Some new properties of the right invertible system in control theory

Abstract

In this paper, we deduce equivalent conditions of controllability, stability and observability of a right invertible system $\{C,A,B\}$ by applying the canonical decomposition of $\{C,A,B\}$ obtained by Wei et al. [Wei, M., Cheng, X., & Wang, Q. (2010). A canonical decomposition of the right invertible system with applications. SIAM Journal on Matrix Analysis and Applications, 31(4), 1958–1981]. We also derive three equivalent sufficient solvability conditions of the non-regular row-by-row decoupling problem, which are generalization of necessary and sufficient conditions of the regular row-by-row decoupling problem.

Introduction

In this paper, we use the following notation. ${\mathbb{R}}^{m\times n}\left({\mathbb{C}}^{m\times n}\right)$ is the set of $m\times n$ matrices with real (complex) entries, ${\mathbb{R}}^m\left({\mathbb{C}}^m\right)$ is the set of $m$-dimensional real (complex) vectors. ${\mathbb{R}}_r^{m\times n}\left({\mathbb{C}}_r^{m\times n}\right)$ is the subset of ${\mathbb{R}}^{m\times n}\left({\mathbb{C}}^{m\times n}\right)$, in which every matrix has rank $r$. $I_k$ denotes the identity matrix of order $k,0_{l\times m}$ the $l\times m$ matrix with zero entries (if no confusion occurs, we will drop the subindex). For a matrix $A,A^T,A^H,\mathrm{rank}\left(A\right),\lambda \left(A\right)$ and $\rho \left(A\right)$ are the transpose, conjugate transpose, rank, spectrum and spectral radius of $A$, respectively.

Wei, Cheng, and Wang (2010) proposed a canonical decomposition of the right invertible system $\{C,A,B\}$. 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

$$\{\begin{array}{c}\dot{x}=Ax+Bu\hfill \\ y=Cx\hfill \end{array}\text{,}x\left(0\right)=x_0,t\ge 0$$

where $x\in {\mathbb{R}}^n,u\in {\mathbb{R}}^p,y\in {\mathbb{R}}^m$ are the state, input, and output vectors of the system, respectively, $A\in {\mathbb{R}}^{n\times n},B\in {\mathbb{R}}^{n\times p},C\in {\mathbb{R}}^{m\times n}$ 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

$$u=Fx+Gv\text{,}$$

where $v\in {\mathbb{R}}^m$ is a new input vector, $F\in {\mathbb{R}}^{p\times n}$ and $G\in {\mathbb{R}}^{p\times m}$. Then the corresponding system is

$$\dot{x}=(A+BF)x+BGv\text{.}$$

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 $F$ 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 $E$ may be singular. Chu and Hung (2006) obtained the necessary and sufficient conditions, under which there exists a matrix pair $\{F,H\}$, 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 $A\in {\mathbb{C}}^{n\times n},B\in {\mathbb{C}}_p^{n\times p},C\in {\mathbb{C}}_m^{m\times n}$. 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 $\{C,A,B\}$. 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.