LocalReliableControlforLinearSystems with Saturating Actuators

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1 LocalRelableControlforLnearSystems wth Saturatng Actuators Shoudong Huang James Lam Bng Chen Dept. of Engneerng Dept. of Mechancal Engneerng Dept. of Mathematcs Australan Natonal Unversty Unversty of Hong Kong Jnzhou Teacher s College Canberra, ACT 2, Australa Pokfulam Road, Hong Kong Jnzhou, Laonng, P.R.Chna shoudong.huang@anu.edu.au james.lam@hku.hk dongshuoch@sna.com Abstract Ths paper consders the problem of local relable control for contnuous-tme lnear systems wth saturatng actuators and dsturbance. The local stablty and the performance of the desgned closed-loop system s guaranteed not only when all control components are operatonal, but also n case of actuator outages n the preselected subset of actuators. Lnear matrx nequalty (LMI) method and teratve LMI (ILMI) method are proposed to desgn state-feedback controllers. The effectveness of our methods s shown by an example. 1. Introducton Relable control s concerned wth the desgn of a closedloop system to mantan key propertes such as stablty and other performances durng sensor or actuator falure. In recent years, consderable attenton has been pad to the desgn problems of relable lnear control systems, and a number of desgn methods have been proposed. For example, Vellette et al. [12] presented a methodology for the desgn of relable H control systems by means of the algebrac Rccat equaton approach. The relable lnear-quadratc state-feedback control was studed n [11]. In practcal systems, lmted power supples manfest as saturatng actuators. Ther presence may lead to serous degradaton of system performance or even nstablty. Recently, the control problems of lnear systems wth saturatng actuators have been wdely studed. However, global and sem-global results can be obtaned only when all the egenvalues of the system matrx have nonpostve real parts. When no assumpton on the openloop system stablty was made, local stablzaton results were obtaned n [1, 3, 8, 1]. Recently, Nguyen and Jabbar [7] and Hu et al. [5] dscussed the analyss anddesgnmethodforlnearsystemssubjecttoboth *ThsworkssupportednpartbytheRGCGrantHKU 713/1P, the Wllam Mong Post-doctoral Research Fellowshp, and the Foundaton for Unversty Key Teacher by the Mnstry of Educaton n P. R. Chna actuator saturaton and dsturbance. However, to our knowledge, there s no results addressng the relable control for lnear systems wth saturatng actuators. Ths paper consders the local relable control problems for lnear systems wth saturatng actuators. No assumpton on the open-loop system stablty s made n our study. As n [9], the saturaton functon consdered n ths paper s a general one. The problem s to fnd smultaneously a state-feedback control law and an assocated doman of safe admssble ntal states such that the local stablty of the desgned closed-loop system s guaranteed not only when all control components are operatonal, but also n case of actuators outages n the preselected subset of actuators. A technque smlar to that n [6] s used. By nvestgatng the propertes of the saturaton functons, a lnear matrx nequalty (LMI) method s presented to solve ths problem. An optmal local relable control problems s also studed. The objectve of ths problem s to obtan a larger stablty regon. Moreover, the local relable H control problem s studed when there s bounded dsturbance. The controller can be obtaned by solvng two matrx nequaltes. Ths paper s organzed as follows. Secton 2 provdes the problem statements. The local relable control problem and the optmal local relable control problem are studed n Secton 3 and Secton 4, respectvely. In Secton 5, the local relable H control problem s dscussed. An example s gven n Secton 6 to llustrate our methods. Secton 7 concludes the paper. 2. Problem Statements The followng notatons are used n ths paper. For a real symmetrc matrx M, M >( ) means that M s postve (sem-)defnte, λ mn (M) and λ max (M) denote the mnmal and maxmal egenvalues of M, respectvely. If not explctly stated, I and denote the dentty matrx and the zero matrx of approprate dmensons, respectvely. Besdes, all matrces are assumed to have compatble dmensons.

