Rational Matrix-valued Functions Research Articles

On efficient computation of rational {1,2}-pseudo-inverses for multivariable rational matrix-valued functions and their applications

In this article we consider the problem of the existence of rational 1,2-pseudo-inverses for rational multivariable matrix-valued functions. We prove that any rational multivariable matrix-valued function has rational 1,2-pseudo-inverse and we describe the set of all 1,2-pseudo-inverses of a given function, in terms of rational free parameters. It is shown that the 1,2-pseudo-inverse can have an effective description relative to the Moore–Penrose pseudo-inverse (that in general is not even rational), thus making it an alternative effective solution to the problem of online matrix inversion of large matrices that depend on real-time measured parameters. The rationality of the 1,2-pseudo-inverse is crucial when it should be realized in the physical real world (e.g. in inverse control), in contrast with realizations in computer algorithms (e.g. in image processing and communication systems) where the rationality is not necessary but the effective description of the pseudo-inverse becomes crucial. The results has applications in control systems, robust control, inverse control, mechanical systems, kinematic chains, kinematic networks, multi-degree-of-freedom systems, image processing, signal processing and communication systems.

Radial parts of Haar measures and probability distributions on the space of rational matrix-valued functions

We consider the space of conjugacy classes of the unitary group with respect to a smaller unitary group . It is known that to every element of we can canonically assign a rational matrix-valued function (the Livshits characteristic function) on the Riemann sphere. We find an explicit expression for the natural measure on obtained as the push-forward of the Haar measure of in terms of characteristic functions.

On Minimal Spectral Factors With Zeroes and Poles Lying on Prescribed Regions

In this paper, we consider a general discrete-time spectral factorization problem for rational matrix-valued functions. We build on a recent result establishing existence of a spectral factor whose zeroes and poles lie in any pair of prescribed regions of the complex plane featuring a geometry compatible with symplectic symmetry. In this general setting, uniqueness of the spectral factor is not guaranteed. It was, however, conjectured that if we further impose stochastic minimality, uniqueness can be recovered. The main result of his paper is a proof of this conjecture.

On maximal weight solutions of two matricial moment problems in the nondegenerate case

We first revisit the question of existence and uniqueness of maximal weight solutions of the truncated trigonometric matrix moment problem (Problem (TTM)) in the nondegenerate case. The starting point of the consideration is a specialized representation of the matrix weight assigned at each point on the unit circle for the solutions of (nondegenerate) Problem (TTM). Our approach is essentially self-contained and different from the existing methods. Moreover, based on this approach we also derive some important weight features of the so-called N-extremal solutions within the whole solution set of (nondegenerate) Problem (TTM) and the discrete structure of them. In the second part, these observations inspire us to look for the corresponding weight features of the solutions of a (nondegenerate) moment problem for rational matrix-valued functions (Problem (R)), where only the theory of orthogonal matrix polynomials on the unit circle is involved.

Null/Pole Interpolation Problem in the Class $$\mathcal {RI}$$ RI of Rational Functions Intertwining Solutions of Linear Differential Equations

We formulate and solve the null/pole interpolation problem for the class \({\varvec{\mathcal {RI}}}\) of rational matrix-valued functions intertwining solutions of linear ODEs with a spectral parameter; such functions appear naturally as transfer functions of overdetermined 2D systems invariant in one direction. The salient new feature, as compared to the usual null/pole interpolation problem for rational matrix-valued functions, is that the null and pole vectors are replaced by the null and pole solutions of the given ODEs for the values of the spectral parameter equal to the prescribed zeroes and poles. As a byproduct we obtain a realization theorem and a Hermitian (conservative) realization theorem.

Rational Cayley inner Herglotz–Agler functions: Positive-kernel decompositions and transfer-function realizations

The Bessmertnyĭ class consists of rational matrix-valued functions of d complex variables representable as the Schur complement of a block of a linear pencil A(z)=z1A1+⋯+zdAd whose coefficients Ak are positive semidefinite matrices. We show that it coincides with the subclass of rational functions in the Herglotz–Agler class over the right poly-halfplane which are homogeneous of degree one and which are Cayley inner. The latter means that such a function is holomorphic on the right poly-halfplane and takes skew-Hermitian matrix values on (iR)d, or equivalently, is the double Cayley transform (over the variables and over the matrix values) of an inner function on the unit polydisk. Using Agler–Knese's characterization of rational inner Schur–Agler functions on the polydisk, extended now to the matrix-valued case, and applying appropriate Cayley transformations, we obtain characterizations of matrix-valued rational Cayley inner Herglotz–Agler functions both in the setting of the polydisk and of the right poly-halfplane, in terms of transfer-function realizations and in terms of positive-kernel decompositions. In particular, we extend Bessmertnyĭ's representation to rational Cayley inner Herglotz–Agler functions on the right poly-halfplane, where a linear pencil A(z) is now in the form A(z)=A0+z1A1+⋯+zdAd with A0 skew-Hermitian and the other coefficients Ak positive semidefinite matrices.

