SUFFICIENT CONDITIONS FOR STABILIZATION BY OUTPUT FEEDBACK IN DESCRIPTOR SYSTEMS

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1 SUFFICIENT CONDITIONS FOR STABILIZATION BY OUTPUT FEEDBACK IN DESCRIPTOR SYSTEMS Elmer Rolando Llanos Villarreal 1, Santos Demetrio Miranda Borjas 2, Walter Rodrigues Martins 3, José Alfredo Ruiz Vargas 4 1,2,3 Departamento de Ciências Exatas e Naturais, Universidade Federal Rural do Semi-árido, UFERSA, Av. Francisco Mota 572, CEP: Costa e Silva Mossoró/RN, Brasil. 1 elmerllanos@ufersa.edu.br 2 santos.borjas@ufersa.edu.br 3 walterm@ufersa.edu.br 4 Faculdade de Tecnologia, Universidade de Brasilia (UNB), Brasília DF, Brasil, vargas@ieee.org Abstract The principal aim of this paper is study the stabilization problem by static output feedback for linear descriptor system. Presented sufficient conditions for the existence of a S-stabilizing output feedback control in terms of two coupled matrix, where the solutions to these equations can be obtained using condition (m + p>q). Keywords: Stabilization, Output Feedback, Descriptor Systems. 1. INTRODUCTION This paper deals with the stabilization for control problem by static output feedback for linear descriptor systems. Consider that an n-dimensional descriptor system consists of a mixture of n q algebraic equations and q first order diferential equations. Descriptor systems arise naturally in the modelling of several dynamic systems commonly used in engineering applications, such as biologic, power and other interconnected systems 7, 11, 16. Besides guaranteeing the closed-loop asymptotic stability, two other properties are desired in practice: closed-loop regularity and absence of impulsive behavior. The problem of computing a suitable static output feedback, from which these closed-loop propierties are verified, is called S- stabilization problem. These three desired properties can be described in terms of the closed-loop eigenstructure: (i) the asymptotic stability is equivalent to have all the finite poles in the left half complex plane; (ii) the absence of impulsive modes is equivalent to have q finite closed-loop; and (iii) the regularity is guaranteed if the system is impulse-free. Thus, the necessary and sufficient conditions for the existence of a S-stabilizing output feedback are obtained as a set of coupled (generalized) Sylvester equations in 8, 9, 3, 1. In case of normal systems, it is shown that output stabilizable (C, A, B)-invariant subspaces are obtained through a pair of coupled Sylvester equations by systems verifying Kimura s condition, thus, two algorithms are proposed to solve the coupled Sylvester equations in 2, 15. This paper deals with the stabilization for control problem by output feedback in descriptor systems. Thus, a new approach for the design of output feedback controller is proposed and the respective output feedback gains are obtained through the solution of the Sylvester equation. Thus, it is presented sufficient conditions under which this problem has a solution. This paper also proposes a new numerically efficient solution algorithm for the coupled design equations to determine the output feedback matrix. Another consequence of this result is that the conditions also provide sufficient conditions to control the problem by output feedback. Of course, in case of these conditions are not met, this does not mean that the system is not stabilizable by output feedback. This paper is organized as follows. The second section presents the problem based on the basic concepts, the necessary and sufficient conditions for the existence of a solution. In section 3, we present some results in the form of Lyapunov equations. The numerical examples illustrate the application of the algorithms which outline the basic steps used to solve the problem. They are presented in the fourth section. Finally, the concluding remarks are presented. Serra Negra, SP - ISSN

