The extended J-spectral factorization for descriptor systems

Abstract

In this paper we study an extended J-spectral factorization problem, i.e., J-spectral factorization problem for descriptor systems which may not have full column normal rank and may exhibit poles and zeros on the extended imaginary axis. We present necessary and sufficient solvability conditions and develop a numerically reliable method for the underlying problem. Our method is implemented by using only orthogonal transformations and has an acceptable computational complexity.

Introduction

Throughout this paper, $J\in {\mathbf{R}}^{p\times p}$ and $J^{\prime }\in {\mathbf{R}}^{k\times k}$ are two given symmetric matrices; ${\mathbf{C}}_{-},{\mathbf{C}}_0$ and ${\mathbf{C}}_{0\mathrm{e}}$ denote open left complex half plane, imaginary axis and extended imaginary axis, respectively; the square pencil $-\mathit{sE}+A$ is stable if it is regular (i.e., $\det (-\mathit{sE}+A)\ne 0$ for some $s\in \mathbf{C}$) and all its finite generalized eigenvalues are on ${\mathbf{C}}_{-}$; ${\mathcal{R}}^{p\times k}(s)$ and ${\mathcal{RL}}_{\infty }^{p\times k}(s)$ denote the sets of $p\times k$ rational matrices and proper rational matrices without poles on ${\mathbf{C}}_0$, respectively; $\begin{array}{c}\hfill \end{array}$ means $G(s)=D+C(\mathit{sE}-A{)}^{-1}B$, and integer k is called its normal rank if ${\max }_{s\in \mathbf{C}}\mathrm{rank}\left[\genfrac{}{}{0em}{}{-\mathit{sE}+A}{C}\genfrac{}{}{0em}{}{B}{D}\right]=n+k$, here $E,A\in {\mathbf{R}}^{n\times n}$.

It is well-known that a large number of quite different types of factorizations of linear systems have been studied in the literature of system theory (Aliev and Larin, 1997, Anderson, 1967, Ball and Ran, 1987, Bart et al., 1986, Chen, 2000, Chen and Francis, 1989, Chen et al., 2004, Chu and Ho, 2005, Clements, 1993, Clements et al., 1997, Clements and Glover, 1989, Green et al., 1990, Green and Limebeer, 1995, Greenberg et al., 1987, Hara and Sugie, 1991, He and Chen, 2002, Katayama, 1996, Kawamoto, 1999, Kawamoto and Katayama, 2003, Kwakernaak and Sebek, 1994, Oara and Varga, 2000, Wangham and Mita, 1997, Xin and Kimura, 1994, Youla, 1961). Among them, the J-spectral factorization problem has played an important role in optimal Hankel-norm model reduction and $H_{\infty }$ optimization (Green and Limebeer, 1995, Ball and Ran, 1987), transport theory (Greenberg et al., 1987), and stochastic filtering (Lindquist & Picci, 1991).

Consider the descriptor system of the form

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

where $E,A\in {\mathbf{R}}^{n\times n}$, $B\in {\mathbf{R}}^{n\times m}$, $C\in {\mathbf{R}}^{p\times n}$, $D\in {\mathbf{R}}^{p\times m}$, and the pencil $-\mathit{sE}+A$ is regular.

Definition 1 (Kawamoto, 1999, Kawamoto and Katayama, 2003)

Given system (1) with all poles in ${\mathbf{C}}_{-}\cup {\mathbf{C}}_{0\mathrm{e}}$. The J-spectral factorization problem for the descriptor system (1) is solvable if $\begin{array}{c}\hfill \end{array}$ has a J-spectral factorization, i.e., there exists an invertible $\Xi (s)\in {\mathcal{R}}^{m\times m}(s)$ such that:

• $G^{\mathrm{T}}(-s)\mathit{JG}(s)={\Xi }^{\mathrm{T}}(-s)J^{\prime }\Xi (s)$.

• All poles and zeros of $\Xi (s)$ lie in ${\mathbf{C}}_{-}\cup {\mathbf{C}}_{0\mathrm{e}}$.

• $G(s){\Xi }^{-1}(s)\in {\mathcal{RL}}_{\infty }^{p\times m}(s)$.