2 Consder the followng lnear system wth saturatng actuators ẋ = Ax + Bf(u)+B w w (1) z = C z x (2) where x R n,u R m,w R r and z R s are the state, control nput, dsturbance and penalty, respectvely. The dsturbance s assumed to be bounded as w T w 1. f: R m R m s an m-dmensonal saturaton functon wth f(u) =[σ 1 (u 1 ),...,σ m (u m )] T and σ : R R are scalar saturaton functons ( =1,...,m). The defnton of scalar saturaton functon s gven as follows. Defnton 1 ([9]) A functon σ : R R s called a scalar saturaton functon f (C1) σ s locally Lpschtz, (C2) sσ(s) ½ >, s 6=, ¾ σ(s) (C3) mn lm s s, lm σ(s) s + >, s (C4) lm nf s σ(s) >. Remark 1 It was ponted out n [9] that functons σ(t) =t, arctan(t), sgn(t)mn{ t, 1} are all scalar saturaton functons under Defnton 1. Moreover, wthout loss of generalty, we may assume that for scalar saturaton functons σ : R R, thereexst >, such that s[σ (αs) sgn(s)mn{ s, }], (3) α 1( =1,...,m). Suppose that a state-feedback control to be desgned must tolerate the outage of certan actuators. Let denote the selected subset of actuators wthn whch outages must be tolerated, and let denote the complementary subset of actuators, wthn whch actuators outages are not taken nto account by the desgn. Let m 1 denote the dmenson of, wthout loss of generalty, suppose = {1,...,m 1 } and = {m 1 +1,...,m}. Let e T ( =1,...,m 1) denotes the th standard bass of the dmenson m 1, e T j (j = m 1 +1,...,m) denotes the (j m 1 )th standard bass of the dmenson m m 1. That s, he T 1,...,eT = I m1, m1 h e T (m 1+1),...,eT m = I (m m1). The matrx B s decomposed as B = B B. In ths paper, we make no assumpton on the stablty of A and consder the followng local relable control problems. Local Relable Control Problem (LRCP): Suppose w =n (1). Gven {A, B, ( =1,...,m)}, fnd a controller u = Kx and a set D R n wth D, such that when x() D, the closed-loop system s asymptotcally stable not only when all control components are operatonal, but also n case of any actuators outages n. Optmal Local Relable Control Problem (OL- RCP): Suppose w =n (1). Gven ( =1,...,m) and A, B, fnd the maxmal D R n together wth the controller u = Kx such that LRCP s solved. Local Relable H Control Problem (LRHCP): Gven {A, B, B w,c z, ( =1,...,m)} and a real number γ >, fnd a controller u = Kx and two sets D, D R n wth D D, such that the followng three condtons are satsfed not only when all control components are operatonal, but also n case of any actuators outages n. (P1) when w =and x() D, the closed-loop system s asymptotcally stable. (P2) when w 6=,w T w 1 and x() D, the trajectory of the closed-loop system wll arrve D and stay n D. (P3) the L 2 -gan of the closed-loop system s less than γ, that s, for each nput w( ) L 2 [, ) wth w T w 1, the response z( ) of the closed-loop system from the ntal state x() = s such that kz(t)k 2 dt γ 2 kw(t)k 2 dt. 3. Local Relable Control In ths secton, we consder the local relable control problem (LRCP). The followng lemma s smlar wth Lemma 1 n [6] and the proof s omtted here. Lemma 1 Suppose f : R m R m s the m-dmensonal saturaton functon n (1), then for