2 2. PRELIMINARIES The considered descriptor time-invariant systems are described by : Eẋ(t) = Ax(t)+Bu(t) (1) y(t) = Cx(t) (2) where: x X R n is the state variable, u U R m is the control variable(input variable), y Y R p is the output variable and E R n n, rank (E) =q< n ; as the other matrices have an appropriate size with rank (B) =m and rank (C) =p. Thus, the control problem is to find a static output feedback control law u(t) =Ky(t), such that the closed-loop system Eẋ(t) = (A + BKC)x(t) (3) is S-stable: regular, assymptotically stable and impulse free. It presents to follow some concepts and definitions from the geometric control theory for the descriptor system. Definition A subspace V, of dimension v, is (C,E,A,B)-invariant output stabilizable subspace, (or simply O.S. (C,E,A,B)-invariant 1 ) if V is (A, E, B)-inner stabilizable and (C, E, A)-outer detectable. For the closed loop system a subspace V O.S. (C,A,E,B)-invariant to be associated does not regulate or have impulsive ways. Consider V R v v such that Im(V )=V and consider T R (n v) v a left annihilator of V, i.e. : Ker (T )=Im(V ). A stronger definition that allows, particularly, regularizable property is considered in 12, as follows : Definition A v-dimensional subspace V X is strongly O.S. (C,A,E,B)-invariant if it is O.S. (C,A,E,B)-invariant and Ker E V = {0}. As in 1, the definition 2.2 is equivalent an existence of matrices (H V R v v, W R m v ) and (H T R n v n v, U R (n v) v ), solutions for the coupled Sylvester Equations : AV EV H V = BW (4) TA H T TE = UC (5) TEV = 0 (6) The definition 2.2 and the equations (4), (5) and (6) play a fundamental role in treating the control problem by output feedback, mainly for Eigenstructure Assignment 14, Output Stabilizable (C,E,A,B)-invariant 13 4 The present study takes the properties of stabilizability and detectability into account through the two following definitions : (i) a subspace (A,E,B)-invariant V is (A,E,B)-inner stabilizable subspace, that is there exist F such that (A + BF) V is asymptotically stable ; and (ii) a subspace (C,E,A)-invariant V is (C, E, A)- outer detectable subspace if there exists, L such that (A + LC) X /V is asymtotically stable. Let V R n v, such that Span(V )=V. Then, the condition Ker E V = {0} can be equivalently replaced by 12: rank(ev )=v (7) It is assumed that (7) is verified, a matrix T R (q v) n is a generalized left annihilator of EV, if TEV = 0 and Ker (T ) Im(E) = {0} 1. Thus, a subspace Span(V ) = V is strongly O.S. (C,A,E,B)-invariant if and only if there exist matrices (H V R v v, W R m v ) and (H T R (q v) (q v), U R (q v) v ) solutions for the coupled-generalized Sylvester equations : AV EV H V = BW, σ(h V ) C (8) TA H T TE = UC, σ(h T ) C (9) TEV = 0 (10) As 1 and 9 the definition 2.2 and equations (8), (9) and (10) play a fundamental role in treating the control problem by static output feedback, mainly by eigenstructure assignment. 3. LYAPUNOV EQUATIONS AND OUTPUT FEEDBACK MATRIX Two following theorems are used in the study of the S-stabilization problem. Thus, the Theorem 3.1 relates the concept of strongly O.S. (C,A,E,B)- invariant subspaces to the existence of an output feedback u(t) =Ky(t), such that solves the S-stabilization problem in the closed-loop system. Theorem There exists an output feedback matrix K : Y U such that σ(e,a + BKC) C and the closed-loop system is regular, asymptotically stable and impulse free if and only if the following conditions are verified for some of the matrices (V R n v,h V R v v,w R m v ), (T R q v n,h T R q v q v,u R n v n v ) and for some scalar positive v n: AV EV H V = BW, σ(h V ) C (11) TA H T TE = UC, σ(h T ) C (12) TEV = 0 (13) Ker (CV ) Ker (W ) (14) Ker (B T ) Ker (U ) (15) where: rank (EV )=rank (TE)=q. Serra Negra, SP - ISSN

3 A quadratic characterization of strongly subspaces O.S. (C,A,E,B)-invariant is obtained by replacing conditions σ(h V ) C and σ(h T ) C by the two equivalent quadratic Lyapunov stability conditions 5 : σ(h T ) C Γ=Γ > 0 such that H T Γ+ΓH T = Q T, Q T = Q T > 0 (16) Theorem 3.2 4, 1 : There exist an output feedback matrix K : Y U, such that σ(e,a+bkc) C, if and only if the following conditions are verified for some positive scalar v n and for some pair of matrices V R n v and T R q v n, such that TEV =0, where rank (EV )=rank (TE)=q: (i) Q V = Q V > 0, Q V R v v, there exists matrices P = P 0, P R n n and Y R m n, such that : AP E + EPA + BY E + EY B = EV Q V V E (17) V PV > 0; TEPE T =0(18) Y = W Π V for some W Π R m v (19) (ii) Q T = Q T > 0, Q T R q v q v, there exist matrices S = S 0, S R n n and Z R n v, such that : A SE + E SA + C Z E + E ZC = E T Q T TE (20) (iii) TST > 0; V E SEV =0(21) Z = T U Γ for some U Γ R q v v (22) Ker CP Ker Y (23) Ker B S Ker Z (24) Remark 3.1 Moreover, another approach based on the Lyapunov equation in 10 can be used for obtained the existence of a stabilizing output feedback in the descriptor