Factorization Of Rational Matrices Research Articles

The H2 control problem: a general transfer-function solution

A general solution of the H2 control problem is presented for linear systems described by rational transfer matrices, not necessarily proper or stable. The control system is considered in the standard configuration, which includes the synthesis model of the plant and the controller. The H2 control problem consists of internally stabilizing the control system while minimizing the H2 norm of its transfer function. The notion of internal stability is based on bounded-input bounded-output stability and means that subsystems defined by any pair of input and output signals within the control system all are bounded-input bounded-output stable. In this manner, the optimal control system is devoid of impulsive as well as non-decaying exponential modes. The solution proceeds in three steps. First, the set of all controllers that internally stabilize the control system is parameterized. Then the subset of the stabilizing controllers that achieve a finite value of the H2 norm of the control system transfer matrix is described, also in parametric form. Finally, the optimal controller is obtained by selecting the parameter that minimizes the norm. The existence of each set of controllers described above is established in terms of the given data. There are plants that cannot be internally stabilized; there are plants that can be internally stabilized but no stabilizing controller renders the H2 norm finite; still there are plants and internally stabilizing controllers that achieve a finite value of the norm among which, however, no one actually minimizes the norm. The mathematical tools applied are doubly coprime, proper stable factorizations of rational matrices. Based on this description of the plant, two synthesis algorithms are derived: the primal and the dual one. The symmetries thus obtained are interesting and useful. The construction of the optimal controller requires two specific operations with proper stable rational matrices, inner--outer factorization and proper stable projection. The solution obtained is general in the sense that no assumptions on the plant are made other than those requiring the outer factors to be square, which can always be achieved.

To Solving Multiparameter Problems of Algebra. 4. The AB-Algorithm and its Applications

The paper continues the investigation of methods for factorizing q-parameter polynomial matrices and considers their applications to solving multiparameter problems of algebra. An extension of the AB-algorithm, suggested earlier as a method for solving spectral problems for matrix pencils of the form A - λB, to the case of q-parameter (q ≥ 1) polynomial matrices of full rank is proposed. In accordance with the AB-algorithm, a finite sequence of q-parameter polynomial matrices such that every subsequent matrix provides a basis of the null-space of polynomial solutions of its transposed predecessor is constructed. A certain rule for selecting specific basis matrices is described. Applications of the AB-algorithm to computing complete polynomials of a q-parameter polynomial matrix and exhausting them from the regular spectrum of the matrix, to constructing irreducible factorizations of rational matrices satisfying certain assumptions, and to computing “free” bases of the null-spaces of polynomial solutions of an arbitrary q-parameter polynomial matrix are considered. Bibliography: 7 titles.

Computation of the (J,J')-Lossless Factorization for General Rational Matrices

J,J'-lossless factorization plays a central role in $H_\infty$-control because it gives a simple and unified framework of $H_\infty$-control from the viewpoint of classical network theory, and it includes the well-known inner--outer factorization of rational matrices, Wiener--Hopf factorization, and spectral factorization of positive rational matrices as special cases. However, up to now, there is still a lack of numerically reliable methods for this important factorization problem in a general setting. In this paper, we present necessary and sufficient solvability conditions and develop a numerically reliable algorithm based on a generalized eigenvalue approach for the J,J'-lossless factorization of general rational matrices.

A non‐standard J‐spectral factorization of rational matrices via zero compensator approach

AbstractThis paper deals with a non‐standard J‐spectral factorization problem (NS‐J‐SFP) that arises in he non‐standard H∞ control problem for a descriptor system. We derive the solvability conditions for the NS‐J‐SFP by adapting the zero compensation technique due to Copeland and Safonov (Internat. J. Robust Nonlinear Control 1992; 2:139) to the descriptor system. It is also shown that the solvability conditions of the NS‐J‐SFP are equivalent to the existence of a semi‐stabilizing solution of a generalized algebraic Riccati equation. Copyright © 2002 John Wiley & Sons, Ltd.

