Delay-dependent Robust Stabilization for Uncertain Singular Systems with Multiple Input Delays

Transcription

1 Vol. 35, No. 2 ACTA AUTOMATICA SINICA February, 2009 Delay-depedet Robust Stabilizatio for Ucertai Sigular Systems with Multiple Iput Delays DU Zhao-Pig 1 ZHANG Qig-Lig 1, 2 LIU Li-Li 1 Abstract I this paper, the problem of delay-depedet robust stabilizatio is ivestigated for sigular systems with multiple iput delays ad admissible ucertaities. First, a improved delay-depedet stabilizatio criterio for the omial system is established i terms of liear matrix iequalities (LMIs). The, based o this criterio, the problem is solved via state feedback cotroller, which guaratees that the resultat closed-loop system is regular, impulse free, ad stable for all admissible ucertaities. Numerical examples are provided to illustrate the effectiveess of the proposed method. Key words Delay-depedet, stability, robust stabilizatio, sigular systems, multiple iput delays Time-delays are ofte ecoutered i various dyamic systems, such as maufacturig systems, ecoomic systems, biological systems, etworked cotrol systems, etc. They are geerally regarded as a mai source of istability ad poor performace i such systems. I the past decades, the problems of robust stability ad robust stabilizatio for liear time-delay systems have bee ivestigated. Commoly, the existig results ca be classified ito two types: delay-idepedet coditios 1 2 ad delay-depedet coditios 3 7. Geerally, the delayidepedet case is more coservative tha the delaydepedet case. I recet years, much attetio has bee focused o the problems of robust stability ad stabilizatio aalysis for sigular time-delay systems. Sigular systems are also referred to as descriptor systems, implicit systems, geeralized state-space systems, differetial-algebraic systems, or semi-state systems 8. May fudametal results based o stadard state-space systems have bee exteded to sigular systems. However, it is kow that the problem for sigular systems is much more complicated tha that for stadard state-space systems. Over the last few years, the delay-idepedet case has bee extesively studied Recetly, the problem of delay-depedet robust stability for ucertai discrete sigular time-delay systems has bee cosidered. Ji 11 solved the problem based o the assumptio that the system was regular ad causal. Ma 12 ad Wag 13 discussed the problem for ucertai discrete sigular time-varyig delay systems. I 12, the delay-depedet robust stabilizatio result was proposed by trasformig the system ito a stadard state-space system. I 13, Wag ivestigated the problem of delay-depedet robust H cotrol for the system based o a fiite sum iequality. For cotiuous sigular time-delay systems, Wu 14, Zhu 15, ad Boukas 16 gave some results o delaydepedet stability ad stabilizatio for sigle state-delay sigular systems. Refereces discussed the problem of delay-depedet H cotrol for sigle time-delay sigular systems. However, to the best of our kowledge, the stability ad stabilizatio problems for cotiuous sigular systems with multiple time-delays have ot yet bee fully ivestigated. Particularly, delay-depedet sufficiet coditios are few eve o-existig i the published works. Received November 