Brief paperSome new properties of the right invertible system in control theory☆
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 matrix with zero entries (if no confusion occurs, we will drop the subindex). For a matrix and are the transpose, conjugate transpose, rank, spectrum and spectral radius of , respectively.
Wei, Cheng, and Wang (2010) proposed a canonical decomposition of the right invertible system . From this decomposition, they studied many properties of the right invertible system. Also by applying this decomposition, in another article Wei, Wang, and Cheng (2010) studied the decoupling and associated pole assignment problem by state feedback.
When we study some important control problems, such as decoupling problem, state-feedback control and the design of a state observer, we need to discuss some basic structure properties including controllability, observability and stability with state feedback; see Chu and Hung (2006), Chu and Malabre (2002), Chu and Mehrmann (1999), Chu and Tan (2002) and Chu and Van Dooren (2006).
Consider the following time-invariant linear system where are the state, input, and output vectors of the system, respectively, are constant matrices.
Controllability and observability are important concepts in control theory, which are first presented by Kalman in Kalman (1960), then again studied in Gilbert (1963) and Silverman and Anderson (1968).
For the control system (1), we usually associate with the following feedback control law where is a new input vector, and . Then the corresponding system is
As we know that state feedback has much superiorities in improving system characteristics, it is often used in system stabilization and decoupling control. Whether there exists a feedback matrix such that the system (3) is stable is determined by the system itself. A real system must be stable and only a stable system can be used in engineering practice. In many articles (Chu and Hung, 2006, Chu and Malabre, 2002, Chu and Mehrmann, 1999, Chu and Tan, 2002, Chu and Van Dooren, 2006, Zheng, 1984), discussion of stability by state feedback takes an important part. So the judgment of whether a system is stable by state feedback is very important.
Another important application of state feedback is the row-by-row decoupling problem. This problem was first presented by Morgan in Morgan (1964). Since then, many scholars have been devoted to the research of this problem and have obtained a lot of useful results; see, e.g., Chu and Mehrmann (1999), Chu and Malabre (2002), Chu and Tan (2002), Descusse and Dion (1982), Descusse, Lafay, and Malabre (1988), Falb and Wolovich (1967), Herrera H and Lafay (1993), Ruiz-León, Orozco, and Begovich (2005), Wei et al. (2010), Wonham (1985) and Zagalak, Lafay, and Herrera H (1993). For the regular case, the solution is first established in Falb and Wolovich (1967), then in Wonham and Morse (1970), Descusse and Dion (1982) and Wonham (1985) by different approaches. Chu and Hung (2006) studied the row-by-row decoupling problem for the descriptor systems, in which the descriptor matrix may be singular. Chu and Hung (2006) obtained the necessary and sufficient conditions, under which there exists a matrix pair , such that the closed-loop transfer matrix is nonsingular and diagonal.
For the non-regular case, available conditions for the existence of solutions are established in particular cases, where restrictive assumptions are enforced to the systems, to the control law, or to the shifted systems; see Descusse et al. (1988) and Herrera H and Lafay (1993). As we know, up until now, there has not been any result to solve the non-regular problem in the general case.
In this paper, we assume that the system (1) is right invertible with . We intend to deduce equivalent conditions of controllability, stability by state feedback and observability of a right invertible system by applying the canonical decomposition of . We also derive sufficient conditions of solvability of the non-regular row-by-row decoupling problem, and deduce two equivalent conditions.
This paper is organized as follows. In Section 2, we provide some preliminary results for further discussion. In Section 3, we derive the controllability conditions of the right invertible system. In Section 4, we derive the stability conditions of the right invertible system by state feedback. In Section 5, we derive the observability conditions of the right invertible system. In Section 6, we derive sufficient solvability conditions for the non-regular row-by-row decoupling problem and equivalent conditions and finally, in Section 7, we conclude the article with some remarks.
Section snippets
Preliminaries
In this section, we provide some preliminary results for further discussion.
For the linear time-invariant system (1), we usually use the following criterion, which we call PBH rank criterion.
Proposition 2.1 The linear time-invariant system (1) is completely controllable if and only ifIn other words, and must be left coprime.
Proposition 2.2 A linear time-invariant system (1) is completely observable if and only ifi.e., and are right coprime.
Controllability conditions of the right invertible system
Notice from Proposition 2.1 that the system is controllable if and only if In this section, we will first derive an equivalent formula of the matrix pencil by applying the canonical decomposition in (8), (9), (10), (11), (12), (13), then deduce equivalent controllability conditions.
Theorem 3.1 Suppose that the right invertible system is given with the canonical decomposition in (8), (9), (10), (11), (12), (13). Then by multiplying a unimodular
Stability conditions of the right invertible system by state feedback
In this section, we will discuss the stability conditions of the right invertible system by state feedback. Notice that from Wonham (1985), there exists such that is stable, if and only if for all , By applying the equivalent formula of in Theorem 3.1, we have the following results.
Theorem 4.1 Suppose that the right invertible system is given with the canonical decomposition in(8), (9), (10), (11), (12), (13). If or and , then the
Observability conditions of the right invertible system
In this section, we will derive the observability conditions of the right invertible system. Notice from Proposition 2.2, that the system is observable if and only if We will deduce the equivalent formula of by applying the canonical decomposition in (8), (9), (10), (11), (12), (13) to obtain the following theorem.
Theorem 5.1 Suppose that the right invertible system is given with the canonical decomposition in (8), (9), (10), (11), (12), (13). Thenin which
Three equivalent sufficient solvability conditions for the non-regular row-by-row decoupling problem
In this section, we will discuss the non-regular row-by-row decoupling problem of the right invertible system with and .
Theorem 2.3 (Falb & Wolovich, 1967), Theorem 2.4 (Descusse & Dion, 1982), and Theorem 2.7 (Wei et al., 2010) all provide necessary and sufficient conditions for solvability of the regular row-by-row decoupling problem. We find that the conditions mentioned in Theorem 2.7 are also sufficient conditions of the non-regular row-by-row decoupling. Furthermore,
Concluding remarks
In this article, we have derived some equivalent conditions of controllability, stability with state feedback and observability of the right invertible system . By applying the canonical decomposition of the right invertible system, we have greatly reduced the matrix dimensions and our equivalent conditions are much simpler, which greatly reduced the difficulties of our further discussion of other control problems.
By applying the canonical decomposition of the right invertible system, we
Acknowledgments
We are grateful to Prof. Ian Peterson, Prof. Delin Chu and two anonymous referees for providing many useful comments and suggestions, which greatly improve the presentation of the article.
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 is a lecturer. From 2012, she moved to Department of Applied Mathematics, Shanghai
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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 is 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.
Musheng Wei, received the B.S. degree in Mathematics in 1982 from Nanjing University, Nanjing, China, and 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 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 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.
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This work was supported by NSFC under grant 11171226, and Shanghai Leading Academic Discipline Project under grant S30405. 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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