any 2I W = dag[w 1,w 2,...,w m ] >,R =dag[r 1,r 2,...,r m ] >, N =dag[n 1,n 2,...,n m ] >, andy =[y 1,y 2,...,y m ] T R m,f {1,...,m}, y r 2, (4) w then 2y T Nf(R 1 y) y T NWR 1 y. The followng theorem gves a soluton of LRCP. Theorem 1 Suppose (A, B ) s stablzable and B has no zero column. If there exst ˆP R n n > and dagonal matrx E R m1 m1 > such that the followng lnear matrx nequalty (LMI) holds A ˆP + ˆPA T B EB T <, (5) then the controller u = R 1 B T ˆP 1 x (6) and the set n o D = x R n : x T ˆP 1 x λ 2 mn(γ ˆP E 1 ) 1 (7)

3 s a soluton of LRCP, where Γ ˆP =dag 2 q 1 e T BT ˆP (8) 1 B e and R > and R > are arbtrarly chosen dagonal matrces such that 2E 1 R. Proof. Denote P = ˆP 1 R, R =, R then the condton (5) becomes PA+ A T P PB EB T P<, (9) and the controller (6) becomes u = R 1 B T Px. Snce actuator outages may occur n, the closed-loop system can be expressed as ẋ = Ax + BNf( R 1 B T Px). (1) where N =dag[i,n ] denotes the gan matrx that may be nserted nto the feedback paths, and N wth N I s a dagonal matrx. Consder the Lyapunov functon V (x) =x T Px, the dervatve of V (x) along the trajectores of system (1) s V (x) = ẋ T Px+ x T P ẋ = x T (PA+ A T P )x +2x T PBNf( R 1 B T Px) Defne W = ER and choose dagonal matrx W wth 2I W > and small enough such that mn mn x T Px : x R n, e T j W R 1 j BT Px ªª =2 j n n oo max mn x T Px : x R n, e T EBT Px =2. (11) Denote W W =. W Snce 2E 1 R >, we have 2I W > and hence 2I W>. From Lemma 1, f, e T EBT Px = T e R 1 BT Px 2 ; j, et j W R 1 BT Px 2 j, (12) then 2x T PBNf(R 1 B T Px) x T PBNWR 1 B T Px. So under condton (12), we have V (x) x T (PA+ A T P )x x T PBNWR 1 B T Px = x T (PA+ A T P PB W R 1 BT P PB N W R 1 BT P )x x T (PA+ A T P PB W R 1 BT P )x = x T (PA+ A T P PB EB T P )x By condton (9), x 6= = V (x) <. Because of (11) and (12), system (1) s asymptotcally stable when the ntal condton x() s n the set D = x R n : x T Px V ª where and V =mn V V =mn x T Px : x R n, et EBT Px ª =2. By the propertes of quadratc forms (see e.g. Lemma 1 n [4]), we have V = mn x T Px : x R n,e T EBT Px =2 ª = (2 ) 2 (e T EBT P )P 1 (PB Ee ) 4 2 = e T EBT PB Ee. Denote Γ P =dag 2 q. (13) e T BT PB e Snce E s dagonal, we have V =mn V = λ 2 mn(γ P E 1 )=λ 2 mn(γ ˆP 1E 1 ). The proof s completed. When the egenvalues of A have non-postve real parts, a relable sem-global stablzaton ([9])resultcanbe obtaned, as shown n the followng corollary. Corollary 1 Suppose the egenvalues of A have nonpostve real parts, (A, B ) s stablzable and B has no zero column, then for any bounded subset B R n, there exsts a controller u = Kx such that B s contaned n the stablty regon of the closed-loop system, not only when all control components are operatonal, but also n case of any actuators outages n. Proof. Snce (A, B ) s stablzable and the egenvalues of A have non-postve real parts, from Lemma 1 n [9],theunquesolutonP ε of Rccat equaton P ε A + A T P ε P ε B B T P ε + εi =, ε > satsfes lm P ε =. ε Because P ε and E = I satsfy condton (9), from Theorem 1, f we choose the controller as u = R 1 R 1 B T P ε x, where R > and R > are arbtrarly chosen dagonal matrces such that 2I R,then D ε = x R n : x T P ε x λ 2 mn(γ Pε ) ª s contaned n the stablty regon. From (13), lm λ mn(γ Pε )=. ε Hence, for any bounded subset B R n, we can choose