29, 2007; i revised form April 13, 2008 Supported by Natioal Natural Sciece Foudatio of Chia ( ) 1. Istitute of Systems Sciece, Northeaster Uiversity, Sheyag , P. R. Chia 2. Key Laboratory of Itegrated Automatio of Process Idustry, Miistry of Educatio, Northeaster Uiversity, Sheyag , P. R. Chia DOI: /SP.J I this paper, we cosider the problem of delaydepedet robust stabilizatio for ucertai sigular systems with multiple iput delays. First, a delay-depedet stabilizatio criterio for the resultat omial system is established i terms of LMIs. The, based o this criterio, the delay-depedet robust stabilizatio criterio is proposed. Fially, two examples are give to show the effectiveess of the preseted results. The rest of this paper is orgaized as follows. I Sectio 2, the problem is formulated. I Sectio 3, the mai results icludig results o stabilizatio ad robust stabilizatio are give. I Sectio 4, umerical examples are preseted to illustrate the proposed results effectively. Fially, the coclusio is give i Sectio 5. Notatios. Throughout this paper, for real symmetric matrices X ad Y, the otatio X Y (respectively, X > Y ) meas that the matrix X Y is semi-positive defiite (respectively, positive defiite). 1 Problem statemet Cosider the followig ucertai sigular system with multiple iput delays: Eẋẋẋ(t) = (A + A)x(t) + (B i + B i)u(t τ i) x(t) = φ(t), t τ, 0 (1) x(t) Rñ is the state, u(t) R m is the cotrol iput, τ i are costat time-delays satisfyig 0 < τ i τ, i = 1, 2,,, τ is the upper boud of τ i, ad φ(t) is a compatible vector valued iitial fuctio. The matrix E Rñ ñ may be sigular ad rak (E) = r ñ is assumed. A, B i, i = 1, 2,, are real costat matrices, ad A, B i, i = 1, 2,, are orm-bouded ucertai parameter matrices, ad are assumed to be of the followig forms: A = DF (t)h, B i = D if i(t)h i (2) D, H, D i, ad H i are kow real costat matrices with appropriate dimesios, ad F (t) R k g, F i(t) R k i g i are ukow parameter matrices satisfyig F T (t)f (t) I, F T i (t)f i(t) I (3) The parametric ucertaities A ad B i are said to be admissible if both (2) ad (3) hold. Without loss of geerality, we ca assume that the matrices E ad A i (1) have the followig forms: E = Ir 0, A = A11 A 12 A 21 A 22 (4)

2 No. 2 DU Zhao-Pig et al.: Delay-depedet Robust Stabilizatio for Ucertai Sigular Systems 163 The purpose of this paper is to develop a delaydepedet robust stabilizatio coditio for ucertai sigular time-delay system (1). The, we ca desig the state feedback cotroller u(t) = Kx(t) for system (1) such that the followig closed-loop system is regular, impulse free, ad stable for all time-delays τ i satisfyig 0 < τ i τ, i = 1, 2,,. Eẋẋẋ(t) = (A + A)x(t) + (B i + B i)kx(t τ i) x(t) = φ(t), t τ, 0 (5) K is a costat matrix to be determied. The omial sigular time-delay system of (5) ca be described as Eẋẋẋ(t) = Ax(t) + B ikx(t τ i) x(t) = φ(t), t τ, 0 (6) Defiitio 1 8, 15, 21. 