When $E=I_n$ and system (1) has no poles and zeros on ${\mathbf{C}}_{0\mathrm{e}}$, the J-spectral factorization of system (1) reduces to the classical spectral factorization (Bart et al., 1986, Anderson, 1967, Youla, 1961). It is understood (Oara & Varga, 2000) that the difficulties in numerically solving the classical spectral factorization problem are related to the existence of poles and zeros on ${\mathbf{C}}_{0\mathrm{e}}$ and to the noninvertibility of $G^{\mathrm{T}}(-s)\mathit{JG}(s)$. If none of these elements are present, the computation of the classical spectral factorization problem reduces to solving a standard algebraic Riccati equation (Green et al., 1990), as stated in the following theorem.

Theorem 2 Green et al., 1990

Given a linear time-invariant system

$$\dot{x}=\mathit{Ax}+\mathit{Bu},y=\mathit{Cx}+\mathit{Du}\text{.}$$

Assume that A is stable, D is of full column rank and the pencil$\left[\genfrac{}{}{0em}{}{-\mathit{sI}+A}{C}\genfrac{}{}{0em}{}{B}{D}\right]$has no finite eigenvalues on${\mathbf{C}}_0$. Then the J-spectral factorization problem for system(2)is solvable if and only if there exists a nonsingular$D_0\in {\mathbf{R}}^{m\times m}$such that$D^{\mathrm{T}}\mathit{JD}=D_0^{\mathrm{T}}{J}^{\prime }{D}_0$and the algebraic Riccati equation

$$A^{\mathrm{T}}P+\mathit{PA}+C^{\mathrm{T}}\mathit{JC}-(\mathit{PB}+C^{\mathrm{T}}\mathit{JD})(D^{\mathrm{T}}\mathit{JD}{)}^{-1}(B^{\mathrm{T}}P+D^{\mathrm{T}}\mathit{JC})=0$$

has a solution P such that$A-B(D^{\mathrm{T}}\mathit{JD}{)}^{-1}(B^{\mathrm{T}}P+D^{\mathrm{T}}\mathit{JC})$is stable. Moreover, under these two conditions, a J-spectral factor$\Xi (s)$is given by

$$\begin{array}{c}\hfill \end{array}$$

Recently, the J-spectral factorization problem for descriptor system (1) with poles/zeros on ${\mathbf{C}}_{0\mathrm{e}}$ has been considered in Kawamoto (1999) and Kawamoto and Katayama (2003) by using the generalized algebraic Riccati equation approach and the zero compensator technique (Copeland & Safonov, 1992).

Theorem 3 (Kawamoto, 1999; Kawamoto & Katayama, 2003)

Given the descriptor system(1). Assume that:

(A1) All the finite generalized eigenvalues of the pencil$-\mathit{sE}+A$lie in${\mathbf{C}}_{-}\cup {\mathbf{C}}_{0\mathrm{e}}$;

(A2) $(E,A,B)$is finite dynamic stabilizable and impulse controllable, i.e.,

$$\mathrm{rank}[-\mathit{sE}+AB]=n,\forall s\in \mathbf{C}⧹{\mathbf{C}}_{-}\text{,}$$

$$\mathrm{rank}[EA\mathcal{N}(E)B]=n\text{,}$$

where$\mathcal{N}(E)$denotes a column orthogonal matrix whose columns span the null space of E. Then the J-spectral factorization problem for the descriptor system(1)is solvable if and only if the pencil$\left[\genfrac{}{}{0em}{}{-\mathit{sE}+A}{C}\genfrac{}{}{0em}{}{B}{D}\right]$is of full column rank for some$s\in \mathbf{C}$and the generalized algebraic Riccati equation

$$\left\{\begin{array}{c}{A}_a^{\mathrm{T}}{X}_a+X_a^{\mathrm{T}}{A}_a+Q_a-X_a^{\mathrm{T}}\left[\begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill (J^{\prime }{)}^{-1}\hfill \end{array}\right]X_a=0,\hfill \\ E_a^{\mathrm{T}}{X}_a=X_a^{\mathrm{T}}{E}_a\hfill \end{array}\right.$$

