Reliable H Controller Design for a Class of Uncertain Linear Systems with Actuator Failures

Transcription

1 954 Inernaional Journal o Conrol, Auomaion, and Sysems, vol. 6, no. 6, pp , December 8 Reliable H Conroller Design or a Class o Uncerain Linear Sysems wih Acuaor Failures Shi-Lu Dai and Jun Zhao* Absrac: his paper is concerned wih he reliable H conroller design problem or uncerain linear sysems agains acuaor ailures. In he design, he H perormance o he closed-loop sysem is opimized during normal operaion (wihou ailures) while he sysem saisies a prescribed H perormance level in he case o acuaor ailures. Single and parameerdependen Lyapunov uncion approaches are applied in designing subopimal reliable H conrollers. Simulaion sudies are presened o demonsrae he eeciveness o he proposed design procedures. Keywords: Acuaor ailures, H conrol, parameer-dependen Lyapunov uncion, reliable conrol.. INRODUCION Manuscrip received April 5, 7; revised December 4, 7; acceped May, 8. Recommended by Ediorial Board member Guang-Hong Yang under he direcion o Edior Jae Weon Choi. his work was suppored by he Naional Naural Science Foundaion o China under Grans 9868, 68744, and Shi-Lu Dai and Jun Zhao are wih Key Laboraory o Inegraed Auomaion o Process Indusry, Minisry o Educaion, Norheasern Universiy, Shenyang 4, P.R. China ( s: dslwm@yahoo.com.cn, zhaoun@ise.neu.edu. cn). * Corresponding auhor. Reliable conrol sysem design problems have araced considerable aenion [-4], since ailures o conrol componens (e.g., sensor ailures or acuaor ailures) oen occur in applicaions. A conrol sysem designed o olerae ailures o conrol componens, while mainaining an accepable level o he closedloop sysems sabiliy/perormance, is called a reliable conrol sysem [], in which a mehodology was inroduced or he design o reliable linear conrol sysems such ha he resuling conrol sysems are sabiliy and saisy an accepable H disurbance aenuaion level no only when all acuaors or sensors are operaional bu also in he case o some admissible acuaor or sensor ouages. A robus and reliable H conrol scheme was developed in [3] via sae-eedback or uncerain linear sysems wih acuaor ailures. In [4], a more pracical model o sensor and acuaor ailures han ouages was considered o develop he reliable H conroller design mehods. Neverheless, in hese reliable design mehodologies, he opimizaion o normal perormance is no explicily addressed when he normal perormance is sacriiced or he reliabiliy (sabiliy) goal. he reliable design should also seek o opimize he normal perormance as sysems are operaing under normal condiions in mos o he ime [5]. Recenly, reliable conrol design mehodologies were developed or linear sysems [5-7], which opimized he normal perormances o he closed-loop sysems while mainaining accepable low-level perormances o he sysems in he even o acuaor ailures. In his paper, we explore he problem o he reliable H conrol or a class o uncerain linear sysems wih exernal disurbances. Dierenly rom he aoremenioned resuls o reliable design, acuaor ailures are described as polyopic uncerainy, which covers he cases o normal operaion, parial degradaions, and ouages. During mos reliable design processes, he normal perormances o he closed-loop sysems are usually sacriiced or he reliabiliy goal. I is, however, undesirable o sacriice signiicanly he normal perormances or he occasional ailure cases in pracice. Moivaed by his ac, we design a sabilizing sae-eedback conroller such ha he normal H perormance is opimized while saisying a prescribed H perormance level in he case o acuaor ailures. wo eicien procedures or designing a subopimal reliable H conroller are presened in erms o linear marix inequaliies (LMIs). In he irs design procedure, a single Lyapunov uncion is used or he acuaor ailure cases. However, he procedure may be conservaive by he use o he single Lyapunov uncion. o reduce is conservaism, he second design procedure employs a parameer-dependen Lyapunov uncion o develop a new design echnique or he acuaor ailure cases.