1) The pair (E, A) is said to be regular if det(se A) is ot idetically zero. 2) The pair (E, A) is said to be impulse free if deg (det(se A)) = rak (E). Defiitio 2. For a give scalar τ > 0, the sigular time-delay system (6) is said to be regular ad impulse free for all costat time-delays τ i satisfyig 0 < τ i τ, i = 1, 2,,, if the pair (E, A) is regular ad impulse free. Defiitio 3. The ucertai sigular time-delay system (1) is said to be robustly stabilizable, if the resultat closed-loop system (5) is regular, impulse free, ad stable for all costat time-delays τ i satisfyig 0 < τ i τ, i = 1, 2,, ad all admissible ucertaities A ad B i. The followig lemmas are used i the proof of the mai results. Lemma Assume that a( ) R a, b( ) R b, ad Ň( ) R a b are defied o the iterval Ω. The, for ay matrices X R a a, Y R a b, ad Z R b b, the followig holds: 2 a T (α)ňb(α)dα Ω T a(α) X Y Ň a(α) b(α) Ω Y T Ň T dα Z b(α) X Y Y T 0. Z Lemma Give a symmetric matrix Ω ad matrices Γ, Ξ with appropriate dimesios, we have Ω + Γ Ξ + Ξ T T Γ T < 0 for all satisfyig T I, if ad oly if there exists a scalar ε > 0 such that Ω + εγγ T + ε 1 Ξ T Ξ < 0 Lemma For symmetric positive-defiite matrix Q ad matrices P ad R with appropriate dimesios, matrix iequality P T R + R T P R T QR + P T Q 1 P holds. 2 Mai results I this sectio, we discuss the coditio of robust stabilizatio for ucertai sigular time-delay system (1). First, we preset a delay-depedet criterio guarateeig system (6) to be regular, impulse free, ad stable, which plays a importat role i obtaiig the delay-depedet coditio for system (1). Theorem 1. For a give state feedback gai K, the omial sigular time-delay system (6) is regular, impulse free, ad stable for all time-delays τ i satisfyig 0 < τ i τ, i = 1, 2,, if there exist matrices Z i > 0, Q i > 0, X i, Y i, i = 1, 2,, ad a osigular matrix P, such that the followig LMIs hold: E T P = P T E 0 (7) Xi Y i Yi T E T 0 Z ie (8) Ω11 Ω 12 Ω T < 0, 12 Ω 22 i = 1, 2,, (9) Γ11 Γ 12 Ω 11 = Γ T 12 Γ 22 Γ 11 = P T A + A T P + ( τx i + Y i + Yi T + Q i) Γ 12 = P T B 1K Y 1, P T B 2K Y 2,, P T B K Y Γ 22 = diag{q 1, Q 2,, Q } Ω 12 = τa, B 1K, B 2K,, B K T Z 1, Z 2,, Z Ω 22 = τdiag{z 1, Z 2,, Z } Proof. The proof of this theorem is divided ito two parts. First, we will show that system (6) is regular ad impulse free, which is equivalet to say that (E, A) is regular ad impulse free. From (4), (7), ad (8), it is easy to see that Yi Zi11 Z i12 P Y i =, Z Y i2 0 i = Zi12 T, P = Z i22 P 2 P 3 (10) From (8) ad (9), we get P T A + A T P + (Y i + Yi T ) < 0 The, we ca obtai A T 22P 3 + P T 3 A 22 < 0, which implies that A 22 is osigular, ad thus the pair (E, A) is regular ad impulse free. Therefore, system (6) is regular ad impulse free. Next, we will show that system (6) is stable. Choose the followig Lyapuov fuctioal cadidate: V (t) = V 1(t) + V 2(t) + V 3(t) V 1(t) = x T (t)e T P x(t) 0 V 2(t) = ẋẋẋ T (α)e T Z ieẋẋẋ(α)dαdβ V 3(t) = τ i t+β x T (α)q ix(α)dα

3 164 ACTA AUTOMATICA SINICA Vol. 35 matrixes P, Z i > 0, Q i > 0 are give i Theorem 1 ad satisfy (7) (9). Usig the Leibiz-Newto formula, we have x(t τ i) = x(t) ẋẋẋ(α)dα. The, system (6) ca be writte as Eẋẋẋ(t) = Ax(t) + B ik x(t) ẋẋẋ(α)dα = ( ) A + B ik x(t) B ik ẋẋẋ(α)dα Thus, the time derivative of V 1(t) is give by V 1(t) = ẋẋẋ T (t)e T P x(t) + x T (t)p T Eẋẋẋ(t) = ) 2x T (t)p (A T + B ik x(t) 2 x T (t)p T B ik ẋẋẋ(α)dα Defiig a( ), b( ), ad Ň as a(α) = x(t), b(α) = ẋẋẋ(α), ad Ň = P T B ik for all α t τ i, t, ad applyig Lemma 1, we ca get ) V 1(t) 2x T (t)p (A T + B