Memory State Feedback Control for Singular Systems with Multiple Internal Incommensurate Constant Point Delays

Vol. 35, No. 2 ACTA AUTOMATICA SINICA February, 2009 Memory State Feedback Control for Singular Systems with Multiple Internal Incommensurate Constant Point Delays JIANG Zhao-Hui 1 GUI Wei-Hua 1 XIE Yong-Fang 1 YANG Chun-Hua 1 Abstract In this paper, the problem of delay-dependent stabilization for singular linear continuous-time systems with multiple internal incommensurate constant point delays SLCS-MIID) is investigated. The condition when a singular system subject to point delays is regular independent of time delays is given and it can be easily tested with numerical or algebraic methods. Based on the Lyapunov-Krasovskii functional approach and the descriptor integral-ineuality lemma, a sufficient condition for delay-dependent stability is obtained. The main idea is to design multiple memory state feedback control laws such that the resulting closed-loop system is regular independently of time delays, impulse free, and asymptotically stable via solving some strict linear matrix ineualities LMIs) problem. An explicit expression for the desired memory state feedback control law is also given. Finally, a numerical example illustrates effectiveness and availability for the proposed method. Key words Singular systems, multiple internal point delays, descriptor integral-ineuality, delay-dependent stabilization, memory state feedback control Singular systems contain a mixture of differential and algebraic euations, the algebraic euations represent the constraints to the solution of the differential part. It is difficult to deal with such singular systems due to their nature complex, particularly when we design control systems for them. Many practical processes [1 3, such as circuit systems, constrained control problems, chemical, large scale systems, etc., can be modeled as singular systems. Singular systems can preserve the structure of physical systems and describe non-dynamic constraints, and finite dynamic and impulsive behavior simultaneously [4. For the past years, analysis and synthesis problems of singular systems have been extensively studied due to the fact that the singular systems provide a more complicated, yet richer, description of dynamical systems than standard state-space systems [5 8. Furthermore, the study of the dynamic performance of singular systems is much more difficult than that of standard state space systems since singular systems usually have three types of modes, namely, finite dynamic modes, impulsive modes, and non-dynamic modes [8, while the latter two do not appear in standard state space systems. Delay systems have attracted much attention over the past decades since time delay is one of the main causes for instability and poor performance of many control systems and is freuently encountered in many industrial processes [9. Delays may be classified as point delays or distributed delays according to their nature, and can also be classified as internals i.e., in the state) or externals i.e., in the input or output) according to the signals that they influence. Point delays may be commensurate if each delay is an integer multiple of a base delay or, more generally, incommensurate if they are arbitrary real numbers [10. The presence of internal delays leads to a large complexity in the resulting system dynamics since the whole dynamical system becomes infinite-dimensional. In addition, this