PARAMETRIC SOLUTIONS TO THE GENERALIZED SYLVESTER MATRIX EQUATION AX ‐ XF = BY AND THE REGULATOR EQUATION AX ‐ XF = BY + R

In this paper, explicit parametric solutions to the generalized Sylvester matrix equation AX ‐ XF = BY and the regulator matrix equation AX ‐ XF = BY + R are proposed without any transformation and factorization. The proposed solutions are presented in terms of the Krylov matrix of matrix pair (A, B), a symmetric operator and the generalized observability matrix of matrix pair (Z, F) where Z is an arbitrary matrix and is used to denote the degree of freedom in the solution. Due to its elegant form and convenient computation, these proposed solutions will play an important role in solving and analyzing these types of equations in control systems theory.     

(J, J′)-lossless factorization based on conjugation

This paper is concerned with a derivation of the state-space form of the (J, J′)-lossless factorization which contains both the inner-outer factorization and the spectral factorization of positive matrices as special cases. Also, the (J, J′)-lossless factorization gives a unified framework of H^∞ control theory. We use the method of conjugation which makes the derivation much simpler than the previous literature, most of which used the technique of (J, J′)-spectral factorization. A necessary and sufficient condition is represented in terms of two Riccati equations one of which is degenerated.     

New Algorithms for Polynomial J-Spectral Factorization

In this paper new algorithms are developed for J-spectral factorization of polynomial matrices. These algorithms are based on the calculus of two-variable polynomial matrices and associated quadratic differential forms, and share the common feature that the problem is lifted from the original one-variable polynomial context to a two-variable polynomial context. The problem of polynomial J-spectral factorization is thus reduced to a problem of factoring a constant matrix obtained from the coefficient matrices of the polynomial matrix to be factored. In the second part of the paper, we specifically address the problem of computing polynomial J-spectral factors in the context of H _∞ control. For this, we propose an algorithm that uses the notion of a Pick matrix associated with a given two-variable polynomial matrix. Date received: January 1, 1998. Date revised: October 15, 1998.     

Necessary and sufficient conditions for existence of J-spectral factorization for para-Hermitian rational matrix functions

For a para-Hermitian rational matrix function G(λ)=J+C(λI_p−A)⁻¹B, where J=J∗ is invertible, and which has no poles and zeros on the imaginary line, we give necessary and sufficient conditions in terms of A,B,C and J for the existence of a J-spectral factorization, as well as an algorithm to obtain the J-spectral factor in case it exists.     

Stability radii of differential algebraic equations with structured perturbations

This paper deals with a formula for computing stability radii of a differential algebraic equation of the form AX^′(t)−BX(t)=0, where A,B are constant matrices. A computable formula for the complex stability radius is given and a key difference between the ordinary differential equation (ODEs for short) and the differential algebraic equation (DAEs for short) is pointed out. A special case where the real stability radius and the complex one are equal is considered.     

Parametric Solutions to the Generalized Discrete Yakubovich‐Transpose Matrix Equation

This paper is concerned with the complete parametric solutions to the generalized discrete Yakubovich‐transpose matrix equation X − AX^TB = CY. which is related with several types of matrix equations in control theory. One of the parametric solutions has a neat and elegant form in terms of the Krylov matrix, a block Hankel matrix and an observability matrix. In addition, the special case of the generalized discrete Yakubovich‐transpose matrix equation, which is called the Karm‐Yakubovich‐transpose matrix equation, is considered. The explicit solutions to the Karm‐Yakubovich‐transpose matrix equation are also presented by the so‐called generalized Leverrier algorithm. At the end of the paper, two examples are given to show the efficiency of the proposed algorithm.     

Solutions to the matrix equation

An explicit solution to the generalized Sylvester matrix equation AX−EXF=BY, with the matrix F being a companion matrix, is given. This solution is represented in terms of the R-controllability matrix of (E,A,B), generalized symmetric operator and a Hankel matrix. Moreover, several equivalent forms of this solution are presented. The obtained results may provide great convenience for many analysis and design problems. A numerical example is used to illustrate the effectiveness of the proposed approach.     

Discrete J-spectral factorization

This paper deals with the J-spectral factorization for general discrete rational matrices. A simple approach based on the Kalman filtering in Krein space is proposed. The main idea is to construct a stochastic state space filtering model in Krein space such that the spectral matrix of the output is equal to the rational matrix to be factorized. The spectral factor is then easily derived by using the generalized Kalman filtering in Krein space, which is similar to the H₂ spectral factorization. Our approach unifies the treatment of the H₂ spectral factorization and the J-spectral factorization. The applications of the derived results in H_∞ and risk-sensitive estimation for both nonsingular and singular systems are demonstrated.     

Further results on controllability and the linear equation Ax = b

The linear equation Ax = b, with A an n × n matrix and b an n × l matrix over a unique factorization domain R, is related to the controllability submodule U of the pair (A, b). It is shown that the above equation has a solution lying in V if, and only if, A is unimodular as an operator on U. An example is given of a matrix which is unimodular as an operator on the controllability submodule, but not as an operator on Rⁿ and sparseness of this occurrence is discussed.     

Frequency domain state feedback and estimation!

