Pole assignment in the regular row-by-row decoupling problem☆
Introduction
In this paper we use the following notation. () is the set of matrices with real (complex) entries, () is the set of -dimensional real (complex) vectors. () is the subset of (), in which every matrix has rank . denotes the identity matrix of order the by matrix of all zero entries. For a matrix and are the transpose, conjugate transpose, rank, spectrum and spectral radius of , respectively. and denote the Moore–Penrose inverse and the -inverse of the matrix , respectively. denotes the real part of the eigenvalues of the matrix . denotes the generic rank of a matrix function , i.e. .
Consider the time-invariant linear system where , the related closed-loop transfer matrix function is
For the control system (1), we associate with the following feedback control rule where is a new input vector, , the related closed-loop transfer matrix function is
In this paper, we assume that the system is right invertible, that is, . Especially, for the regular case , this system is invertible. Furthermore, without loss of generality, we assume that has full column rank. In this article we consider the case that all matrices are complex.
The row-by-row decoupling problem (RRDP) was first proposed by Morgan in Morgan (1964). In frequency domain, the RRDP is to determine necessary and sufficient conditions under which there exists a matrix pair , such that the closed-loop transfer matrix is nonsingular and diagonal. The row-by-row decoupling is usually required for ease of system operations, for example, in the process and chemical industries, see Smith (2001), Stefanovski (2001) and Wang (2003).
As one of the most challenging problems in linear system control theory, since the RRDP was risen, many experts have focused on the solutions of the problem, and have obtained many deep and systematical results, from which researchers have revealed the structure of the linear system.
For the regular RRDP , Falb and Wolovich (1967) first obtained a necessary and sufficient condition and characterized a set of decoupling state feedbacks, based on the nonsingularity of a matrix constructed from the system matrices. Then, Wonham and Morse (1970) derived more general conditions by geometric approach based on the concept of controllability subspace. Descusse and Dion (1982) obtained an equivalent solvability condition based on the infinite structure of the system. Wei, Wang, and Cheng (2010) obtained another equivalent solvability condition based on the canonical decomposition of the right invertible system.
For the non-regular RRDP , available conditions for the existence of solutions have been established in particular cases, restrictive assumptions are enforced to the systems, to the control law, or to the shifted systems, see, e.g., Descusse, Lafay, and Malabre (1985), Descusse, Lafay, and Malabre (1988), Herrera and Lafay (1993), Zagalak, Lafay, and Herrera (1993) and references cited therein.
On the other hand, during the recent years, Chu and Hung (2000), Chu and Hung (2006), Chu and Mehrmann (2001), Chu and Tan (2002a) and Zuniga, Ruiz-León, and Henrion (2003) proposed numerical reliable methods for various problems related to the system decoupling problems. Furthermore, Chu and Tan (2002a) and Garcia and Malabre (1994) also discussed the row-by-row decoupling problem with stability (RRDPS).
In this article we are concerned with pole assignment in the regular RRDP. For the time-invariant linear system (1) and any given poles , pole assignment problem is to determine the state feedback matrix in (3) such that the eigenvalues of the closed-loop control system satisfy This problem was briefly discussed in Morse and Wonham (1971). Then, Koussiouris (1980) developed a method for assigning arbitrary all non-fixed poles in the case that the system is controllable, Ruiz-León, Orozco, and Begovich (2005) characterized the family of all attainable transfer function matrices for the decoupled closed-loop system under the conditions that the system is controllable, and the orders of both the invariant zeros of the system and the row invariant zeros of the system are the same.
In Wei et al. (2010), Wei et al. proposed a canonical decomposition of the right invertible system . By applying this canonical decomposition, in this paper, we will discuss the pole assignment in the regular RRDP. We first derive the explicit formulas of all solutions of the regular row-by-row decoupling problem, then derive a matrix and prove that, the system is controllable if and only if , where is a parameter obtained in the canonical decomposition. We then define so called decoupling controllable vectors, with different cases of and the dimension of the decoupling controllable vector subspace and characterize all attainable transfer function matrices for the decoupling and pole assignment in the regular RRDP.