fact increases the difficulty when studying basic properties, such Received October 23, 2007; in revised form February 29, 2008 Supported by the Key Program of National Natural Science Foundation of China 60634020), Doctoral Program Foundation of Ministry of Education of China 20050533028, 20070533132), Natural Science Foundation of Hunan Province 06JJ5145), and Program for New Century Excellent Talents in University NCET-07-0867) 1. School of Information Science and Engineering, Central South University, Changsha 410083, P. R. China DOI: 10.3724/SP.J.1004.2009.00174 as, controllability, observability, stability, and stabilization, compared with the delay-free case since the transfer functions consist of a transcendent numerator and denominator uasi-polynomials [11. Due to these reasons, the stabilization of multiple internal incommensurate constant point delay systems is much more challenging than that of the delay-free case [12. Time delays have naturally an effect on the dynamics of singular systems. It is concluded that their dynamics would be more complex because time delays may exist in the differential euations and/or algebraic euations. It should be pointed out that the stability problem for singular time delay systems is much more complicated than that for regular systems because it reuires to consider not only asymptotically stability, but also regularity and impulse free at the same time [13. The latter two need not be considered in regular systems. Very recently, a lot of efforts have been devoted to the investigation of the stability and stabilization problem of singular time-delay systems [14 22. However, to the best of authors knowledge, there are few contributions to the problems of delay-dependent stabilization of singular linear continuous-time systems with multiple internal incommensurate constant point delays SLCS-MIID). This fact motivated the authors to develop delay-dependent controllers for SLCS-MIEID based on state feedback control. The singular systems dealt with are assumed to be regular, i.e., they satisfy a consistency condition ensuring the existence of at least a solution for each given admissible function of initial conditions and control input. For SLCS- MIID, the condition of regularity depends on the time delays and the condition is not easy to test. In this paper, we deal with the system structure and the formulation of the consistency condition for the system subject to point delays to be regular in the time-invariant case. An euivalent condition of regularity is given, which is independent of time delays. In state feedback control synthesis problems for linear time delay systems, two typical classes of state feedback controllers would be delay-independent memoryless state feedback controllers and delay-dependent memory state feedback controllers. It is well known that the linearuadratic optimal control has a memory state feedback form. From the view point of the infinite-dimensional state space of time-delay systems, memory controllers are natural state feedback controllers, and we can expect that mem-