ik x(t) + τ ix T (t)x ix(t) + 2 ẋẋẋ T (α)e T Z ieẋẋẋ(α)dα + x T (t)(y i P T B ik) ẋẋẋ(α)dα x T (t) P T A + A T P + 2 (τ ix i + Y i + Yi T ) x(t) + ẋẋẋ T (α)e T Z ieẋẋẋ(α)dα + x T (t)(p T B ik Y i)x(t τ i) matrixes X i, Y i, ad Z i satisfy (8). Sice V 2(t) ad V 3(t) yield the relatio V 2(t) = V 3(t) = 0 τ i ẋẋẋ T (t)e T Z ieẋẋẋ(t) ẋẋẋ T (t + β)e T Z ieẋẋẋ(t + β)dβ = τ i Ax(t) + B ikx(t τ i) T Z i Ax(t) + B ikx(t τ i) ẋẋẋ T (α)e T Z ieẋẋẋ(α)dα (11) {x T (t)q ix(t) x T (t τ i)q ix(t τ i)} (12) we have the time derivative of V (t) alog the trajectory of system (6) as V (t) = V 1(t) + V 2(t) + V 3(t) ξ T (t) Ω ξ(t) (13) ξ(t) = x T (t), x T (t τ 1), x T (t τ 2),, x T (t τ ) T Ω11 Ω12 Ω = Ω 22 Ω T 12 Ω 11 = P T A+A T P + Ω 12 = ( τx i+y i+yi T +Q i+ τa T Z ia) P T B 1K Y 1 + Y + τa T Z ib 1K,, P T B K τa T Z ib K Ω 22 = diag{q 1, Q 2,, Q } + τk T B1 T Z ib 1K τk T B T 1 Z ib K τk T B T Z ib 1K τk T B T Z ib K It follows that the iequality Ω < 0 guaratees V (t) < 0 for all o-zero ξ(t). Hece, Ω < 0 guaratees that system (6) is stable for all time-delays τ i satisfyig 0 < τ i τ, i = 1, 2,,. By Schur complemet, Ω < 0 is equivalet to LMI (9). The, we have the desired result immediately. Theorem 1 presets a stabilizatio result for the omial system of (1). Now, we are i a positio to preset the result o the problem of delay-depedet robust stabilizatio for system (1). Theorem 2. For a give state feedback gai K, the ucertai sigular time-delay system (1) is robustly stabilizable for all time-delays τ i satisfyig 0 < τ i τ, i = 1, 2,, ad all admissible ucertaities if there exist matrices Z i > 0, Q i > 0, X i, Y i, a osigular matrix P, ad scalars ε, ε i, i = 1, 2,,, such that (7), (8), ad the followig LMI hold: ˆΩ 11 = ˆΩ 11 ˆΩ12 ˆΩ13 ˆΩ T 12 ˆΩ T 13 ˆΩ 22 ˆΩ T 23 ˆΩ23 ˆΩ 33 ˆΓ11 ˆΓ12 ˆΓ T 12 ˆΓ 22 < 0, i = 1, 2,, (14) ˆΓ 11 = Γ 11 + εh T H ˆΓ 12 = Γ 12 ˆΓ 22 = diag{ Q 1 + ε 1K T H1 T H 1K, Q 2 + ε 2K T H T 2 H 2K,, Q + ε K T H T H K} ˆΩ 12 = Ω 12 ˆΩ 13 = P, 0,, 0 T D, D 1, D 2,, D ˆΩ 22 = Ω 22 ˆΩ 23 = τz 1, Z 2,, Z T D, D 1, D 2,, D ˆΩ 33 = diag{εi, ε 1I, ε 2I,, ε I}

4 No. 2 DU Zhao-Pig et al.: Delay-depedet Robust Stabilizatio for Ucertai Sigular Systems 165 ad Γ 11, Γ 12, Ω 12, Ω 22 are defied i (9). Proof. Replace A ad B i, i = 1, 2,, i (9) with A+ DF (t)h ad A i + D if i(t)h i, i = 1, 2,,, respectively. The, we have Ω + U T F T (t)v + V T F (t)u + Ui T Fi T (t)v i + V T i F i(t)u i < 0 (15) U = H, 0,, 0 2 V = D T P, 0,, 0, τd T Z 1,, τd T Z U i = 0,, 0, H ik, 0,, 0 i 2 i V i = Di T P, 0,, 0, τd i T Z 1,, τd i T Z Ω11 Ω 12 ad Ω = Ω T is defied i (9). Accordig to 12 Ω 22 Lemma 2, (15) is equivalet to Ω + εu T U + ε 1 V T V + ε iui T U i + ε 1 i Vi T V i < 0 From Schur complemet ad Theorem 1, we ca get Theorem 2 easily. Remark 1. As show i Theorems 1 ad 2, whe B ik ad B ik are set to be A i ad A i, respectively, A i ad A i are matrices with appropriate dimesios, ad A i is a admissible ucertaity, the system (5) is a sigular system with multiple state delays ad admissible ucertaities. That is to say, our results are valid to some sigular systems with multiple state delays ad admissible ucertaities. I the followig, we are i a positio to preset the result o the cotroller desig for