + I * D ( ε + I a ( ε * * + I * D P ( ε + I where Ξ = QA + AQ M B BM + I b E L ( ε (7 (8 (7a = L B BL + A P + P A C N NC (8a and Q = P (7b L = M Q K = P N, (7c (8b where * denotes the transposed elements n the symmetrc postons. Proof. Before proceedng, we recall the followng matrx neualty, whch wll be needed throughout the proof of heorem. Lemma (Lee et al. [8]. Gven constant matrces D and F, symmetrc constant matrx S and unknown constant matrx F of approprate dmenson satsfyng the constrant F F < R. he followng two propostons are euvalent: S + DFE + E F D ε R E S + ( E D for some ε < > ε I D Consder the Lyapunov functon canddate ( (, ( ( ( ( ( V x t e t = V x t + V e t (9 Wth V x t x t P x t P Q (3 ( ( ( ( ( ( ( ( = ; = >, V e t = e t P e t ; P > (3 = { } = { } ( ( ( ɺ ( ( ( ɺ ( P dag P P P (3 P dag P P P (33 he tme dervatve of ( Vɺ x t = x t P x t + x t P x t = V x t along the trajectory of (4 s ( ( + ( + ( ( + ( ( ( x t H P PH x t (34 µ ( ξ ( t x t P B B t L e t H = A + A t B + B t L (35 Wth ( ( Usng Lemme and Assumpton, the followng neualty holds: 96

4 Output Feedback Robust Stablzaton of the Decoupled Multple Model ( ( + ( ( ( ( + ( + ( ( + ( x t P B B t L e t x t P P DD P x t e t L B B E E L e t (36 akng account of (36 n (34, the tme dervatve of ( ( ( x( t V x t s as follows: Vɺ x ( t ( H P + PH + P + PDD P x ( t µ ( ξ ( t + e ( t L ( t( B B + E E L e( t he tme of ( ( Vɺ ( e( t = eɺ ( t P e( t + e ( t P eɺ ( t = V e t along the trajectory of (6 s ( ( Σ + Σ ( e ( t P ( A ( t B ( t L x( t (37 e t P P e t (38 µ ( ξ ( t + Σ = A KC + B t L (39 Wth ( Usng Lemme and Assumpton, the followng neualty holds: e t P A t B t L x t ( ( ( ( ( ( ( ( + e t P DD P e t ( ( ( ( ( x t E E L E E L x t Vɺ he tme-dervatve of ( ( ( e( t V e t s then as follows: ( (( Σ + Σ + ( ( ( ( (4 e t P P P DD P e t (4 µ ( ξ ( t + x t ( E E L E E L x( t Fnally, by combnng (34 and (4, the tme dervatve of ( (, ( ( (, e( t V x t e t can be wrtten as follows: Vɺ x t µ ξ = ( ( ( x t H P PH P P DD P R x t + e ( t( Σ P + P Σ + P DD P + S e( t (4 ( ( ( R Wth = E E L E E L (43 S = L B B + E E L (44 H P + PH + P + PDD P + R (45 Assumng that the frst sum s negatve defnt,.e. Or euvalently P + PDD P + R + ( ( ( ( P ( A A ( t ( B B ( t L A + A t B + B t L P (46 hen, applyng Assumpton to (4 yelds ϒ + PDF t E E L + E E L F t D P < (47 ( ( ( ( ϒ = P + PDD P + R + A P + P A L B P PBL (48 Accordng to Lemma, the above matrx neualty (49 holds for all F ( t satsfyng ( ( there exsts a constant ε such that: F t F t I f and only f ( E E L ( PD ε I ϒ + ( E E L PD (49 ε I By rewrtng PDD P + R n the followng form : ( E E L ( PD I ( E E L PD I hen (49 becomes ( ε + (5 ( E E L ( P D I Γ + ( E E L P D ( ε + I (5 Γ = P + A P + P A L B P PBL (5 Let Q = P M = L Q, Pro-and-post multplyng both sdes of (5 by Q = results n P ( ε ( + I E Q E M Ξ + ( E Q E M D ( ε + I D (53 Ξ = QA + AQ M B BM + I (54 Applyng Schur complement to (53 results n the frst LMI, (7, n heorem. he second LMI (8 can be establshed through a smlar procedure. Assume that the second sum n E. (4 s negatve defnte: Σ P + P Σ + P DD P + S < (55 Or euvalently ( ( ( ( P DD P + S + A KC + B t L P + P A KC B t L + (56 then, applyng Assumpton to (56 yelds Ψ + P DF t E L + E L F t D P < (57 ( ( ( ( ( ( Ψ = < (58 P DD P S A KC P P A KC Accordng to lemme, the above matrx neualty (58 holds : ε I E L Ψ + ( E L P D ε I (59 ( P D By rewrtng P DD P + L E E L n the followng form I E L P DD P + L E E L = ( E L P D I ( P D (6 hen (59 becomes ( ε + I E L Π + ( E L P D ( ε + I( P D 97