system. For guarantee the stabilization, the same equations are used with the following conditions Q = C C, and V, W obtained from the theorem 3.1 with Y=WV and T obtained from the theorem 3.1 such that satisfied the equations 12 and 13. This paper shows that based coupled Sylvester equations and the coupled Lyapunov like equations can also be used to obtain the sufficient conditions for the existence of a S-stabilizing output feedback. The Theorem is presented as follows: Theorem 3.3 : There exists an output feedback matrix K : Y U such that σ(e,a + BKC) C, if the sufficient following conditions are verified for some positive scalar v n and for some pair of matrices V R n v and T R q v n, that such TEV =0, where rank (EV )=rank (TE)=q: (i) Let Q = C C, Q R n n, there exist matrices Pc = Pc 0, Pc R n n and Y = WV R m n, such that : AP ce + EP ca = EC CE (25) Pd = V P cv > 0 ; TEPcE T =0 (26) Ker CP c Ker Y (27) Proof: Sufficiency: Consider that (E,A + BKC) is S-stable. It is known that there exists an unique symmetric nonnegative definite matrix P, such that (A + BKC)PE + EP(A + BKC) PBR 1 B P + ECQC E =0 (28) for real symmetric nonnegative definite matrix R. For F defined for we obtain F = KC (29) K (B PB + R)K = SS (30) After the substitution from (29) to (26), the equation (28) is obtained. From detectability of (E,A,C) the existence of L such that (E,A + LC) is guaranteed. Thus, also A+BKC C (KC) is detectable since A + LC =(A + BKC)+L BC (KC) Therefore, from (28), considering the previous arguments, the stability (E,A + BKC) is obtained. 4. ALGORITHMIC ASPECTS The theorem (3.2) is propose in a new numerically efficient solution algorithm to determine the output feedback matrix in descriptor systems. The resulted algebraic of theorems (3.1) and (3.2) are used for the construction of the subspaces strongly O.S. (C,A,E,B)-invariant leading to the calculation of output feedback matrix K The Syrmos-Lewis algorithm For the calculation of the output feedback that stabilizes the closed-loop system, when m + p > q. The eigenvalues in closed-loop arbitrary assignment near the set Λ=Λ T Λ V, where Λ T = {λ 1,...,λ q p } and Λ V = {λ q p+1,...,λ q } are symmetrical sets of pre-specified eigenvalues. The system (E,A,B,C) is considered stronlgy controllable and stronlgy detectable 1. Step 1: Choose a matrix H T R q p q p such that σ(h T ) = Λ T C and solve the Sylvester equation (12) to find the matrix T R q p n, such that ( ) TE rank = q C (31) Serra Negra, SP - ISSN

4 Step 2: Solve the Sylvester equation (11), for some matrix H V R p p such that σ(h V )=Λ V C carrying in consideration which the matrix V has verified the coupling condition (13) and to take it in account which rank (EV )=v (or Ker (TE)=Ker (E) Im(V ), where represents the direct sum) 2. Step 3: For the construction, the matrix V has verified that rank (CV )=p and a matrix K is computed by: K = W (CV ) 1 (32) Remark 4.1 The steps 1 and 2 are solving using standart technical for eigenstructure assignment. Consider the matrices H T R q p q p and H V R p p : Step 1: Find t j C n and u j C p, such that : A t j u λj E j =0 j =1,...,q p (33) C As matrix lines T R (q p) n, denoted by T j, are formed from the vector t j, as follows: if λ j R, then T j = t j ; if λ j C, considered λ j+1 = λ j and { T j = Re (t j ) T j+1 = Imag (t j ). Step 2 : Determine v i C n and w i C n, such that : A λie B vi =0 i = q p +1,...,q (34) TE 0 w i The matrices V and W are constructed only with real elements, and it is used to calculate K. In particular: if λ i C, consider λ i+1 = λ i and { Vi = Re (v i ), V i+1 = Imag (v i ) W i = Re (w i ), W i+1 = Imag (w i ) where V i and W i denoted the columns the matrices V and W, respectively. The degrees of freedom in the choice of V satisfying the coupling condition TEV =0, also can be used to ensure the obtain of K such that KCV = W. q 2 The condition (31) guaranteed, in particular, that rank (TE)=, 4.2. Approach using the Lyapunov Algorithm From the previous techniques in section 4.1, the step 1 under the condition which the system is strongly detectable is always possible to construct a matrix T that verified the condition (31), using standard technical for eigenstructure assignment. Thus, the q p finite poles is obtained. The step 2 is solving the equations of Lemma 3.3, where Pd = V P cv > 0 and T EP ce T =0. Thus, in the step 3 the other finite poles are obtained with KCPc = Y. Example 4.1 : Consider the following data 6 : E = ; A = B = ; C = The corresponding descriptor system with finite poles is given by: σ(e, A)={0.0, 0.0, 1.0}. The system (C,E,A,B) is both stronlgy controllable and observable. In a first step, the eigenstructure assignment is used to find the eigenvalues that are given for : Λ T = { 1} Λ V = { 3.5}. Step 1 : For λ 1 = 1, determined T that verified (33) and such that (31) is also verified and (A,B,T,E) have not invariant zeros : T = Step 2 : For λ 2 = 3.5, determined V and W using (34) : V = ; Step 3 : W = ; Serra Negra, SP - ISSN

