Solution to Sylvester equation associated to linear descriptor systems

Abstract

This paper presents the solution of the constrained Sylvester equation associated to linear descriptor systems. This problem has been recently studied in Castelan and da Silva [On the solution of a Sylvester equation appearing in descriptor systems control theory, Systems Control Lett. 54 (2005) 109–117], where sufficient conditions for the existence of the solution are given. In the present paper, a simple and direct method is developed to solve this problem. This method shows that the conditions given in Castelan and da Silva [On the solution of a Sylvester equation appearing in descriptor systems control theory, Systems Control Lett. 54 (2005) 109–117] are necessary and sufficient.

Introduction

Many problems in control and systems theory are solved by computing the solution of Sylvester equations. As it is well known, these equations have important applications in stability analysis, in observers design, in output regulation with internal stability, and in the eigenvalue assignment (see, e.g. [2], [4], [5], [6]). Sylvester equations associated with descriptor systems (or generalized Sylvester equations) have received wide attention in the literature, see [2], [4], [5], [6]. Recently, sufficient conditions for the existence of the solution to these equations under a rank constraint have been given in [2]. The present paper considers the problem studied in [2]. A new approach to solve these equations is developed, necessary and sufficient conditions are presented.

Consider the linear time-invariant multivariable system described by

$$E\dot{x}=\mathit{Ax}+\mathit{Bu}\text{,}$$

$$y=\mathit{Cx}\text{,}$$

where $x\in {\mathbb{R}}^n$ is the state vector, $y\in {\mathbb{R}}^p$ the output vector, and $u\in {\mathbb{R}}^m$ the input vector. The matrices $E\in {\mathbb{R}}^{n\times n}$, $A\in {\mathbb{R}}^{n\times n}$, $B\in {\mathbb{R}}^{n\times m}$, and $C\in {\mathbb{R}}^{p\times n}$ are known constant matrices, with $\mathrm{rank}(E)=q<n$, $\mathrm{rank}(B)=m$, and $\mathrm{rank}(C)=p<q$.

Consider Problem 1 studied in [2], which can be formulated as follows.

Let $\mathcal{D}$ be a region in the open left half complex plane, $\mathcal{D}\subseteq {\mathbb{C}}^{-}$, symmetric with respect to the real axis. The problem is to find matrices $T\in {\mathbb{R}}^{(q-p)\times n}$, $Z\in {\mathbb{R}}^{(q-p)\times n}$, and $H_T\in {\mathbb{R}}^{(q-p)\times (q-p)}$, such that

$$\mathit{TA}-H_T\mathit{TE}=-\mathit{ZC}\text{,}$$

with $\sigma (H_T)\subset \mathcal{D}$, under the rank constraint

$$\mathrm{rank}\left[\begin{array}{c}\hfill \mathit{TE}\hfill \\ \hfill \mathit{LA}\hfill \\ \hfill C\hfill \end{array}\right]=n\text{,}$$

where $\sigma (H_T)$ is the spectrum of $H_T$ and $L\in {\mathbb{R}}^{(n-q)\times n}$ is any full row rank matrix satisfying $\mathit{LE}=0$.

The Sylvester equation (2) has a close relation with many problems in linear control theory of descriptor systems, such as the eigenstructure assignment [6], [8], and the state observer design. The observer design problem can be formulated as follows [2].

For the descriptor system ((1a), (1b)), consider a reduced-order observer of the form

$$\dot{z}(t)=H_{T}z(t)+\mathit{TBu}(t)-\mathit{Zy}(t)\text{,}$$

$$\widehat{x}(t)=\mathit{Sz}(t)+\overline{N}\overline{y}+\mathit{Ny}(t)\text{,}$$

where $z\in {\mathbb{R}}^{(q-p)}$ is the state of the observer and $\overline{y}\in {\mathbb{R}}^{(n-q)}$ is a fictitious output. As shown in [2], if Problem 1 is solved for some matrices T, Z, and $H_T$ and if we compute the matrices S, $\overline{N}$, and N such that

$$[S\overline{N}N]\left[\begin{array}{c}\hfill \mathit{TE}\hfill \\ \hfill \mathit{LA}\hfill \\ \hfill C\hfill \end{array}\right]=I\text{,}$$

then, for observer ((4a), (4b)) we have:

• ${\lim }_{t\to \infty }(z(t)-\mathit{TEx}(t))=0$,

• for $\overline{y}(t)=-\mathit{LBu}(t)$, the estimated state $\widehat{x}(t)$ satisfies ${\lim }_{t\to \infty }(x(t)-\widehat{x}(t))=0$.

Remark 1

Notice that for $L=0$, Problem 1 is reduced to finding matrices $T\in {\mathbb{R}}^{(n-p)\times n}$, $Z\in {\mathbb{R}}^{(n-p)\times n}$, and $H_T\in {\mathbb{R}}^{(n-p)\times (n-p)}$, such that

$$\mathit{TA}-H_T\mathit{TE}=-\mathit{ZC}\text{,}$$

with $\sigma (H_T)\subset \mathcal{D}$, under the rank constraint

$$\mathrm{rank}\left[\begin{array}{c}\hfill \mathit{TE}\hfill \\ \hfill C\hfill \end{array}\right]=n\text{.}$$

These conditions are those required for the observer design with order $(n-p)$, see [4], [5], [8].