ucertai sigular time-delay system (1). First, a cotroller desig method for system (6) is give. Theorem 3. The omial sigular time-delay system (6) is regular, impulse free, ad stable for all time-delays τ i satisfyig 0 < τ i τ, i = 1, 2,, if there exist matrices Z i > 0, Q i > 0, N, X i, Ỹi, i = 1, 2,, ad a osigular matrix X, such that the followig LMIs hold: X T E T = EX 0 (16) Xi Ỹ i Ỹi T X T E T + EX E T 0 (17) ZiE Θ11 Θ 12 Θ T < 0, i = 1, 2,, (18) 12 Θ 22 I this case, a desired state feedback gai is give by Θ 11 = ˇΓ11 ˇΓ12 ˇΓ T 12 K = NX 1 ˇΓ 22 ˇΓ 11 = AX + X T A T + ( τ X i + Ỹi + Ỹ i T + Q i) ˇΓ 12 = B 1N Ỹ1, B1N Ỹ2,, BN Ỹ ˇΓ 22 = diag{ Q 1, Q 2,, Q } Θ 12 = τax, B 1N, B 2N,, B N T I, I,, I Θ 22 = τdiag{ Z 1, Z 2,, Z } Proof. Usig Schur complemet, (9) is show to be equivalet to Θ11 Θ12 < 0 (19) Θ 22 Θ 11 = Γ11 Γ12 Γ T 12 Θ T 12 Γ 22 Γ 11 = P T A + A T P + ( τx i + Y i + Yi T + Q i) Γ 12 = P T B 1K Y 1, P T B 2K Y 2,, P T B K Y Γ 22 = diag{q 1, Q 2,, Q } Θ 12 = τa, B 1K, B 2K,, B K T I, I,, I Θ 22 = τdiag{z 1 1, Z 1 2,, Z 1 } The, pre- ad post-multiplyig both sides of iequality (19) by diag{p T,, P T, I,, I} ad its traspose, +1 ad defiig X = P 1, Xi = X T X ix, Ỹi = XT Y ix, Qi = X T Q ix, Zi = Z 1 i, N = KX, we ca obtai iequality (18). Pre- ad post-multiplyig (7) by X T ad X, respectively, we ca get (16) easily. Pre- ad post-multiplyig both sides of iequality (8) by diag{x T, X T } ad its traspose, we have Xi Ỹ i Ỹ T i X T E T Z 1 i EX From (4), (10), ad (20), we have X T E T Z 1 i EX = Usig Lemma 3, we ca get 0, i = 1, 2,, (20) P T 1 Z i11p 1 0 (21) X T E T + EX E T ZiE = P T 1 + P 1 1 (Z i11 Z i12z 1 i22 ZT i12) P T 1 (Z i11 Z i12z 1 i22 ZT i12)p 1 X T E T Z 1 i EX The, if Xi Ỹ i Ỹ T i X T E T + EX E T ZiE 0 we ca get iequality (20). It is easy to see that (16) (18) imply (7) (9), respectively. Therefore, accordig to Theorem 1, we ca see for all time-delays τ i satisfyig 0 < τ i τ, i = 1, 2,,

5 166 ACTA AUTOMATICA SINICA Vol. 35 that system (6) is regular, impulse free, ad stable, ad the correspodig state feedback gai is K = NX 1. Next, based o Theorem 3, we give the followig theorem. Theorem 4. The ucertai sigular time-delay system (1) is robustly stabilizable for all time-delays τ i satisfyig 0 < τ i τ, i = 1, 2,, ad all admissible ucertaities if there exist matrices Z i > 0, Q i > 0, N, X i, Ỹi, a osigular matrix X, ad scalars ε, ε i, i = 1, 2,,, such that (16), (17) ad the followig LMI hold: ˆΘ 11 ˆΘ12 ˆΘ13 ˆΘ T 12 ˆΘ 22 ˆΘ23 ˆΘ T 13 ˆΘ T 23 ˆΘ 33 < 0, i = 1, 2,, (22) I this case, a desired state feedback cotroller is give by ˆΘ 11 = Γ11 Γ12 Γ T 12 u(t) = NX 1 x(t) Γ 22 Γ 11 = AX + X T A T + εdd T + ( τ X i + Ỹi + Ỹ i T + Q i) + ε id idi T Γ 12 = B 1N Ỹ1, B2N Ỹ2,, BN Ỹ Γ 22 = diag{ Q 1, Q 2,, Q } ˆΘ 12 = τax, B 1N, B 2N,, B N T I,, I + εd T, 0,, 0 T τd T,, τd T + ε idi T, 0,, 0 T τd i T,, τd i T ˆΘ 22 = τdiag{ Z 1, Z 2,, Z } + ε τd T,, τd T T τd T,, τd T + ε i τd i T,, τd i T T τd i T,, τd i T ˆΘ 13 = diag{x T H T, N T H1 T, N T H2 T,, N T H T } ˆΘ 23 = 0 ˆΘ 33 = diag{εi, ε 1I, ε 2I,, ε I} Proof. Replace A ad B i, i = 1, 2,, i (18) with A + DF (t)h ad B i + D if i(t)h i, i = 1, 2,,, respectively. Similarly to the proof of Theorem 2, we ca get this theorem easily. Remark 2. From the proof of these theorems, we ca see that all the obtaied results are expressed i terms of LMIs ivolvig o decompositio of system matrices, which makes the desig method relatively simple ad reliable. 