5 Determined K such that KCV = W : K = ; A + BKC = The corresponding closed-loop system (E,A + BKC) has the desired generalized eigenvalues.. Example 4.2 : Consider the result obtained in the previous example. For the construction of the subspace V = Im V is strongly O.S. (C,A,E,B)-invariant in relation the system (1), (2) used in example 4.1. In this case, are used H V = 3.5, and Π=1, that implies in Q V = From the corresponding matrices V obtained : P = VV = Y = WV = and W, it is that verified a part (i) of Theorem 3.2. The corresponding S-stabilizing output feedback, K = The finite poles in closed-loop system are given by : σ(e,a + BKC) = 1.00 where the eigenvalue 1 corresponds to step 1. Example 4.3 : Consider the result obtained in the previous example (4.1). In the step 1, for λ 1 = 1, determined T that verified (33) and such that (31). In the step 2, the equations are solving of Theorem (3.3), such that: Pc = Pd = V P cv = The corresponding S-stabilizing output feedback, K = W (CV Pd) 1 ; ; K = The finite poles in closed-loop system are given by : σ(e,a + BKC) ={ , } where the eigenvalue 1 corresponds to step CONCLUDING REMARKS Consider the stabilization of control problem by output feedback in descriptor systems. Thus, a new approach for construction the output feedback controller was proposed, and the respective output feedback matrix was obtained through the solution of Sylvester equation. This paper show that based coupled Sylvester equations and the coupled Lyapunov like equations can also be used to obtain sufficient conditions for the existence of a S-stabilizing output feedback. Thus, a theorem was presented for the construction of the output feedback matrix. Some base results had been presented in this paper that basis the algorithms and theoretical results. The characterization of solution Lyapunov Equations in two steps still will be explored in the searche future to obtain poles regional assignment. Aiming some opens questions for future studies among may consider the possible extension of the outlined technique to treat less restrictive cases than the m + p>qcase. References 1 E. B. Castelan, "Estabilização de Sistemas Descritores por Realimentação de Saídas via Subespaços Invariantes," SBA, Sociedade Brasileira de Automática, Vol. 16 No. 4, pp , E. B. Castelan, J. C. Hennet and E. R. Ll. Villarreal, "Quadratic Characterization and Use of Output Stabilizable Subspaces," IEEE Trans. Automatic. Control, Vol. 48, No. 4, pp , E. B. Castelan and J. C. Hennet and E. R. Ll. Villarreal, "Output Feeedback Design by Coupled Lyapunov-Like Equations" (Paper No. 575) 15 th IFAC World Congress on Automatic Control, pp. 1-6, Barcelona, Pr., E. B. Castelan and E. R. Ll. Villarreal and Sophie Tarbouriech, "Quadratic Characterization and Use of Output Stabilizable Subspaces in Descriptor Systems," Proceedings 1st IFAC Symposium on System Structure, Prague, pp , Praga, C. T. Chen, "Linear System Theory and Design," Holt, Rinehart and Winston, D. L. Chu and H. C. Chan and D. W. C. Ho, Necessary and Suficient Conditions for the Output Feedback Regularization of Descriptor Systems, IEEE Trans. Automat. Contr., Vol. 44, No. 2, pp , L. Dai, "Singular Control System", Springer-Verlag, , G. R. Duan, Eigenstructure assignment in descriptor systems via output feedback: a new complete parametric approach, Int. J. Control, Vol.72, No. 4, pp , L. R. Fletcher, Eigenstructure Assingment by Output Feedback in Descriptor Systems, IEE Proceedings, Vol. 135, No. 4, pp , J. Y. Ishihara and M. H. Terra, "A New Lyapunov equation for Discretetime Descriptor Systems," American Control Conference, Denver Colorado, USA, Vol. 6 pp , June F. L. Lewis, A Survey of Linear Singular Systems, Circuits Systems Signal Process, Vol. 5, No. 1, pp.3-36, Serra Negra, SP - ISSN

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