2 Reliable H Conroller Design or a Class o Uncerain Linear Sysems wih Acuaor Failures 955. PROBLEM SAEMEN Consider he uncerain linear sysem described by x () = Ax () () + Bu () () + Gω, () () () z () = Cx () () + Du () () + D () ω(), n where x () R is he sae vecor, u () R is he conrol inpu, ω () L [, ) is he exernal p disurbance signal, and z () R is he conrolled oupu. Suppose ha sysem marices have he ollowing orm A () B () G () A B G C () D () D () = C D D ω ω H + F ()[ E E E 3], H ω m () where A, B, G, C, D, Dω, H i ( i =, ), and E i ( i=,, 3) are consan marices o appropriae dimensions, and F () is an unknown marix uncion saisying F () F() I, wih he elemens being Lebesguemeasurable. he ailure model is described by F u () = αu(), α = diag{ α,, α }, (3) α α α, =,,, m (4) where α and α represen he lower and upper bounds, respecively. Denoe he marix se Ω β = { βl βl = diag{ βl, βl,, βlm}, β = α or α, =,,, m}, L m (5) where L=,,, n β, n β =, and l m is he number o acuaor ailures. I is clear ha he marix α resides wihin he marix polyope wih he veriices Ω β, i.e., α Ω α wih nβ Ω α = { α α = θ β, θ, θ = }. L L L L L= L= l nβ (6) Consider sysem () wih he ollowing saeeedback conroller u () = Kx (), (7) m n where K R is a consan marix gain o be designed. hen, he closed-loop sysem wih acuaor ailures (3) is given by x () = Ac (, α) x() + G() ω(), (8) z() = C (, α) x() + D () ω(), c ω where A (, α) = A() + B() αk and C (, α ) = C() c c + D () α K. Obviously, when α = I, sysem (8) becomes he ollowing sysem x () = Ac () x() + G() ω(), (9) z() = Cc () x() + Dω () ω(), wih Ac () = A() + B() K and Cc () = C() + D() K. For convenience, sysem (8) and sysem (9) are said o be he reliable and sandard closed-loop sysem, respecively. he reliable H conrol problem: Consider sysem (). Given consan scalars γ γ >, design a conroller (7) such ha: i) he sandard closed-loop sysem (9) is asympoically sable and z () < γ ω() holds or all z (), ω () saisying (9), where. denoes he usual L (, ) -norm; ii) he reliable closed-loop sysem (8) is asympoically sable and under he zero iniial condiion, z () < γ () ω holds or all z (), ω () saisying (8) and all probable α Ω α. I is worh poining ou here migh exis a se o conrollers such ha above reliable conrol problem is solvable. Here, he obecive o his paper is o design a conroller (7) such ha: i) he above reliable conrol problem is solvable; ii) or a given consan scalar γ > in he acuaor ailure cases, he level γ > or he closed-loop sysem in he normal case is minimized. I he above obecive is achieved, conroller (7) is said o be a subopimal reliable H conroller. he ollowing lemma will be used in he developmen o he main resuls. Lemma [8]: For given marices Y, M and N o appropriae dimensions and a scalar η >, Y + MΞ N + N Ξ M < holds or all Ξ saisying Ξ Ξ ηi i and only i here exiss a consan ε > such ha η Y + ε MM + N N <. ε 3. RELIABLE H CONROLLER DESIGN his secion presens wo eicien procedures or designing subopimal reliable H conroller or sysem (). In he design procedures, single Lyapunov uncion and parameer-dependen Lyapunov uncion are employed, respecively, o develop he procedures