No. 2 JIANG Zhao-Hui et al.: Memory State Feedback Control for Singular Systems with Multiple 175 ory controllers achieve better performances than memoryless controllers. An inevitable issue of memory controller syntheses is the difficulty arising from their infinite dimensionality in computations and implementations. In this paper, we discuss a synthesis problem of memory controllers which stabilize the SLCS-MIID, and propose a synthesis procedure based on LMIs. The syntheses of memory controllers for linear time-delay systems have been discussed from various viewpoints [23 24. We discuss here a synthesis problem from the viewpoint of stabilization. With these motivations, we consider the delay-dependent stabilization problem for SLCS-MIID based on memory state feedback control. To prove the stability, we introduce a descriptor integral-ineuality that can be used to study the delay-dependent stabilization problems of singular time-delay systems. Using the Lyapunov-Krasovskii functional techniue combined with LMI techniues, we design a delay-dependent memory state feedback controller for singular time-delay system, which guarantees that the closed-loop system is regular independently of the delays, impulse free, and asymptotically stable. The delaydependent stability criterion is derived in terms of LMIs, and the solutions provide a parameterized representation of the memory state feedback controller. The LMIs can be easily solved by various efficient convex optimization algorithms. Notations. I and 0 denote the appropriately dimensioned identity matrix and zero matrix, respectively. diag{ } is a block-diagonal matrix. The symmetric terms [ in a symmetric [ matrix are denoted by, e.g., X Y X Y = Z Y T. Z 1 Problem statement Consider a class of SLCS-MIID described by Eẋxt) = A 0xt) + A jxt h j) + B ju jt) xt) = φt), t [ h, 0 xt) R n represents the state, u jt) R m j is the control input, A 0, A j, and B j are real matrices of orders compatible with the dimensions of those vectors, and j = 1, 2,, is the internal point delays. φt) is a compatible vector valued continuous different initial function. The maximum delay h is defined as h = max 1 j h j). The singular matrix E R n n with rank E) = r < n gives the singular character to system 1) compared to the case E = I n standard system). The unforced singular delay system of 1) can be written as Eẋẋẋt) = A 0xt) + A jxt h j) 2) In order to guarantee that system 2) is regular and impulse-free, the following definitions and lemmas are given. Definition 1. System 2) is said to be regular if there exists a constant s C such that detse j=0 Aje h j s ) 0. The fact that det se j=0 Aje h j s ) 0 depends on the internal point delays h j, makes Definition 1 hard to be testified. An alternative characterization of regularity is given as follows. First, the generic rank in C of a complex matrix Qs) is defined as max s C rank[qs)). The 1) following proposition presents a condition for regularity of system 2) which is euivalent to Definition 1. Proposition 1. System 2) is said to be regular independently of the delays h j if the rank[e, Āj = n Ā j = [A 0, Āj1 with Āj1 = [A1, A2,, A. Proof. From Definition 1, a straight forward calculation yields se A je h j s = j=0 [E, Āj[sIn, In, e h 1s I n,, e hs I n T 3) Thus, from 3), we can get that s C, det se j=0 Aje h j s ) 0 is euivalent to rank[e, Āj = n. Since rank[si n, I n, e h1s I n,, e hs I n = n, s C, max s C se j=0 Aje h j s ) = rank[e, Āj. So, rank [E, Āj = n max s CsE j=0 Aje h j s ) = n detse j=0 Aje h j s ) 0 for s C. Lemma 1 [14. Suppose the pair E, A 0) is said to be regular and impulse free. Then, the solution to 2) exists and is impulse free and uniue on [0, ). Lemma 2 [25. The pair E, A 0) is said to be regular, impulse free, and stable if and only if there exists a matrix P such that EP T = P E T and P T A T 0 P < 0. For system 