4 ε suffcently small such that B D ε, the proof s completed. 4. Optmal Local Relable Control In ths secton, we consder the optmal local relable control problem (OLRCP). By Theorem 1, OLRCP s to maxmze D = x R n : x T Px λ 2 mn(γ P E 1 ) ª = ( x R n : x T Px 1 λ 2 max(eγ 1 P ) = x R n : x T λ 2 max(eγ 1 P )P x 1 ª subject to (9). Snce P, E satsfy (9) f and only f for any a>, P = ap, Ẽ = 1 ae also satsfy (9), and λ 2 max(eγ 1 P )= 1 aλ2max(ẽγ 1 P ), λ 2 max(eγ 1 P )P = λ2max(ẽγ 1 P ) P, (14) wthout loss of generalty, we can assume λ 2 max(eγ 1 P )= 1. Now OLRCP s to maxmze D = x R n : x T Px 1 ª subject to (9) and E Γ P. (15) If the problem to maxmze the largest ball that contaned n D s consdered, then we should mnmze λ max (P ) and the followng result can be obtaned. Theorem 2 Suppose ˆP R n n >, dagonal matrx E R m1 m1 >, λ R > are the solutons of the followng matrx nequalty (MI) problem max λ subject to (5) and ˆP λi (16) E Γ ˆP 1. (17) Then the largest ball contaned n the stablty regon can be obtaned by B max = x R n : x T x λ ª. (18) The correspondng controller s gven by (6) where R > and R > are arbtrarly chosen dagonal matrces such that 2E 1 R. Proof. If we denote ˆP = P 1, then (9) and (15) become (5) and (17). To mnmze λ max (P ) s equvalent n to maxmze λ that o satsfes (16). Snce D = x R n : x T ˆP 1 x 1, the largest ball B max can be obtaned by (18). The proof s completed. The MI problem n Theorem 2 s not an LMI problem because Γ ˆP 1 n (17) s not lnear n ˆP. So we suggest usng the followng teratve LMI (ILMI) algorthm to desgn the controller. ) Algorthm ILMI: Step 1 Choose ˆP R n n >, λ R > and a tolerance η R >. " # 2 Step 2 Compute Γ ˆP 1 =dag. q e T BT 1 ˆP B e Step 3 Solve LMI problem max λ subject to (5), (16) and ˆP ˆP (19) E Γ ˆP 1 (2) to obtan ˆP. Step 4 If λ λ λ < η, go to Step 6, else go to Step 5. Step 5 Let ˆP = ˆP, λ = λ, go to Step 2. Step 6 Obtan the controller and the largest ball by (6) and (18). Remark 2 By (14), a choce of the ntal ˆP to guarantee the feasblty of the LMI problem n Step 3 s as follows. Solve LMI (5) to obtan ˆP and E, thenlet ˆP = λ 2 max(eγ 1 ˆP 1) ˆP. Remark 3 Snce ˆP ˆP mples Γ ˆP 1 Γ ˆP 1, (19) and (2) mply (17). Hence ˆP,E and λ obtaned n Step3satsfes the MIs n Theorem 2. Moreover, once the LMI problem n Step 3 s feasble for a ˆP, t s also feasble for the next terate defned n Step 5. Furthermore, the postve sequence {λ} obtaned n Algorthm ILMI s non-decreasng. If t s bounded, then the algorthm converges. If λ +, then the system s relably sem-global stablzable and the algorthm can be stopped when a requred λ s obtaned. Remark 4 If the problem to maxmze the volume of D s consdered, then we should mnmze log det(p ) nstead of λ max (P ) and smlar results as n Theorem 2 can be obtaned. 5. Local Relable H Control In ths secton, we consder the local relable H control problem (LRHCP). The followng theorem gves a soluton of the problem. Theorem 3 Suppose (A, B ) s stablzable and B has no zero column. For a gven scalar γ >, fthere exst P R n n >, dagonal matrx E R m1 m1 > and scalars β 1, β 2 >, 1 > ε > such that the followng matrx nequaltes hold PA+ A T P PB EB T P + β 1 γ PB 2 w BwP T + 1 β C T (21) 1 z C z <, PA+ A T P PB EB T P +β 2 PB w B T wp + 1 εβ 2 λ 2 mn (ΓP E 1 ) P <, (22)