3 956 Shi-Lu Dai and Jun Zhao or he acuaor ailure cases. he subopimal reliable conrollers can be obained hrough solving convex opimizaion problems wih LMI consrains. 3.. Single Lyapunov uncion approach Following bounded real lemma (BRL) [9], i is clear ha or sysem (9) wih a scalar γ >, i here exiss a marix X = X > such ha Ac() X + XA c() G() XCc () * γ I Dω ( ) <, * * γ I () where he symbol * represens he ransposed elemens in he symmeric posiions, hen sysem (9) is asympoically sable and saisies z () < γ ω(). For a given scalar γ >, i here exiss a marix X = X > such ha Φ(, α) G ( ) XCc (, α) * γ I Dω ( ) <, * * γ I () where Φ (, α) = Ac(, α) X + X Ac (, α), hen sysem (8) is asympoically sable and saisies z () < γ () ω. In order o obain reliable H conrollers, we need o solve inequaliies () and (). However, inequaliies () and () are no convex. o solve his non-convex problem, le us se X = λ X, () where λ > is a scalar variable. Since inequaliy () is ininie-dimensional, in general, he problem o direc esing is easibiliy is no racable. Using he characerisic o he polyopic uncerainy, his problem can be ransormed ino he es only a he verices Ω β. he ollowing resul gives a soluion o he reliable H conrol problem. heorem : Consider he closed-loop sysem (8). For given scalars γ γ >, he conroller (7) solves he reliable H conrol problem i here exiss a soluion ( ε, ε, λ, X, Y ) o he ollowing: M G M XE Y E * * * M33 * * * εi 3 + γ I Dω E3 <, (3) N L NL λg N3L XE + Y E * λγ : I λdω λe3 = * * N33L (4) * * * λε I <, L=,,, n, where M = AX + XA + BY + Y B + εhh, 3 = + +ε, 33 = γ + ε, L = + + βl + βl + ε, 3L = + βl + ε, 33L = λγ + ε. M XC Y D H H M I H H N AX XA B Y Y B H H N XC Y D H H N I H H In his case, he sae-eedback gain is given by K = YX. (5) Proo: By applying Lemma and Schur complemen ormula and inroducing he new variable Y = KX, i is easy o veriy ha inequaliy () holds i inequaliy (3) holds. By perorming congruence ransormaion o () / / / by diag{ λ I, λ I, λ I}, considering () and (5), and using Lemma, we can conclude ha a suicien condiion or () is Nc( α) λg Nc3( α) Nc4( α) * λγ I λdω λe3 Nc ( α): = * * Nc33 * * * λε I <, (6) wih N ( α) = AX + XA + BαY + Y αb + ε H H, c c3( α) = + α + ε, c4( α) = + α, c33 = λγ + ε. N XC Y D H H N XE Y E N I H H Since α Ω, we have n β α N(α) = θn. (7) c L L L= hen, i is clear ha inequaliy (6) holds i inequaliy (4) holds. his complees he proo. Remark : Consider sysem (8). I we choose a proper value o γ, hen subopimal reliable H conroller o he orm (7) can be deermined by solving he ollowing convex opimizaion problem: min γ ε,ε,γ,x,y,λ subec o (3), (4), X >. β

4 Reliable H Conroller Design or a Class o Uncerain Linear Sysems wih Acuaor Failures 957 he reliable conroller gain is given by (5). Remark : Remark conains he uning scalar parameer λ. he opimum value can be ascerained by he approach in []. Applying a numerical opimizaion algorihm, such as he program minsearch in he opimizaion oolbox o Malab, a numerical soluion o his problem is obained. Since he single Lyapunov marix X = λ X solves inequaliy (4) or all values o α, he single Lyapunov uncion ( ) V x = x () X x() works or all probable α Ω α. he design procedure above is simple bu migh be conservaive due o employing he single Lyapunov uncion ( ) V x = x () X x(). o reduce his conservaism, we will resor o a parameer-dependen Lyapunov uncion or he acuaor ailure cases in he nex subsecion. I should be menioned ha parameer-dependen Lyapunov uncions have been exensively used o solve he H conrol problem or coninuous-ime polyopic sysems in he lieraure, see, e.g., [, ] and reerences herein, however, hese sudies neiher consider he uncerainy orm (), nor ackle he reliable conrol issue. 