On Positive Stable Realization for Continuous Linear Singular Systems

Transcription

1 Int. Journal of Math. Analysis, Vol. 8, 2014, no. 8, HIKARI Ltd, On Positive Stable Realization for Continuous Linear Singular Systems Muhafzan Department of Mathematics, Faculty of Mathematics and Natural Science Andalas University, Kampus Unand Limau Manis Padang, Indonesia, Copyright c 2014 Muhafzan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, a study on the positive stable realization problem for continuous linear singular systems is established. A necessary and sufficient condition for existence of a positive stable realization of an improper transfer matrix is proposed. It is showed that the existence of a positive stable realization of an improper transfer matrix is equivalent to existence of positive stable realization of a strictly proper transfer matrix which corresponding to it. Mathematics Subject Classification: 93C05, 93D15, 93D20 Keywords: Positive stable realization, singular system, transfer matrix 1 Introduction The realization of a transfer function is a classical problem in system theory. Such a problem has a complete solution in the case of standard linear systems. However, when constraints on the realization, i.e. on positivity, are imposed, the problem becomes much more complicated. An overview on the positive realization problem for standard linear systems is given in [1], [3]. Currently, Kaczorek [5] propose a procedure to compute the positive stable realization for the case of discrete time, whereas the stability of positive standard linear systems is discussed in [7] for the case of continuous time.

2 396 Muhafzan The positive stable realization problem of the more general linear systems, i.e. singular systems, is also already discussed in [6] for the case of discrete time and in [8] for the case of continuous time without a stability condition. Unfortunately, the positive stable realization for a transfer matrix need not always exist. Therefore, the criteria for existence a positive stable realization of a transfer matrix becomes important to be established. In this paper, we discuss a necessary and sufficient condition for existence of positive stable realization for continuous linear singular system. The following notation will be used: R - the set of real numbers, R n m - the set of n m real matrices, R+ n m - the set of n m real matrices with nonnegative entries, R n m - the set of n m real matrices with nonpositive entries, R n = R n 1, M n - the set of n n Metzler matrices, I n - the n n identity matrix and O - the zero matrix. 2 Background Material and Problem Statement Consider the singular continuous linear system Eẇt) =Kwt)+Lut), w0) = w 0 vt) =Mwt), 1) where wt) R n, ut) R m, vt) R p denote state, control and output vector, respectively, and E,K R n n,l R n m,m R p n. If E is nonsingular, then the system 1) is reduced in to a standard system ẇt) =Awt)+But), w0) = w 0 vt) =Mwt), 2) where A = E 1 K and B = E 1 L. In what follows we give some useful definitions and theorems. Definition 1 [9] The standard system 2) is called positive if for any initial state w0) = w 0 R n + and all control ut) Rm + with t 0, then wt) Rn + and vt) R p +. Furthermore, it is called stable if lim wt) =0 for all w 0 R n t +. The standard system 2) is called stable positive if it is positive and stable. The criterion for positive stable of the standard system 2) has been reported by some authors, see [1], [4], [9]. Theorem 1 [4, 9] The standard system 2) is positive stable if and only if A M ns, B R+ n m, and M R p n +, 3) where M ns denotes the set of n n stable Metzler matrices.

3 On positive stable realization for continuous linear singular systems 397 The transfer matrix of the standard system 2) is given by Gs) =M si n A) 1 B R p m s). 4) The transfer matrix Gs) is called proper if and only if lim Gs) s =K Rp m, 5) and it is called strictly proper if K =O. Otherwise, the transfer matrix is called improper. Definition 2 [4] Matrices A R n n,b R n m,m R p n are called a positive stable realization of transfer matrix Gs) if they satisfy 4) and the system standard 2) is stable positive. In [4] is stated that the matrices A, B, M are a positive stable realization of the transfer matrix Gs) if and only if 3) holds. It is now assumed that the matrix E is singular and there exists a complex number s such that det se K) 0. 6) Definition 3 [9] The singular system is called positive if for any admissible initial state w0) = w 0 R n + and all control ut) Rm + such that uj) t) = d j ut) R m dt j + with t 0, j =1, 2,...,q, then wt) R n + and vt) R p +. It is called stable if lim wt) =0 for all admissible w 0 R n +. t The transfer matrix of the descriptor system 1) is given by T s) =M se K) 1 L R p m s). 7) Definition 4 The matrices E,K R n n,l R n m,m R p n are called a positive realization of transfer matrix T s) if they satisfy 7) and the singular system 1) is positive. Moreover, these matrices E, K, L, M are called a positive stable realization of transfer matrix T s) if they are a positive realization and the singular system 1) is stable. If the nilpotency index q of the matrix E is greather than or equal to 1, then the transfer matrix 7) is improper and can be always written as the sum of strictly proper transfer matrix T sp s) and the polynomial transfer matrix T pl s) [6], namely, T s) =T sp s)+t pl s). 8)