The primary purpose of this paper is to present a frequency domain compensation scheme which is completely analogous to the time domain (state-space) employment of a Luenberger state estimator for approximating a linear state variable feedback (l.s.v.f.) design. To accomplish this objective we (i) employ a ‘ factorization ’ of the rational transfer matrix, T(s), of a given system as the product R(s)P(s)^?1, where R(s) and R(s) are ‘ relatively right prime ’ polynomial matrices, (ii) present a frequency domain characterization of l.s.v.f. in terms of its effect on T(s) (actually on R(s) and P(s)), and (iii) extend the notion of the classical ‘ eliminant matrix ’ of two polynomials and its determinant, the ‘ resultant ’, to the matrix case. An example is employed to illustrate the various steps involved and some comparisons are then made to other related investigations.     

Discrete J-spectral factorization of possibly singular polynomial matrices

We develop two state-space algorithms for discrete-time J-spectral factorization of possibly singular, possibly non-column-reduced, and with possible zeros at the origin para-hermitian matrices, by assignment of matrices in optimal LQ return difference equality. The column degrees of the spectral factor equal the column degrees of the given matrix and the dimension of a discrete-time algebraic Riccati system equals the sum of the column degrees. Necessary and sufficient condition for the least-order property of our state-space realizations is column reduction of the given matrix.     

Nonlinear local J-lossless conjugation and factorization

A new definition of nonlinear local J-lossless factorization is introduced, which plays a crucial role in nonlinear H^∞ control theory. Sufficient (and in two special cases also necessary) conditions for the existence of this factorization and state-space formulae of the factor systems are given here. The main tools for the J-lossless factorization are the local right and left J-lossless conjugations, introduced in this paper. The former corresponds to the standard linear J-lossless conjugation, while the latter has no counterpart in the linear theory where it is completely dual to the former one and hence conceptually redundant. In the nonlinear case, however, this duality is much weaker and therefore the left J-lossless conjugation is essential for solving the local J-lossless factorization of unstable systems. This factorization requires a transformation of the given system to a special form and solving two independent Hamilton-Jacobi partial differential equations. Solutions of the two Hamilton-Jacobi equations have to satisfy a simple coupling condition.     

Fast QR factorization of low-rank changes of Vandermonde-like matrices

This paper is concerned with the solution of linear systems with coefficient matrices which are Vandermonde-like matrices modified by adding low-rank corrections. Hereafter we refer to these matrices as modified Vandermonde-like matrices. The solution of modified Vandermonde-like linear systems arises in approximation theory both when we use Remez algorithms for finding minimax approximations and when we consider least squares problems with constraints. Our approach relies on the computation of an inverse QR factorization. More specifically, we show that some classical orthogonalization schemes for m×n, m≥n, Vandermonde-like matrices can be extended to compute efficiently an inverse QR factorization of modified Vandermonde-like matrices. The resulting algorithm has a cost of O(mn) arithmetical operations. Moreover it requires O(m) storage since the matrices Q and R are not stored. Received: January 1997 / Accepted: November 1997     

A new solution to the generalized Sylvester matrix equation

This note deals with the problem of solving the generalized Sylvester matrix equation AV-EVF=BW, with F being an arbitrary matrix, and provides complete general parametric expressions for the matrices V and W satisfying this equation. The primary feature of this solution is that the matrix F does not need to be in any canonical form, and may be even unknown a priori. The results provide great convenience to the computation and analysis of the solutions to this class of equations, and can perform important functions in many analysis and design problems in control systems theory.     

A new factorization of the mass matrix for optimal serial and parallel calculation of multibody dynamics

This paper describes a new factorization of the inverse of the joint-space inertia matrix M. In this factorization, M ^?1 is directly obtained as the product of a set of sparse matrices wherein, for a serial chain, only the inversion of a block-tridiagonal matrix is needed. In other words, this factorization reduces the inversion of a dense matrix to that of a block-tridiagonal one. As a result, this factorization leads to both an optimal serial and an optimal parallel algorithm, that is, a serial algorithm with a complexity of O(N) and a parallel algorithm with a time complexity of O(logN) on a computer with O(N) processors. The novel feature of this algorithm is that it first calculates the interbody forces. Once these forces are known, the accelerations are easily calculated. We discuss the extension of the algorithm to the task of calculating the forward dynamics of a kinematic tree consisting of a single main chain plus any number of short side branches. We also show that this new factorization of M ^?1 leads to a new factorization of the operational-space inverse inertia, Λ ^?1, in the form of a product involving sparse matrices. We show that this factorization can be exploited for optimal serial and parallel computation of Λ ^?1, that is, a serial algorithm with a complexity of O(N) and a parallel algorithm with a time complexity of O(logN) on a computer with O(N) processors.     

A Toeplitz algorithm for polynomial J-spectral factorization

A block Toeplitz algorithm is proposed to perform the J-spectral factorization of a para-Hermitian polynomial matrix. The input matrix can be singular or indefinite, and it can have zeros along the imaginary axis. The key assumption is that the finite zeros of the input polynomial matrix are given as input data. The algorithm is based on numerically reliable operations only, namely computation of the null-spaces of related block Toeplitz matrices, polynomial matrix factor extraction and linear polynomial matrix equations solving.