5 where Γ P s defned n (13), then the controller u = R 1 R 1 B T Px (23) and the sets D = x R n : x T Px λ 2 mn(γ P E 1 ) ª (24) D = x R n : x T Px ελ 2 mn(γ P E 1 ) ª (25) s a soluton of LRHCP, where R > and R > are arbtrarly chosen dagonal matrces such that 2E 1 R. Proof. Denote R R =, R then the controller (23) becomes u = R 1 B T Px. and the closed-loop system can be expressed as ẋ = Ax + BNf( R 1 B T Px)+B w w (26) z = C z x (27) Notce that (21) s stronger than (9), when w =, closed-loop stablty s ensured by Theorem 1. Now we consder the trajectory of the closed-loop system when w 6=. The dervatve of V (x) along the trajectores of system (26) s V (x) = x T (PA+ A T P )x +2x T PB w w +2x T PBNf( R 1 B T Px). x D, (12) holds and β 2 >, V (x) 1 w T w β 2 = x T (PA+ A T P )x +2x T PBNf( R 1 B T Px) +2x T PB w w 1 w T w β 2 x T PA+ A T P x x T PBNWR 1 B T Px +2x T PB w w 1 w T w β 2 = x T (PA+ A T P PB W R 1 BT P PB N W R 1 BT P )x à T Ã! pβ2 BwPx T p w! 1 pβ2 BwPx T p 1 w β2 β2 +β 2 x T PB w B T w Px x T PA+ A T P + β 2 PB w BwP T x x T (PB W R 1 BT P + PB N W R 1 BT P )x x T PA+ A T P PB EB T P + β 2PB w BwP T x Now from (22), V (x) 1 w T 1 w β 2 εβ 2 λ 2 mn(γ P E 1 ) xt Px Snce w T w 1, we have V (x) 1 1 β 2 εβ 2 λ 2 mn(γ P E 1 ) xt Px. If x T Px > ελ 2 mn(γ P E 1 ),.e. x/ D, then V (x) <. Nowwehaveprovedthat x D \D, V (x) <. So when x() D, the trajectory of the closed-loop system wll arrve D and stay n D. Now consder the L 2 -gan of the closed-loop system. Smlar to the above, x D, β 1 >, V (x) γ2 w T w + 1 z T z β 1 β 1 = x T (PA+ A T P )x +2x T PBNf( R 1 B T Px) +2x T PB w w γ2 w T w + 1 x T Cz T C z x β 1 β 1 x T (PA+ A T P PB EB T P + β 1 γ 2 PB wbwp T + 1 Cz T C z )x. β 1 By condton (21), t R, dv (x(t)) dt + 1 β 1 kz(t)k 2 γ2 β 1 kw(t)k 2. Integratng the above nequalty on the nterval [,T] yelds R V (x(t )) V (x()) + 1 T β 1 kz(t)k2 dt R γ2 T β 1 kw(t)k2 dt. Snce w( ) L 2 [, ), wehavez( ) L 2 [, ). Takng x() = and lettng T yelds kz(t)k 2 dt γ 2 kw(t)k 2 dt. The proof s completed. Remark 5 MI (22) contans λ 2 mn(γ P E 1 ) whch s nonlnear n P and E. However, as n Secton 4, wecan assume λ mn (Γ P E 1 )=1and the problem can be solved by an teratve LMI method smlar to Algorthm ILMI. Remark 6 When = {1,...,m} (no possble actuator falures), the problems consdered n ths paper are smlar wth that n[5]. However, the methods used n [5] may not be sutable for relable control problems. 6. Example Consder the system n (1) and (2) wth A = ,B= 1 1 B w = 1 1 1,C z = = 2 = 3 = 4 =2.,