has a semi-stabilizing solution$X_a$ (for the related definition, we refer to (Kawamoto, 1999, Kawamoto and Katayama, 2003)), here

$$E_a=\left[\begin{array}{cc}\hfill E\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right],A_a=\left[\begin{array}{cc}\hfill A\hfill & \hfill B\hfill \\ \hfill 0\hfill & \hfill I\hfill \end{array}\right]\text{,}$$

$$Q_a=\left[\begin{array}{cc}\hfill C^{\mathrm{T}}\mathit{JC}\hfill & \hfill C^{\mathrm{T}}\mathit{JD}\hfill \\ \hfill D^{\mathrm{T}}\mathit{JC}\hfill & \hfill D^{\mathrm{T}}\mathit{JD}-J^{\prime }\hfill \end{array}\right],X_a=\begin{array}{c}\hfill \begin{array}{ccc}\hfill n\hfill & \hfill \hfill & \hfill m\hfill \end{array}\hfill \\ \hfill \left[\begin{array}{cc}\hfill X_{11}\hfill & \hfill X_{12}\hfill \\ \hfill X_{21}\hfill & \hfill X_{22}\hfill \end{array}\right]\begin{array}{c}\}n\hfill \\ \}m\hfill \end{array}\hfill \end{array}\text{.}$$

Furthermore, in this case, a J-spectral factor$\Xi (s)$is given by

$$\begin{array}{c}\hfill \end{array}$$

It is easy to know that the pencil $\left[\begin{array}{cc}\hfill -\mathit{sE}+A\hfill & \hfill B\hfill \\ \hfill C\hfill & \hfill D\hfill \end{array}\right]$ is of full column normal rank is a necessary condition for the J-spectral factorization problem of the descriptor system (1). So, Definition 1 excludes system (1) without full column normal rank. The condition (i) in Definition 1 implies that $G^{\mathrm{T}}(-s)\mathit{JG}(s)$ is nonsingular and has a constant inertia that is the same as the inertia of matrix $J^{\prime }$ for any $s\in {\mathbf{C}}_0$ that is not a pole of system (1). Obviously, if the descriptor system (1) is not of full column normal rank, then $G^{\mathrm{T}}(-s)\mathit{JG}(s)$ is always singular and its inertia always contains some 0 for any $s\in {\mathbf{C}}_0$ that is not a pole of system (1). Therefore, in order to include all descriptor systems without full column normal rank in the J-spectral factorization problem, Definition 1 has to be modified. This is the first motivation of the present work.

Assume that $G^{\mathrm{T}}(-s)\mathit{JG}(s)$ has a constant inertia $(\mu ,\tau ,\nu )$ for almost $s\in {\mathbf{C}}_0$, then a natural extension of the condition (i) in Definition 1 is that

$$G^{\mathrm{T}}(-s)\mathit{JG}(s)={\Xi }^{\mathrm{T}}(-s)\left[\begin{array}{ccc}\hfill I_{\mu }\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -I_{\nu }\hfill \end{array}\right]\Xi (s)\text{,}$$

equivalently,

$$G^{\mathrm{T}}(-s)\mathit{JG}(s)={\widehat{\Xi }}^{\mathrm{T}}(-s)J^{\prime }\widehat{\Xi }(s)\text{,}$$

$$\widehat{\Xi }(s)=\left[\begin{array}{ccc}\hfill I_{\mu }\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill I_{\nu }\hfill \end{array}\right]\Xi (s),J^{\prime }=\left[\begin{array}{cc}\hfill I_{\mu }\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -I_{\nu }\hfill \end{array}\right]\text{,}$$

where $\widehat{\Xi }(s)$ is of full row rank. This consideration leads us to generalize Definition 1.