1), the following memory state feedback control law is adopted u jt) = K j1xt) + K j2xt h j), j = 1,, 4) When we apply control law 4) to system 1), the resulting closed-loop system is given by Eẋxt) = Āxt) + A jxt h j) + B j2xt h j) xt) = φt), t [ max{h, h }, 0 5) Ā = A0 + B j1, Bj1 = B jk j1, Bj2 = B jk j2, and h = max 1 j h j). The aim of this paper is to develop a new delaydependent stabilization method that provides the control gain, K j1 and K j2, of the control law 4) such that the closed-loop system 5) is regular independently of the delays, impulse free, and asymptotically stable. For this purpose, the following lemmas are introduced. Lemma 3 [15. Let xt) R n be a vector-valued function with first-order continuous-derivative entries. Then, the following descriptor integral-ineuality holds for arbitrary matrices E, M 1, M 2, Y, X = X T > 0, and a scalar h 0, ẋx T s)e T XEẋxs)ds ξ T t)υξt)+hξ T t)y T X 1 Y ξt) t h 6) ξt) = [ x T t) x T t h) T [, Y = M1 M 2 [ M T Υ = 1 E + E T M 1 M1 T E + E T M 2 M2 T E E T M 2 Lemma 4 [26. For a given a matrix A R m n, there exists arbitrary full rank matrix P R m m or Q R n n such that ranka) = rankp A) = rankaq).

176 ACTA AUTOMATICA SINICA Vol. 35 2 Main results Theorem 1. For given scalars h > 0 and h > 0, if there exist symmetric and positive definite matrices X, Y j, Y j, R, R, and arbitrary matrices M 1j, M 2j, M ij, M 2j such that rank [E, T 0 = n 7) [ T 0 = EX T = XE T 0 8) Ξ11 Ξ12 Ξ13 Ξ = Ξ 22 0 < 0 9) Ξ 33 B j1 A 1 A B12 B2 Ξ 11 = Θ ht 0 T h T0 T hr 1 0, ℵ = [ 0 0 h R 1 Ξ T 12 = [ H T ℵ T ℵ T, H = [ hh1 T hh T Ξ T 13 = [ H T ℵ T ℵ T, H = [ h H 1 T h H T H j = [ M 1j L ℵ, H j = [ M 1j ℵ L L = [ { M2j, j = i r 1j r ij r j, rij = L = [ r 1j r ij r j Ξ 22 = diag{ hr,, hr} Ξ 33 = diag{ h R,, h R } Θ = Θ 11 = X Θ11 Θ12 Θ13 Θ 22 0 Θ 33 ) B j1 +, r ij = { M 2j, j = i ) T B j1 X + [Y j + Y j + M 1j T + M1j)E T + E T M 1j + M ij) Θ 12 = [ Π 1 Π, Θ13 = [ Π 1 Π Θ 22 = diag{ 1,, }, Π j = XA j M T 1jE + E T M 2j Θ 33 = diag{ 1,, }, j = Y j + M T 2jE + E T M 2j Π j = X B j2 M T 1j E + E T M 2j, j = Y j + M T 2j E + E T M 2j then the closed-loop system 5) is regular independently of the delays and impulse free and asymptotically stable. Proof. Suppose 7) 9) hold for symmetric and positive definite matrices X, Y j, Y j, R, R, and any matrices M 1j, M 2j, M ij, M 2j. Then, from 9), it is easy to see that X ) B j1 + ) T B j1 X < 0 10) By Proposition 1 and Lemma 2, it follows from 7), 8), and 10) that system 5) is regular independently of the delays and impulse free. Next, we shall examine the stability of the singular delay system 5). To this end, we choose a Lyapunov-Krasovskii functional candidate as with V t) = V 1t) + V 2t) + V 3t) + V 4t) V 1t) = x T t)xext) V 2t) = x T s)y jxs)ds + t h j V 3t) = V 4t) = 0 h 0 h t+θ t+θ t h j ẋẋẋ T s)e T REẋẋẋs)dsdθ ẋẋẋ T s)e T R Eẋẋẋs)dsdθ x T s)y j xs)ds Then, the time derivative of V t) along the trajectory 5) satisfies with V t) = V 1t) + V 2t) + V 3t) + V 4t) 11) V 1t) + V 2t) = η T t)ψηt) 12) V 3t) + V 4t) = η T t)t T 0 hr + h R )T 0ηt) ℵ 1 13) Ψ = Ψ 11 = X Ψ11 Ψ12 Ψ13 Ψ 22 0 Ψ 33 ) B j1 + ) T B j1 X + Ψ 12 = [ XA 1 XA Ψ 13 = [ X B 12 X B 2 Ψ 22 = diag{ Y 1,, Y } Y j + Y Ψ 33 = diag{ Y 1,, Y } η T t) = [ x T t) η T 1 t) η T 2 t) η T 1 t) = [ x T t h 1) x T t h ) η T 2 t) = [ x T t h 1) x T t h ) ℵ 1 = ẋx T s)e T REẋxs)ds + t h t h ẋx T s)e T R Eẋxs)ds By applying Lemma 3, it is clear that the following is true, i.e., ℵ 1 η T t) Ψ + ℵ 2)ηt) 14) Ψ = Ψ 11 Ψ12 Ψ13 Ψ22 0 Ψ33 j )