5 (6 Π = L B BL + A P + P A C K P P KC (6 Let N = P K We can rewrte (6 n the followng form: ( ε + I E L + ( E L P D ( ε + I( P D (63 = L B BL + A P + P A C N NC (64 Applyng Schur complement to (63 results n the second LMI, (8, n heorem. In order to show clearly mplementaton of the observer-based decouple mult-model controller, the desgn procedures are gven as follows: Frst, we solve LMI (7 n the varables Q and M. Once gans L have been calculated from (7b, condtons (8 become lnear n P and N, and can be easly resolved usng the LMI tool to determne gans K from (8b. IV. COMPUER SIMULAION Consder the decoupled multple model wth L = submodels wth dfferent dmensons (n = 3 and n =, gven by: A =.3.4., A =, B C D E E = [ ] B = [ ] = [ ] C = [ ] = [... ], D = [ ] = [... ], E = [.. ],.3.5.6,.4.3,.5.3.5,.4.3,.., =., E =.3, Here, the decson varable ξ ( t s the nput sgnal ( u t.he weghtng functons are obtaned from normalzed Gaussan functons: µ ξ = η ξ η ξ j j = = c L ( η ξ exp ξ σ, Wth the standard devaton σ =.6 and the centers c =.3 and c =.3. Usng LMI optmzaton algorthm [6] to solve LMIs (3-(34, feedback gan and observer gan matrces can be obtaned as L = , L = [ ] [ ] K = [ ] he ntal values of states are chosen x( = [ ], xˆ ( = [ ]. Fgs. - llustrate the closed-loop system behavours. he smulaton results show that the decoupled multmodel based, controller through mult-observer s robust aganst norm-bounded parametrc uncertantes. he state estmaton errors are gven by Fg. 3. Fg. 4 allows the comparson of the nomnal model states and estmated model states. From the latter, one can see that the syntheszed observer and decoupled mult-model controller showed ther effectveness. he evoluton of the control law s gven n Fg. 5. V. CONCLUSION In ths paper, we have developed a new robust decoupled mult-model controller desgn methodology for non lnear systems represented by multple models wth parameter uncertantes under the condtons that the state varables are unavalable for measurement. he basc approach s based on the rgorous Lyapunov stablty theory, and the basc tool s lnear matrx neualty (LMI. Some suffcent condtons for robust stablzaton of the decoupled mult-model are formulated n the LMIs format. he smulaton results have shown the effectveness of the proposed control desgn method. REFERENCES [] E. H. Mamdan and S. Asslan, An experment n lngustc synthess wth a fuzzy logc controller, Internatonal Journal of Man-Machne Studes, vol. 7, pp. 3, January 975. []. akag and M. Sugeno, Fuzzy dentfcaton of systems and ts applcatons to modelng and control, IEEE rans. Systems, Man & Cybernetcs, vol. SMC-5 (, pp. 6 3, Jan-Feb [3] L.K. Wong, F.H.F. Leung and P.K.S. am, Fuzzy model-based desgn of fuzzy logc controllers and ts applcaton on combnng controllers, IEEE rans. Indust. Elect, vol. 45 (3, pp. 5 59, 998. [4] D. Flev, Fuzzy modelng of complex systems, Internatonal Journal of Approxmate Reasonng, vol. 5 (3, pp. 8 9, 99. [5] R. Orjuela, Contrbuton l estmaton d état et au dagnostc des systèmes représentés par des multmodèle, PhD. thess, Insttut Natonal Polytechnue de Lorrane (INPL, Nancy, France, 8. [6] R. Orjuela, B. Marx, D. Maun and J. Ragot, State estmaton of nonlnear dscrete-tme systems based on the decoupled multple model approach. 