Factorization of Rational Matrices in Sampled-Data Systems Design

Special features of factorization procedures for two types of rational matrices encountered in i>H2 and i>H∞ optimization problems for sampled-data systems are considered. A rigorous estimate of the characteristic polynomial of the i>H2-optimal closed-loop system is given.

Computation ofJ-inner-outer factorizations of rational matrices

A new numerically reliable computational approach is proposed to compute the factorization of a rational transfer function matrix G as a product of a J-lossless factor with a stable, minimum-phase factor. In contrast to existing computationally involved ‘one-shot’ methods which require the solution of Riccati or generalized Riccati equations, the new approach relies on an efficient recursive poles and zeros conjugation technique. The resulting factors always have minimal order descriptor representations. The proposed approach is completely general being applicable whenever G is proper/strictly proper or not, or of full column/row rank or not. It is also applicable to solve the canonical J-spectral factorization problem. © 1998 John Wiley & Sons, Ltd.

Computation of J-inner–outer factorizations of rational matrices

Computation of J-inner–outer factorizations of rational matrices

On irreducible factorizations of rational matrices and their applications

This paper is an extension of our studies of the computational aspects of spectral problems for rational matrices pursued in previous papers. Methods of solution of spectral problems for both one-parameter and two-parameter matrices are considered. Ways of constructing irreducible factorizations (including minimal factorizations with respect to the degree and size of multipliers) are suggested. These methods allow us to reduce the spectral problems for rational matrices to the same problems for polynomial matrices. A relation is established between the irreducible factorization of a one-parameter rational matrix and its irreducible realization used in system theory. These results are extended to the case of two-parameter rational matrices. Bibliography: 15 titles.

Optimal solutions to a two-block ℋ∞ problem

In this paper a generator of all optimal solutions to two-block ℋ∞ problems is derived. The derivation is in frequency domain and hinges on geometrical arguments. The results involve noncanonical Wiener-Hopf factorization of rational matrices.

On factorizations of rational matrices

It is shown that any complex rational matrix G(s) can be factorized over the complex field as a product of n complex rational matrices each of which has McMillan degree 1, where n is the McMillan degree of G(s). When confined to the real field, G(s) with real coefficient can be factorized as a product of a sequence of real national matrices with McMillan degree 1 or 2, where the McMillan degrees of the factors add up to n. >

On the factorization of rational matrices depending on a parameter

In 1961 Youla published his paper “On the factorization of rational matrices”. He proved that any proper rational para-Hermitian matrix that is positive definite on the imaginary axis can be factorized as the product of a proper rational matrix, stable with respect to the closed right half plane, and its adjoint. In this paper it is proved that for any positive definite, nonstrictly proper matrix this factorization can be given depending analytically on the original matrix, in a sufficiently small neighbourhood. This result is applied to the problem of metrizing the space of transfer matrices of linear systems, in accordance with Vidyasagar's graph topology.

Infinite frequency interpretations of minimal bases

The relevance of minimal bases in infinite frequency considerations is shown to lie more in the aspect of factorizations of rational matrices than in their characterization of the absence of zeros (finite and infinite). Accordingly the paper distinguishes between relatively prime and minimal factorizations and establishes an important structural property of the latter. This property is demonstrated to be extremely useful in a study of the effect of constant output feedback on the infinite poles and zeros of a transfer function matrix.

Correction to 'On the Factorization of Rational Matrices'

Correction to 'On the Factorization of Rational Matrices'

On the factorization of rational matrices

Many problems in electrical engineering, such as the synthesis of linear n ports and the detection and filtration of multivariable systems corrupted by stationary additive noise, depend for their successful solution upon the factorization of a matrix-valued function of a complex variable <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</tex> . This paper presents several algorithms for affecting such decompositions for the class of rational matrices <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G(p)</tex> , i.e., matrices whose entries are ratios of polynomials in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</tex> . The methods employed are elementary in nature and center around the Smith canonic form of a polynomial matrix. Several nontrivial examples are worked out in detail to illustrate the theory.