No. 2 JIANG Zhao-Hui et al.: Memory State Feedback Control for Singular Systems with Multiple 177 Ψ 11 = M1j T + M 1j T )E + E T M 1j + M 1j) Ψ 12 = [ T 1 T, Ψ13 = [ T 1 T Ψ 22 = diag{γ 1,, Γ }, Ψ33 = diag{γ 1,, Γ } T j = M T 1jE + E T M 2j, T j = M T 1j E + E T M 2j Γ j = M2jE T E T M 2j, Γ j = M 2j T E E T M 2j ℵ 2 = Hj T hr 1 H j + H j T h R 1 H j) Substituting 14) into 13) gives us V 3t)+ V 4t) η T t) Ψ+T T 0 hr+h R )T 0+ℵ 2)ηt) 15) Combining 11) 15) yields V t) η T t)θ + T T 0 hr + h R )T 0 + ℵ 2)ηt) 16) From 16), it is clear that Ξ < 0 guarantees V t) < 0 by the Schur complement. According to the Lyapunov- Krasovskii functional theorem, the closed-loop system 5) is asymptotically stable. As K j1 and K j2 are design matrices, Ξ is nonlinear in the design parameters K j1, K j2, and X. Thus, in this case, 9) cannot be solved directly by LMI toolbox. To obtain a controller gain, K j1 and K j2, from 7) 9), the following theorem is given. Theorem 2. For given numbers h > 0, h > 0, λ j, λ j, µ j 0, and µ j 0, if there exist symmetric and positive matrices X, Ȳj, Ȳ j, R, R, and arbitrary matrices K j1 and K j2 such that ℵ 3 = rank[e, A 0 X + ℵ3, T 0 = n 17) XE = E T X 0 18) Σ 11 Σ 12 Σ 13 Σ 14 Σ 15 Σ 22 0 0 0 Σ 33 0 0 Σ 44 0 Σ55 < 0 19) B j Kj1, T 0 = [A 1,, A, B 1 K12,, B K2 Σ 11 = Ω1 hωt 2 h Ω T 2 h R 0,ΣT 12 = [ HT ℵ T ℵ T h R Σ T 13 = [ HT ℵ T ℵ T H = [ h H 1 T h H T, H = [ T T h H 1 h H H j = [ 0 L ℵ, H j = [ 0 ℵ L L = [ r 1j r ij r j, rij = { R, j = i L = [ { R r 1j r ij r j, r, j = i ij = Σ T 14 = Σ T 15 = [ ℵ T 4 ℵ T ℵ T, ℵ 4 = [ X X Σ 22 = diag{ h R,, h }{{ R } } Σ 33 = diag{ h R,, h R } Σ 44 = diag{ Ȳ1,, Ȳ}, Σ55 = diag{ Ȳ 1,, Ȳ Ω11 Ω12 Ω13 Ω 12 = [ 1 Ω 1 = Ω 22 0, Ω 33 Ω 13 = [ 1 Ω 11 = A 0 X + ℵ3 + A 0 X + ℵ3) T λ 2 jµ 2 j Ȳ j + λ 2 j µ 2 j Y j λ jµ 1 j A j Ȳ j + A j Ȳ j) T ) + λ jµ 1 j B j Kj2 + B j Kj2) T ) Ω 22 = diag{ 1,, }, Ω 33 = diag{ 1,, } j = µ 1 j A j Ȳ j + XE T + λ jµ 2 j Ȳ j + λ jµ 1 Ȳ je T j = µ 1 j B j Kj2 + XE T + λ jµ 2 j Ȳ j + λ jµ 1 j j = µ 2 j Ȳ j µ 1 j EȲj µ 1 j Ȳ je T j = µ 2 j Ȳ j µ 1 j EȲ j µ 1 j Ȳ j E T Ω 2 = [ ℵ 5 ℵ 6 ℵ 7 ℵ 6 = [ µ 1 1 A1Ȳ1 µ 1 A Ȳ ℵ 5 = A 0 X + ℵ3 λ jµ 1 j A j Ȳ j + λ jµ 1 j ℵ 7 = [ µ 1 1 B 1 K12 µ 1 B K2 j Ȳ j E T B j Kj2) then the memory state feedback controller 4) guarantees that the closed-loop system 5) is regular independently of the delays, impulse free, and asymptotically stable. The controller gains K j1 and K j2 can be obtained by solving 17) 19), and K j1 = K X 1 j1, K j2 = K j2 Ȳ 1 j. Proof. To cast the problem of designing a stabilizing controller 4) into the LMI formulation, let X = X 1, Ȳ j = Y 1 j, Ȳ j = Y 1 j, R = R 1, R = R 1, K j1 = K j1 X, Kj2 = K j2 Ȳ j and define a full rank matrix P = diag{i, X, I,, I, Ȳ 1,, Ȳ }; then [E, T 0P = [E, A 0 X + ℵ3, T 0. From Lemma 4, we can conclude that rank[e, T 0P ) = rank[e, T 0) = n. That is to say 17) is euivalent to 7). By pre- and post-multiplying 8) by X, we can conclude that 18) holds. Define the following matrices [ [ X 0 A0 + W =, Ā = B j1 Ā 11 W 1 W 2 Ā 12 Ā 22 { } U = diag Y j + Y j ), Y 1,, Y, Y 1,, Y W1 T = [ M11 T M1 T M 11 T M 1 T W 2 = diag{m 21,, M 2, M 21,, M 2} Ā 11 = [ A 1 A B12 B2 Ā T 12 = [ E T E T E T E T A 13 = diag{ E,, E, E,, E} Then, Ξ = W T Ā + ĀT W + U H j = [ 0 V ℵ W, H j = [ 0 ℵ V W }