3.. Parameer-dependen Lyapunov uncion approach Le us give an equivalen represenaion o inequaliy (). Lemma : Given a consanγ >, he ollowing saemens are equivalen. i) here exiss a marix X = X > saisying inequaliy (). ii) here exis a scalar δ >, marices Q and X = X > such ha Ψ () X -Q + δac() Q G() Q Cc () * δ( Q+ Q ) δq C c ( ) <, * * γ I Dω ( ) * * * γ I (8) where Ψ () = Ac() Q+ Q Ac (). Proo: ii) i): I is clear ha muliplying (8) by J on he le and by J on he righ, respecively, yields (), where J I A c() = I. C() c I i) ii): I here exiss a marix X > such ha () holds, hen here exiss a suicienly small scalar δ > such ha Ac() X+ XAc () G () XCc () * γ I Dω ( ) * * γ I Ac() X Ac() X δ + X, < Cc() X Cc() X which is equivalen o Φ() X X + δac() X G() XCc () * δ( X + X) δxcc ( ) <, * * γ I Dω ( ) * * * γ I where Φ () = Ac() X + XAc (). Hence, here exis a scalar δ > and marix Q = X such ha (8) holds. his complees he proo. Unlike in (), wih he inroducion o he exra variables Q and δ, he marix X is separaed rom he marices Ac () and Cc () in (8). his propery is useul and criical, which can be used o design conroller or he acuaor ailure cases by employing a parameer-dependen Lyapunov uncion. his is shown in he ollowing heorem. heorem : For some given scalars γ γ >, he conroller (7) solves he reliable H conrol problem i here exiss a soluion ( ε 3, ε 4, λ, δ, δ, XQYX,,, L, L=,, n β ) o he ollowing: M M G M4 M5 * δ( Q+ Q ) M4 δ M5 * * γ I Dω (9) * * * M 44 * * * * ε3i <, NL NL λg N4L N5L * NL N4L δ N5L * * λγ I λdω () * * * N44L * * * * λε 4I <, where M = AQ+ Q A + BY + Y B + ε3hh, M = X Q + δ ( AQ+ BY ),

5 958 Shi-Lu Dai and Jun Zhao 4 = + + ε3, 5, δ 44 = γ + ε3, L = + + βl + βl + ε4, L = L + δ + βl M Q C Y D H H M = Q E + Y E M4 = ( Q C + Y D ), M I H H N AQ Q A B Y Y B H H N X Q ( AQ B Y), 4L = + βl + ε4, 5L βl, L N Q C Y D H H N = Q E + Y E N = δ ( Q+ Q ), L L N4 = δ ( Q C + Y β D ), 44L = λγ + ε4. N I H H In his case, he sae-eedback gain is given by K = YQ. () Proo: he proo is similar o ha o heorem by Lemma, and is, hus, omied or conciseness. Remark 3: Consider sysem (8). Choosing a proper value o γ, he subopimal reliable H conroller o he orm (7) can be deermined by solving he ollowing convex opimizaion problem: min γ ε 3,ε 4,γ,X,XL, Q, Y,λ, δ, δ subec o (9), (), X >, XL >, L=,,, nβ. he reliable conroller gain is given by (). 4. SIMULAION SUDIES o illusrae he eeciveness o he proposed design procedures, reliable conrollers are applied o he laeral conrol o an AV-8A Harrier VOL aircra in a hover model. he linearized unsable laeral dynamic model wih parameers or his aircra is aken rom []. he sae-space model or his aircra is given by x () = ( A+ A ()) x () + Bu () + Gω (), where A() = sin(). A. Deine he conrolled oupu: z () = Cx() + Du (), where C = and D =. For comparison purpose, we design hree dieren conrollers: he opimal sandard H conroller; and he subopimal reliable H conrollers by he irs and second design procedures given in Secion 3, respecively. By he approach proposed in [8], he obained minimum disurbance aenuaion level is * γ s =.55 or sandard conroller. Nex, reliable conrollers are designed o caer boh he normal and he acuaor ailure cases. Here, our main purpose is o design subopimal reliable H conrollers such ha he laeral sick perurbaion is olerable o ailure and is conrol eeciveness can vary rom he normal case (% eeciveness) o he complee ouage case (% eeciveness). Le γ = and he iniial value λ =.. Applying Remark and he minsearch in Remark gives he minimum disurbance aenuaion level * γ =.4588 wih λ =.67. Le he iniial value ( δ, λ, δ ) = (.3,.,.). Applying Remark 3 * yields γ =.49, which is beer han ha given by he irs design procedure. Assume he exernal disurbance signal.5 ω() = [ ω], ω = e sin(.4π +.5 π ). Under he iniial sae variables [ ] x =..5., Figs. and show he sae responses or he corresponding closed-loop sysems wih sandard conroller and reliable conroller in he case o laeral sick perurbaion ñeíf O ñ=nmq M JO JQ JS JU M NM OM PM QM íeëéåf Fig.. he sae responses o sandard conroller wih ailure. ñeíf S Q O M JO M NM OM PM QM íeëéåf Fig.. he sae responses o reliable conroller wih ailure.