3 Examples To demostrate the effectiveess of our method, we briefly cosider the followig two examples. Example 1 demostrates the effectiveess of the obtaied criterio i Theorem 1 i that it may readily be used to fid a solutio to the problem proposed. Example 1. Cosider the followig sigular system with multiple iput delays: Eẋẋẋ(t) = Ax(t) + B 1Kx(t τ 1) + B 2Kx(t τ 2) E = B 1 = K = diag{1, 1} 0.5 0, A = 0 1, B 2 = Accordig to Theorem 1 ad usig Matlab LMI toolbox, it is foud that this system is regular, impulse free, ad stable for all time-delays τ i satisfyig 0 < τ i , i = 1, 2. However, the methods i fall short of obtaiig this delay-depedet coditio. Example 2. Cosider the followig ucertai sigular system with iput delay: Eẋẋẋ(t) = (A + A)x(t) + (B 1 + B 1)Kx(t τ 1) (23) E =, A = , B 1 = 1.5 ad the ucertai matrices ca be described by A = DF (t)h, B 1 = D 1F 1(t)H 1, ad with A 0.2, B D = D 1 =.2 cos t 0 F (t) = F 1(t) = H = 0 1, H 1 = 0 si t 1 1 I this case, accordig to Theorem 4 ad usig Matlab LMI toolbox, it is foud τ = For example, whe τ = 1.5, the solutios to system (23) are show below X =, X1 = Ỹ 1 =, Z1 = Q 1 = ε = , ε 1 = , N = ad the correspodig state feedback cotroller is u(t) = x(t) However, the methods i are ot feasible to this example. Remark 3. Theorems i this paper exted the results i 14 16, which deal with the problem of stability for sigle time-delay systems, to multiple time-delay systems. That is to say, our results are more geeral ad are a improvemet over the previous oes.

6 No. 2 DU Zhao-Pig et al.: Delay-depedet Robust Stabilizatio for Ucertai Sigular Systems Coclusio The problem of delay-depedet robust stabilizatio for sigular systems with multiple iput delays ad admissible ucertaities has bee ivestigated. First, a delaydepedet stabilizatio criterio for the omial system is preseted. The, based o this criterio, the problem is solved via state feedback cotroller, thus guarateeig the sigular system with multiple iput delays ad admissible ucertaities to be robustly stabilizable, ad a explicit expressio of the desired state feedback cotroller is also give. The obtaied results are expressed i terms of LMIs ivolvig o decompositio of system matrices, makig the desig method relatively simple ad reliable. Two examples are give to show the effectiveess of the proposed method ad the improvemet over some existig methods. Refereces 1 Luo J S, va de Bosch P P J. Idepedet of delay stability criteria for ucertai liear state space models. 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Correspodig author of this paper. duzhaopig98@163.com ZHANG Qig-Lig Professor at Northeaster Uiversity. His research iterest covers sigular systems, etworked cotrol systems, ad robust cotrol. qlzhag@mail.eu.edu.c LIU Li-Li Ph. D. cadidate at the Istitute of Systems Sciece, Northeaster Uiversity. She is also a lecturer at Northeaster Uiversity. Her research iterest covers oliear systems, etworked cotrol systems, ad sigular systems. liulili @sia.com