6 The open-loop system s unstable because spec(a) = { ±.8918, ±.9949} *C. Suppose = {1, 2, 3}, = {4}. It s easy to see that B has no zero column. Moreover, t s easy to test that (A, B ) s stablzable. Consder optmal relable control problem, usng Algorthm ILMI we obtan B max = x R n : x T x λ = ª and the correspondng controller can be chosen as u = x For the local relable H control problem, P = , E = dag [ , , ], β 1 = , β 2 =1.213, ε =.5367 s a feasble soluton of the MIs (21) and (22). Choose R =dag[.41,.28,.38], R =1, we obtan the controller u = x and the two sets D = x R 4 : x T Px.1866 ª, D = x R 4 : x T Px.12 ª. 7. Concluson Local relable control for contnuous-tme lnear systems wth saturatng actuators were studed n ths paper. The state-feedback control law and estmated stablty regon of the closed-loop system can be obtaned smultaneously by solvng an LMI problem. An teratve LMI method s gven to desgn state feedback controllers such that the stablty regon of the closed-loop system s enlarged. The local relable H control problem s also studed and t s related wth the soluton of two matrx nequaltes. It should be noted that the number of LMIs (MIs) wll not ncrease when the dmenson of the system ncreases, whch s dfferent from the results n [2], [3] and [5]. Thus our method s lkely to be more useful to hgh dmensonal systems. Further research works wll be focused on the optmal local relable H control problems for lnear systems wth saturatng actuators. References [1] J. M. Gomes da Slva Jr and S. Tarbourech. Contractve polyhedra for lnear contnuous-tme systems wth saturatng controls. In Proc Amercan Control Conference, pages , San Dego, Calforna, [2] D. Henron and S. Tarbourech. LMI relaxaton for robust stablty of lnear systems wth saturatng controls. Automatca, 35(6): , [3] D. Henron, S. Tarbourech, and G. Garca. Output feedback robust stablzaton of uncertan lnear systems wth saturatng controls: an LMI approach. IEEE Transactons on Automatc Control, 44(11): , [4] H. Hnd and S. Boyd. Analyss of lnear systems wth saturaton usng convex optmzaton. In Proc. of the 37th IEEE Conference on Decson and Control, pages 93 98, Tampa, [5] T. Hu, Z. Ln, and B. M. Chen. An analyss and desgn method for lnear systems subject to actuator saturaton and dsturbance. Automatca, 38(2): , 22. [6] S. Huang and J. Lam. Saturated lnear-quadratc regulaton of uncertan lnear systems: stablty regon estmaton and controller desgn. Int. J. Control, 75(2):97 11, 22. [7] T. Nguyen and F. Jabbar. Dsturbance attenuaton for systems wth nput saturatons: an LMI approach. IEEE Transactons on Automatc Control, 44(4): , [8] C. Pttet, S. Tarbourech, and C. Burgat. Stablty regon for lnear systems wth saturatng controls va crcle and Popov crtera. In Proc. of the 36th IEEE Conference on Decson and Control, pages , San Dego, Calforna, USA, [9] A. Saber, Z. Ln, and A. R. Teel. Control of lnear systems wth saturatng actuators. IEEE Transactons on Automatc Control, 41(3): , [1] L. Scble and B. Kouvartaks. Stablty regon for a class of open-loop unstable lnear systems: theory and applcaton. Automatca, 36(1):37 44, 2. [11] R. J. Vellette. Relable lnear-quadratc state-feedback control. Automatca, 31(1): , [12] R. J. Vellette, J. V. Medanc, and W. R. Perkns. Desgn of relable control systems. IEEE Transactons on Automatc Control, 37(3):29 34, 1992.