Definition 4

${\Xi }^{(+)}(s)\in {\mathcal{R}}^{m\times k}(s)$ is called the Moore–Penrose inverse of $\Xi (s)\in {\mathcal{R}}^{k\times m}(s)$ if

$$\Xi (s){\Xi }^{(+)}(s)\Xi (s)=\Xi (s),{\Xi }^{(+)}(s)\Xi (s){\Xi }^{(+)}(s)={\Xi }^{(+)}(s)\text{,}$$

$$\Xi (s){\Xi }^{(+)}(s)=(\Xi (s){\Xi }^{(+)}(s){)}^{\sim },{\Xi }^{(+)}(s)\Xi (s)=({\Xi }^{(+)}(s)\Xi (s){)}^{\sim }\text{.}$$

Definition 5

Given the descriptor system (1) with all poles in ${\mathbf{C}}_{-}\cup {\mathbf{C}}_{0\mathrm{e}}$. The extended J-spectral factorization for system (1) is solvable if $\begin{array}{c}\hfill \end{array}$ has an extended J-spectral factorization, i.e., there exists a $\Xi (s)\in {\mathcal{R}}^{k\times m}(s)$ such that the conditions (i) and (ii) in Definition 1 hold, and furthermore $G(s){\Xi }^{(+)}(s)\in {\mathcal{RL}}_{\infty }^{p\times k}(s)$. Here k is the normal rank of $G(s)$.

Remark 1

In Definition 5, if $G(s)$ is of full column normal rank and $G^{\mathrm{T}}(-s)\mathit{JG}(s)$ is nonsingular, then $k=m$ and ${\Xi }^{(+)}(s)={\Xi }^{-1}(s)$. Consequently, Definition 5 reduces to Definition 1. Hence, the J-spectral factorization in Kawamoto (1999) and Kawamoto and Katayama (2003) is a special case of our extended J-spectral factorization.

The motivations for our interest to the extended J-spectral factorization also include:

• It is much more difficult to compute the semi-stabilizing solution $X_a$ of the generalized algebraic Riccati equation (5) than the stabilizing solution of the algebraic Riccati equation (3), and the overall semi-stabilizing solution $X_a$ of (5) is not necessary for the J-spectral factorization because only its blocks $X_{21}$ and $X_{22}$ contribute to the factorization (see Theorem 3). This indicates that the numerical computation of the J-spectral factorization for the descriptor system (1) with poles/zeros on ${\mathbf{C}}_{0\mathrm{e}}$ is certainly worthy of further investigation.

• It has been shown in Oara and Varga (2000) that the classical spectral factorization in the general setting relies essentially on the stabilizing solution of a standard algebraic Riccati equation. However, the approach in Oara and Varga (2000) does not work for our extended J-spectral factorization problem. The reasons for this include (i) the classical spectral factorization problem in Oara and Varga (2000) is always solvable, but the extended J-spectral factorization problem may not be solvable, the algebraic Riccati equations in Oara and Varga (2000) are always solvable provided some related matrix pairs are controllable (stabilizable) or observable (detectable), however, this is not true for their analogies with indefinite parameter matrices J and $J^{\prime }$; (ii) more important matter is that the algebraic Riccati equations in Oara and Varga (2000) may have no analogies with parameter matrices J and $J^{\prime }$. Hence, the approach in Oara and Varga (2000) cannot be used to solve our extended J-spectral factorization problem.

• A general approach has been proposed in Clements and Glover (1989), Clements et al. (1997) and Clements (1993) for the classical spectral factorization problem of a paraconjugate Hermitian rational matrix which is Hermitian positive semi-definite on ${\mathbf{C}}_0$. But it was pointed out in Oara and Varga (2000) that “the methods derived there, although some of them are algorithmic and very general, are far from numerically sound computational procedures”. In addition, this approach is also not applicable to our extended J-spectral factorization problem.

Motivated by the observations above, in this paper we study the computation of the extended J-spectral factorization which contains the J-spectral factorization in Kawamoto (1999) and Kawamoto and Katayama (2003) as a special case. We will present necessary and sufficient solvability conditions and develop a numerically reliable algorithm for the extended J-spectral factorization. The main features of our method include that (i) it only involves two lower-dimensional generalized eigenvalue problems and is implemented by only orthogonal transformations, thus, our method is numerically reliable; (ii) it avoids the semi-stabilizing solutions of any generalized algebraic Riccati equation of the form (5); (iii) our method has a computational complexity which is cubic in the state dimension of the associated system.