6 Reliable H Conroller Design or a Class o Uncerain Linear Sysems wih Acuaor Failures 959 complee ouage. From Fig., i can be seen ha sandard conroller can no mainain he closed-loop sysem sabiliy, while reliable conroller mainains he resuling closed-loop sysem sabiliy (see Fig. ). 5. CONLUSIONS In his paper, he subopimal reliable H conroller design procedures have been proposed or uncerain linear sysems wih exernal disurbances. wo eicien procedures or designing a subopimal reliable H conroller have been presened. he subopimal reliable conroller design procedures or he AV-8A Harrier VOL aircra wih laeral unsable dynamic model parameers have been given o illusrae he applicabiliy o he design procedures proposed. Simulaion resuls urher demonsrae he necessiy and eeciveness o reliable conrollers. REFERENCES [] R. J. Veillee, J. V. Medanic, and W. R. Perkins, Design o reliable conrol sysems, IEEE rans. on Auomaic Conrol, vol. 37, no. 3, pp. 9-34, 99. [] Q. Zhao and J. Jiang, Reliable sae eedback conrol sysem design agains acuaor ailures, Auomaica, vol. 34, no., pp. 67-7, 998. [3] C. J. Seo and B. K. Kim, Robus and reliable H conrol or linear sysems wih parameer uncerainy and acuaor ailure, Auomaica, vol. 3, no. 3, pp , 996. [4] G.-H. Yang, J. L. Wang, and Y. C. Soh, Reliable H conroller design or linear sysems, Auomaica, vol. 37, no. 3, pp ,. [5] J.-S. Yee, G.-H. Yang, and J. Wang, Reliable oupu-eedback conroller design or discree ime linear sysems: an ineracive LMI approach, Proc. American Conrol Con., pp. 35-4,. [6] F. Liao, J. Wang, and G.-H. Yang, Reliable robus ligh racking conrol: An LMI approach, IEEE rans. on Conrol Sysem echnology, vol., no., pp ,. [7] G.-H. Yang, J. L. Wang, Y. C. Soh and K.-Y. Lou, Reliable sae eedback conrol synhesis or uncerain linear sysems, Asian Journal o Conrol, vol. 5, no., pp. 3-38, 3. [8] L. Xie, Oupu eedback H conrol o sysems wih parameer uncerainy, Inernaional Journal o Conrol, vol. 63, no. 4, pp , 996. [9] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Marix Inequaliies in Sysems and Conrol heory, SIAM, Philadelphia, PA, 994. [] U. Shaked, An LPD approach o robus H and H saic oupu-eedback design, IEEE rans. on Auomaic Conrol, vol. 48, no. 5, pp , 3. [] Y. He, M. Wu, and J.-H. She, Improved Bounded-Real-Lemma represenaion and H conrol o sysems wih polyopic uncerainies, IEEE rans. on Circuis and Sysems-II, vol. 5, no. 7, pp , 5. [] E. M. Jaarov, Robus sliding mode conrollers design echniques or sabilizaion o mulivariable ime-delay sysems wih parameer perurbaions and exernal disurbances, Inernaional Journal o Sysems Science, vol. 36, no. 7, pp , 5. Shi-Lu Dai received he B.E. degree in hermal Engineering and Dynamics in and he M.E. degree in Navigaion, Guidance and Conrol in 6, rom Norheasern Universiy, Shenyang, China. He is currenly a Ph.D. candidae in Norheasern Universiy, China. Since November, 7, as a Visiing Suden, he has also been wih he Deparmen o Elecrical and Compuer Engineering, he Naional Universiy o Singapore. He is recipien o Chinese Governmen Scholarship rom China Scholarship Council (CSC), 7-9. His curren research ineress lie in robus conrol, neworked embedded sysems and swiched sysems. Jun Zhao received he B.S. and M.S. degrees in Mahemaics rom Liaoning Universiy, Shenyang, China, in 98 and 984, respecively, and he Ph.D. degree in Conrol heory and Applicaions rom he Norheasern Universiy, Shenyang, in 99. From 99 o 993, he was a Posdocoral Fellow a he Norheasern Universiy, where since 994, he has been wih he School o Inormaion Science and Engineering, and is currenly a Proessor. From February 998 o February 999, he was a Senior Visiing Scholar a he Coordinaed Science Laboraory, Universiy o Illinois, Urbana-Champaign. From November 3 o May 5, he was a Research Fellow in he Deparmen o Elecronic Engineering, Ciy Universiy o Hong Kong. Since November 6, he has also been wih he Deparmen o Inormaion Engineering, Research School o Inormaion Sciences and Engineering, he Ausralian Naional Universiy, Canberra, A.C.., Ausralia, as a Fellow. His curren research ineress include swiched sysems, hybrid conrol, nonlinear sysems, and robus conrol.