4 398 Muhafzan It is well known that under the transformation [ ] y1 t) wt) =P y 2 t) 9) for a nonsingular matrix P R n n, with y 1 t) R n 1, y 2 t) R n n 1 and the assumption 6), the system 1) can be always decomposed into the following system [2]: [ ] [ ] [ In1 O K1 O L1 O N ][ ẏ1 t) ẏ 2 t) = O vt) = [ M 1 I n n1 ][ y1 t) y 2 t) M 2 ] [ y 1 t) y 2 t) ] + L 2 ] ut) 10) where K 1 R n 1 n 1,L 1 R n1 m,l 2 R n n1) m,m 1 R p n 1,M 2 R p n n1), and N is a nilpotent matrix. The positive stable realization problem for the singular system can be stated as follows. Given an improper transfer matrix T s) R p m s), find the matrices Ẽ, K R n n, L R n m, M R p n, with detẽ) =0, such that and the system T s) = M sẽ K) 1 L, Ẽẇt) = Kwt)+ Lut) vt) = Mwt) 11) is positive and stable. 3 Result The main result of this paper is given in the following theorem. Theorem 2 If N, L2,I n n1, M 2 be a realization of polynomial transfer matrix T pl s) R p m s), where L 2 R n n1) m, M 2 R p n n 1) n n +, N R 1 ) n n 1 ) +, with N q 1 O, N q = O, then there exists a positive stable realization of the improper transfer matrix T s) R p m s) of the form [ ] [ ] [ ] In1 O Ẽ = O N, K K1 O =, O L L1 =, L M = [ ] M1 M2 12) 2 I n n1 if and only if K1, L 1, M 1 be a positive stable realization of strictly proper transfer matrix T sp s) R p m s).

5 On positive stable realization for continuous linear singular systems 399 Proof. Suppose that there exists a positive stable realization of the improper transfer matrix T s) R p m s) of the form 12), then the system 11) is positive and stable, or equivalently the both systems ẇ 1 t) = K 1 w 1 t)+ L 1 ut) v 1 t) = M 1 w 1 t) 13) and Nẇ 2 t) =w 2 t)+ L 2 ut) v 2 t) = M 14) 2 w 2 t), [ ] w1 t) where wt) =, are positive and stable. In accordance to Theorem 1 w 2 t) we have K 1 M n1s, L 1 R n 1 m +, M 1 R p n 1 +. Therefore, these matrices are a positive stable realization of strictly proper transfer matrix T sp s) R p m s). Otherwise, let K 1, L 1, M 1 be a positive stable realization of strictly proper transfer matrix T sp s) R p m s), then the system 13) is stable positive. Furthermore, since N, L 2,I n n1, M 2 be a realization of polynomial transfer matrix T pl s), then T pl s) = M 2 s N In n1 ) 1 L2. We will show that the matrices Ẽ, K, L, M are a positive stable realization of the improper transfer matrix T s). Observe that T sp s)+t pl s) = M 1 sin1 K ) 1 1 L1 + M ) 1 2 s N In2 L2 = [ ] [ si n1 K ] 1 [ ] 1 O L1 M1 M2 O s N I n n1 L 2 = [ ] [ ] [ ]) [ ] In1 O K1 M1 M2 s O N O L1 O I n n1 L 2 = M K) 1 sẽ L = T s). This shows that Ẽ, K, L, M be a realization of transfer matrix T s). The solution of 14) is given by q 1 w 2 t) = j=1 N j L2 u j) t). 15) It is obvious that w 2 t) R n n 1 + due to N j L2 R n n 1) m, and w 2 t) 0 when t. Therefore Ẽ, K, L, M is a positive stable realization for T s).

6 400 Muhafzan 4 Conclusion A criterion for existence of a positive stable realization of an improper transfer matrix for continuous linear singular systems has been established. It is showed that the existence of a positive stable realization of an improper transfer matrix is equivalent to existence of positive stable realization of a strictly proper transfer matrix that corresponding to it. ACKNOWLEDGEMENTS. This work was supported by Institute for Research and Community Development, Andalas University, Indonesia throught Grant No. 003/H.16/PL/HB-MT/III/2010. References [1] L. Benvenuti and L. Farina, A Tutorial on the Positive Realization Problem, IEEE Trans. on Automatic Control, ), 5: [2] G. R. Duan, Analysis and Design of Descriptor Linear Systems, Springer, New York, [3] L. Farina, Necessary Condition for Positive Realizability of Continuous Time Linear Systems, Systems and Control Letters, ), [4] T. Kaczorek, Computation of Positive Stable Realization for Linear Continuous Time Systems. Bull. Pol. Acad. Sci. Techn., ), 3: [5] T. Kaczorek, Positive Stable Realization of Discrete Time Linear Systems. Bull. Pol. Acad. Sci. Techn., ), 3: [6] T. Kaczorek, Positive Realization for Descriptor Discrete Time Linear Systems, Acta Mechanica et Automatica, ), 2: [7] Muhafzan and I. Stephane, On Stabilization of Positive Linear Systems, Applied Mathematical Sciences, ), 37: [8] Muhafzan, Existence of a Positive Realization for Generalized Continuous Linear State Space Systems, Int. Journal of Math. Analysis, ), 56: [9] E. Virnik, Stability Analysis of Positive Descriptor Systems, Linear Algebra and Its Applications, ), Received: February 10, 2014