Computation of the transfer function matrix and its Laurent expansion of generalized two-dimensional systems

An algorithm is developed for the computation of the transfer function matrix of a two-dimensional system, which is given in its generalized form. The algorithm is a recursion in terms of the original system matrix and does not require the inversion of a two-variable polynomial matrix. An algorithm for the evaluation of the Laurent expansion of the inverse of a two-variable polynomial matrix is also presented.     

Generalized Cayley-Hamilton theorem for polynomial matrices with arbitrary degree

We give an alternative, more convenient expression of the well known Cayley-Hamilton theorem when polynomial matrices of arbitrary degree are involved. Based on the results of a recently developed algorithm (Fragulis et al. 1991) for the computation of the inverse of a polynomial matrix, certain relationships among the coefficient matrices of the given polynomial matrix are obtained. We also propose two ways of finding the powers of a polynomial matrix: one in terms of its coefficient matrices and the other making use of the generalized Cayley-Hamilton theorem. These methods are of closed form and are easily implementable in a digital computer.     

Recursive formulation of the matrix Padé approximation in packed storage

The Extended Euclidean algorithm for matrix Padé approximants is applied to compute matrix Padé approximants when the coefficient matrices of the input matrix polynomial are triangular. The procedure given by Bjarne S. Anderson et al. for packing a triangular matrix in recursive packed storage is applied to pack a sequence of lower triangular matrices of a matrix polynomial in recursive packed storage. This recursive packed storage for a matrix polynomial is applied to compute matrix Padé approximants of the matrix polynomial using the Matrix Padé Extended Euclidean algorithm in packed form. The CPU time and memory comparison, in computing the matrix Padé approximants of a matrix polynomial, between the packed case and the non-packed case are described in detail.     

Necessary and sufficient conditions stability forn-input,n-output convolution feedback systems with a finite number of unstable poles

This paper considersn-input,n-output convolution feedback systems characterized byy = G astr eande = u - Fy, where the open-loop transfer functionhat{G}contains a finite number of unstable multiple poles andFis a constant nonsingular matrix. Theorem 1 gives necessary and sufficient conditions for stability. A basic device is the following: the principal part of the Laurent expansion ofhat{G}at the unstable poles is factored as a ratio of two right-coprime polynomial matrices. There are two necessary and sufficient conditions: the first is the usual infimum one, and the second is required to prevent the closed-loop transfer function from being unbounded in some small neighborhood of each open-loop unstable pole. The latter condition is given an interpretation in concepts of McMillan degree theory. The modification of the theorem for the discrete-time case is immediate.     

An algorithm for inverting rational matrices

We propose an algorithm for computing the inverses of rational matrices and in particular the inverses of polynomial matrices. The algorithm is based on minimal state space realizations of proper rational matrices and the matrix inverse lemma and is implemented as a MATLAB¹ function. Experiments show that the algorithm gives accurate results for typical rational matrices that arise in analysis and design of linear multivariable control systems. Illustrative examples are given.     

快速次优固定区间Wiener平滑器算法

为了克服带相关噪声控制系统的最优固定区间Kalman平滑算法要求较大计算负担的缺点,应用Kalman滤波方法,基于CARMA新息模型,由稳态最优Kalman平滑器导出了带相关噪声控制系统的最优固定区间Wiener递推状态平滑器,它带有系数阵指数衰减到零的高阶多项式矩阵.用截断系数矩阵近似为零的项的方法提出了相应的快速次优固定区间Wiener平滑算法.它显著地减少了计算负担,便于实时应用,还给出了截断误差公式和选择截断指标的公式.仿真例子说明了快速平滑算法的有效性.     

The division of polynomial matrices

A division of polynomial matrices is presented in this note. The polynomial matrices are written in the propers-power expansion forms. Then, the division can be accomplished by equating the coefficient matrices of the polynomial matrices. The algorithm is iterative and easy to handle.     

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.     

Rational matrix structure

We bring together in this paper, in a unified way, certain results developed by various people regarding the pole/zero structure of a rational matrix and the structure of the vector-space generated by its columns. Pole and zero structure, at finite and infinite arguments, is compactly described by using elementary ideas from the language of valuation theory. The concept of column-reducedness of a rational matrix at some argument is introduced, and shown to determine when its pole/zero structure is simply that of its columns taken separately. We describe a procedure that operates on the Laurent expansion of a given rational matrix at the argument of interest in order to transform the matrix to one that is column-reduced at this argument but has the same pole/zero structure. The occurrence of such a "structure-extraction" algorithm in various contexts in system theory is pointed out. Special properties of rational bases (for a rational vector space) that are column-reduced at all arguments are noted, somewhat extending what is already well-known for minimal polynomial bases for such a space.     

Stability of a matrix polynomial in continuous systems

Two sufficient conditions under which the roots of the determinant of a given (m times m) matrix polynomial ofnth order lie in the open left-half plane have been obtained. The first condition is given in terms of the positive definiteness of an (mn times mn) symmetric matrix, while the second condition is given in terms of the positive definiteness of an (m times m) matrix that is a function ofs, Res leq 0. These conditions are represented in terms of rational functions of the coefficient matrices of the given matrix polynomial. Therefore, the explicit computation of the determinant polynomial is not required.     

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.     

Infinite-frequency structure and a certain matrix Laurent expansion

A method for determining the Smith-McMillan form at infinity of a rational matrix is derived by considering the Laurent expansion at infinity of the matrix. This method is used to provide a new test for the absence of infinite zeros in a rational matrix and a new formula for calculating the highest degree among the largest minors of a polynomial matrix.     

Structured system models Part 3. An application to feedback compensator design

The controllability and observability indices are studied and applied to the feedback compensator design. The compensator design method uses polynomial matrices as system models. As the main result, a new algorithm is introduced for the construction of a first candidate for the feedback compensator. A new algorithm is also given for constructing a state-space model from polynomial matrix models. Such a realization is needed if there is originally only a polynomial matrix model for the system.     

Operational matrices of piecewise linear polynomial functions with application to linear time-varying systems

Application of the piecewise linear polynomial functions expansion is extended to linear time-varying systems. With the treatment of the product of two time functions, two types of operational matrix are developed. By applying these operational matrices, the dynamic equations are transformed into a set of algebraic equations. A recursive algorithm is derived and the system equations can be solved with very low dimensional matrix inversions. This represents a considerable saving of computer memory capacity and computing time.     

Column and diagonal‐reduction of polynomial matrices by orthogonal transformation

Many of the applications of polynomial matrices in real world systems require column‐ or diagonally‐reduced polynomial matrices. If a given polynomial matrix is not column‐ or diagonally‐reduced, Callier or Wolowich algorithms, which use unimodular transformations, can be applied for column‐ or diagonal‐reduction, respectively, as a pre‐processing step in the applications. However, Callier and Wolowich algorithms may be unstable, from a numerical viewpoint, because they use elementary column and row operations. The purpose of this paper is to present sufficient conditions for existence of a constant orthogonal transformation of the given polynomial matrix so that it becomes column‐ or diagonally‐reduced. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Parameter identification of time-delay systems via exponential

A delay matrix D is derived and used along with the exponential Fourier operational matrix of integration in a new algorithm for parameter identification of LTI delayed systems. The main advantage of this method over similar algorithms is that Fast Fourier Transform (FFT) can be employed for determining expansion coefficients. Therefore, it reduces the computing time considerably. A second advantage is that the Fourier delay and integration matrices are simpler than their counterparts associated with other orthogonal functions. This further reduces compulations. An example is given which shows that the algorithm gives accurate parameter estimates.