4 th Internatonal Conference on Informatcs n Control, Automaton and Robotcs, ICINCO 7, 7. [7] R. Orjuela, B. Marx, D. Maun and J. Ragot, A decoupled multple model approach for state estmaton of nonlnear systems subject to delayed measurements. 3 rd IFAC Advanced Fuzzy and Neural NetworkWorkshop, Valencennes, France, 9-3 octobre, 8. [8] L. Xe, Output feedback H control of systems wth parameter uncertantes, Internat. J. Control, vol. 63 (4, pp , 996. [9] P. Gahnet, A. Nemrovsk, A. Laub and M. Chlal, LMI Control oolbox. Natck, MA: he Math Works, 995. [] A. Akhenak, Concepton d observateurs non lnéares par approche multmodèle: applcaton au dagnostc, Ph. D. thess, Insttut Natonal Polytechnue de Lorrane (INPL, France, 4. [] M. Rodrgues, Dagnostc et commande actve tolérante aux défauts applués aux systèmes décrts par des mult-modèles lnéares, Ph. D. thess, Unversté Henr Poncaré, France, 5. [] S. ong and H. L, Observer-based robust fuzzy control of nonlnear systems wth parametrc uncertantes, Fuzzy Sets and Systems, vol. 3, pp , Oct.. [3] H.J. Lee, J.B. Park and G. Chen, Robust fuzzy control of nonlnear systems wth parametrc uncertantes, IEEE rans. Fuzzy Systems, vol. 9(, pp ,. [4] K. anaka,. Ikeda and H.O. Wang, Fuzzy regulators and fuzzy observers: relaxed stablty condtons and LMI-based desgns, IEEE rans. Fuzzy Systems, vol. 4 (, pp. 5 65, 998. [5] X.J. Ma and Z.Q. Sun, Analyss and desgn of fuzzy controller and fuzzy observer, IEEE rans. Fuzzy Systems, vol. 9 (, pp. 4 5,

6 Output Feedback Robust Stablzaton of the Decoupled Multple Model [6] J. Yoneyama and M. Nshkawa et al., Output stablzaton of akag-sugeno fuzzy systems, Fuzzy Sets and Systems, vol., pp ,. [7] R. Orjuela, B. Marx, D. Maun, and J. Ragot. State estmaton for nonlnear systems usng a decoupled multple model, Internatonal Journal of Modellng Identfcaton and Control, vol. 4 (, pp , 8. [8] H.J. Lee, J.B. Park and G. Chen, Robust fuzzy control of nonlnear systems wth parametrc uncertantes, IEEE rans. Fuzzy Systems, vol. 9 (, pp ,. [9] A. ahar and M.N. Abdelkrm, Multmodel H loop shapng control for lnear sngularly perturbed systems, th Internatonal conference on Scences and echnues of Automatc control and computer engneerng, Sousse, unsa,. [] A. ahar and M.N. Abdelkrm, Multmodel H loop shapng control of a DC motor under varable loads, 8th Internatonal Mult-conference on Systems, Sgnals and devces, Sousse, unsa,. [] D. Ichalal, B. Marx, J. Ragot and D. Maun, Advances n observer desgn for akag-sugeno systems wth unmeasurable premse varables, th Medterranean conference on Control and Automaton (MED, Barcelona, Span,. []. A. Johansen, Operatng regme based process modelng and dentfcaton, Ph. D. thess, Norwegan Insttute of echnology, rondhem, Norway, 994. [3] M. Ltaef, A. Messaoud and R. Ben Abdennour, Optmal systematc determnaton of models' base for multmodel representaton: Real tme applcaton, Internatonal Journal of Automaton and Computng, vol., pp , x.6.4. x. x tme(sec tme(sec Fg.. Closed-loop response of submodel tme(sec x x tme(sec Fg.. Closed-loop response of submodel tme(sec e.4. e.4. e tme(sec tme(sec Fg. 3. State estmaton errors of submodel tme(sec 99

7 e e tme(sec Fg. 4. State estmaton errors of submodel tme(sec.5..5 u tme(sec Fg. 5. Control nput