The paper is organized as follows. In Section 2, we provide some preliminary results; in Section 3, we derive all solutions of the regular RRDP; in Section 4, we study the decoupling and pole assignment in the regular RRDP; in Section 5, we provide some numerical examples; and finally in Section 6, we conclude this paper with some remarks.
Section snippets
Preliminaries
In this section, we state some preliminary results for our further discussion. The first result is the canonical decomposition of the invertible system under the conditions of the regular RRDP. By combining the canonical decomposition of the right invertible system obtained in Wei et al. (2010) and the necessary and sufficient solvability conditions of the regular RRDP obtained in Wei et al. (2010), we have
Theorem 2.1 Suppose that is an invertible system with and the regularWei et al., 2010, Wei et al., 2010
All solutions of the regular RRDP
In this section, we will derive formulas of all solutions of the regular RRDP based on the results obtained in the previous section. We first consider properties related to the matrix when . Suppose that are different nonzero eigenvalues of , and has a Jordan canonical form for , in which are Jordan blocks for the eigenvalue of order , and
Pole assignment in the regular RRDP
In this section, we will discuss pole assignment in the regular RRDP. To simplify the analysis and without loss of generality, if , we assume . We first derive equivalent conditions for the invertible system being controllable when . Lemma 4.1 Under the conditions and notation of Theorem 2.1, if furthermore, , thenwhere
Proof In Theorem 3.1 of Shen and Wei (2012), set and for . □
Lemma 4.2 Under the
Numerical examples
The following example is presented in order to illustrate the results of this paper.
Example 5.1 Consider a system with here and , According to Theorem 2.1, the regular RRDP is solvable. We consider the following four different cases. Case I: We take then and the system is controllable. By some computations, we have and .
Concluding remarks
In this article, we have derived all solutions of the regular row-by-row decoupling problem by applying the canonical decomposition of the invertible system . Based on these formulas of the solutions, we have discussed the decoupling and pole assignment problem and characterized all attainable transfer function matrices in general cases. Finally, we have also provided numerical examples to illustrate our results.
So far we have a better understanding of properties and techniques of the
Acknowledgments
We are grateful to Professor Ian R. Petersen, Professor Delin Chu, and two anonymous referees for providing many useful comments and suggestions, which greatly improved the presentation of the article.
Musheng Wei, received the B.S. degree in Mathematics in 1982 from Nanjing University, Nanjing, China, received the M.S. and Ph.D. degrees in Applied Mathematics in 1984 and 1986, respectively, both from Brown University, RI, USA. Between 1986 and 1988, he was a post doctoral fellow at IMA in the University of Minnesota, the Ohio State University, and Michigan State University. Between 1988 and 2008, he obtained a position at the Department of Mathematics, East China Normal University, Shanghai,
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Musheng Wei, received the B.S. degree in Mathematics in 1982 from Nanjing University, Nanjing, China, received the M.S. and Ph.D. degrees in Applied Mathematics in 1984 and 1986, respectively, both from Brown University, RI, USA. Between 1986 and 1988, he was a post doctoral fellow at IMA in the University of Minnesota, the Ohio State University, and Michigan State University. Between 1988 and 2008, he obtained a position at the Department of Mathematics, East China Normal University, Shanghai, China, where he was a professor. From 2008 he moved to the College of Mathematics and Science, Shanghai Normal University, Shanghai, China as a professor. His research interests include numerical algebra, matrix analysis, scattering theory, signal processing, control theory, and scientific computing.
Dongmei Shen, received the B.S. degree in Computational Mathematics in 1999 from Nanjing Normal University, Jiangsu, China, and the M.S. and Ph.D. degrees in Numerical Mathematics in 2005 and 2012 from East China Normal University, Shanghai, China and Shanghai Normal University, Shanghai, China, respectively. Between 1999 and 2012, she joined the School of Science, Nantong University, Jiangsu, China, where she was a lecturer. From 2012, she moved to Department of Applied Mathematics, Shanghai Finance University, Shanghai, China as a lecturer. Her research interests include linear system and matrix analysis.
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This work was supported by NSFC under grant 11171226. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Delin Chu under the direction of Editor Ian R. Petersen.
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