178 ACTA AUTOMATICA SINICA Vol. 35 V = [ v 1j v ij v j, vij = V = [ v 1j v ij v j { I, j = i, v ij = { I, j = i When M 1j = λ jx, M 2j = µ jy j, M 1j = λ jx, M 2j = µ jy j, µ j 0 and µ j 0, it is obvious that W is invertible, and [ X W 1 0 =, W1 = [ W11 W12 W 1 W2 W 11 = [ λ 1µ 1 1 Ȳ 1 λ µ 1 Ȳ W 12 = [ λ 1µ 1 1 Ȳ 1 λ µ 1 Ȳ W 2 = diag{µ 1 1 Ȳ 1,, µ 1 Ȳ, µ 1 1 Ȳ 1,, µ 1 Ȳ } Define T = diag{w 1, I, I, R 1,, R 1, R 1,, R 1 }. Then, Σ12 Σ13 T T ΞT = Σ 22 0 < 0 20) Σ 33 = 1 = 1 hωt 2 h Ω T 2 h R 0 h R 11 = Ω 11 + 11 Ω12 Ω13 Ω 22 0 Ω 33 XY j + Y j ) X Applying Schur complements, we find that 20) is euivalent to 19). 3 Numerical example In this section, a numerical example is presented to demonstrate the validity of the results described above. Consider the SLCS-MIID 1), with the following parameters, E = 1 0 0 0 1 0 0 0 0, A 0 = 0.2 0.1 0.5 0.25 0 0.2 0 1 2 0.4 0.2 0.6 1.2 0.1 0 A 1 = 0 0.5 0, A 2 = 0 1.1 0.1 0 0 2 0.1 0 0.1 B 1 = [ 1 0 0 T, B 2 = [ 1 1 0 T In this example, we assume that the maximum point time delays are h = 3.66, h = 0.8, λ j, µ j, λ j, µ j are chosen as λ 1 = 2.8953, λ 1 = 1.0451, λ 2 = 0.4451, µ 1 = 10.7188, µ 1 = 0.1651, and µ 2 = 2.3654. By using Matlab LMI control toolbox to solve the feasible problems 17) 19), a stabilizing memory state feedback control law can be obtained as u 1t) = [ 0.5624 0.4637 2.7508 xt) + [ 0.1326 1.0238 1.6542 xt h 1) u 2t) = [ 0.0035 0.0476 0.0039 xt) + [ 0.1128 0.8675 0.0324 xt h 2) According to Theorem 1 and Theorem 2, controller 4) with gains, K j1 and K j2, given previously, guarantees that the closed-loop systems 5) is regular independently of the delays, impulse free, and asymptotically stable. The simulation result is shown in Fig. 1. It can be observed from the figure that the system is stable. Fig. 1 4 Conclusions Simulation results for the closed-loop system In this paper, the delay-dependent stabilization problem of SLCS-MIID has been studied. The main contribution of this study is to obtain the control law, which guarantees that the closed-loop system 5) is regular independently of the delays, impulse free, and asymptotically stable. This is done by using the Lyapunov-Krasovskii functional approach combined with a descriptor integral-ineuality. The numerical example shows that the proposed controller can achieve desired design purposes. References 1 Stevens B L, Lewis F L. Aircraft Modeling, Dynamics, and Control. New York: Wiley, 1991 2 Campbell S L. Singular Systems of Differential Euations. Pitman, Marshfield: MA, 1980 3 Campbell S L. Singular Systems of Differential Euations II. Pitman, Marshfield: MA, 1982 4 Verghese G, Levy B, Kailath T. A generalized state-space for singular systems. IEEE Transactions on Automatic Control, 1981, 264): 811 831 5 Stefanovski J. LQ control of descriptor systems by cancelling structure at infinity. International Journal of Control, 2006, 793): 224 238 6 Gui Wei-Hua, Jiang Zhao-Hui, Xie Yong-Fang. Decentralized robust H control of singular large-scale systems based on descriptor output feedback. Systems Engineering and Electronics, 2006, 285): 736 740 in Chinese) 7 Wang J, Sreeram V, Liu W. An improved H suboptimal model reduction for singular systems. International Journal of Control, 2006, 797): 798 804 8 Dai L. Singular Control Systems. New York: Springer, 1989 9 Richard J P. Time-delay systems: an overview of some recent advances and open problems. Automatica, 2003, 3910): 1667 1694 10 De La Sen M. Quadratic stability and stabilization of switched dynamic systems with uncommensurate internal point delays. Applied Mathematics and Computation, 2007, 1851): 508 526

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