Para-orthogonal rational matrix-valued functions on the unit circle

Para-orthogonal rational matrix-valued functions on the unit circle

On a class of extremal solutions of a moment problem for rational matrix-valued functions in the nondegenerate case I

The main theme of this paper is the discussion of a parametrized family of solutions of a finite moment problem for rational matrix-valued functions in the nondegenerate case. We will show that each member of this family is extremal in several directions concerning some point of the open unit disk. These investigations are inspired by the authors' paper [23], where a similar topic is studied in the context of the matricial Caratheodory problem. We will see that larger parts of the results presented there can be extended to the rational case studied here.The main theme of this paper is the discussion of a parametrized family of solutions of a finite moment problem for rational matrix-valued functions in the nondegenerate case. We will show that each member of this family is extremal in several directions concerning some point of the open unit disk. These investigations are inspired by the authors' paper [23], where a similar topic is studied in the context of the matricial Caratheodory problem. We will see that larger parts of the results presented there can be extended to the rational case studied here (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

On a class of extremal solutions of a moment problem for rational matrix-valued functions in the nondegenerate case II

The main theme of this paper is the discussion of a family of extremal solutions of a finite moment problem for rational matrix functions in the nondegenerate case. We will point out that each member of this family is extremal in several directions. Thereby, the investigations below continue the studies in Fritzsche et al. (in press) [1]. In doing so, an application of the theory of orthogonal rational matrix functions with respect to a nonnegative Hermitian matrix Borel measure on the unit circle is used to get some insights into the structure of the extremal solutions in question. In particular, we explain characterizations of these solutions in the whole solution set in terms of orthogonal rational matrix functions. We will also show that the associated Riesz–Herglotz transform of such a particular solution admits specific representations, where orthogonal rational matrix functions are involved.

More on a class of extremal solutions of a moment problem for rational matrix-valued functions in the nondegenerate case

The main subject of the paper is an in-depth analysis of Weyl matrix balls which are associated with a finite moment problem for rational matrix functions in the nondegenerate case. Thereby, the investigations tie in with preceding studies on a class of extremal solutions of the moment problem in question. We will point out that each member of this class is also extremal concerning the parameters of Weyl matrix balls. The considerations lead to characterizations of these particular solutions within the whole solution set of the problem. In doing so, an application of the theory of orthogonal rational matrix functions with respect to a nonnegative Hermitian matrix Borel measure on the unit circle is used to get that insight.

Parallel solution of large-scale algebraic Bernoulli equations with the matrix sign function method

We investigate the numerical solution of algebraic Bernoulli equations (ABE) via the Newton iteration for the matrix sign function. Bernoulli equations are nonlinear matrix equations arising in control and systems theory in the context of stabilisation of linear systems, coprime factorisation of rational matrix-valued functions, as well as model reduction. The algorithm proposed here is easily parallelisable and thus provides an efficient tool to solve large-scale problems. We report the parallel performance and scalability of our parallel implementations on a cluster of Intel Xeon processors.

Solution of a multiple Nevanlinna–Pick problem for Carathéodory functions via orthogonal rational functions in the matrix case

We consider an interpolation problem of Nevanlinna–Pick type for matrix-valued Carathéodory functions, where the values of the functions and its derivatives up to certain orders are given at finitely many points of the open unit disk. For the non-degenerate case, i.e., in the particular situation that a specific block matrix (which is formed by the given data in the problem) is positive Hermitian, the solution set of this problem is described in terms of orthogonal rational matrix-valued functions. These rational matrix functions play here a similar role as Szegő's orthogonal polynomials on the unit circle in the classical case of the trigonometric moment problem. In particular, we present and use a connection between Szegő and Schur parameters for orthogonal rational matrix-valued functions which in the primary situation of orthogonal polynomials was found by Geronimus. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Dual pairs of orthogonal systems of rational matrix-valued functions on the unit circle

SummaryWe analyse certain dual pairs of orthogonal systems of rational matrix-valued functions with poles not located on the unit circle. The rational matrix-valued functions belonging to such kind of pairs are recursively connected similar as the functions of the first and the second kind in the classical theory of orthogonal polynomials on the unit circle. We present some basic facts on these dual pairs and, in particular, we show that the rational matrix-valued functions in question admit specific integral representations.

On Hilbert modules of rational matrix-valued functions and related inverse problems

We study a moment problem for rational matrix-valued functions. In particular, we will characterize the case that there is a unique solution. A second main theme is to handle some inverse problems for rational matrix-valued functions. Hereby, we will essentially use several representation formulas for reproducing kernels of the underlying Hilbert modules of rational matrix-valued functions.

On a moment problem for rational matrix-valued functions

The papers deals with a finite moment problem for rational matrix-valued functions. We present a necessary and sufficient condition for the solvability of the problem. In the nondegenerate case we construct a particular solution which has interesting extremal properties.

On a theorem of Hermite and Hurwitz

The Hermite-Hurwitz theorem computes the degree, over R , of a real rational function ƒ in terms of the signature of an associated quadratic form—known today as the Hankel matrix of ƒ. This formula, which Hermite was led to by his work on the problem of representing integers as sums of squares, gave rise to striking applications in the theory of equations and in the stability theory of ordinary differential equations. In this paper, this theorem and various generalizations to the matrix-valued case are discussed and described in terms of signature formulae. These include its relation to stability theory and the matrix Hermite-Hurwitz theorem of Bitmead-Anderson as applied to questions of circuit synthesis. This also includes a global form of Hörmander's signature formula for the Maslov index of a rational loop in a Lagrangian Grassmannian, due to Byrnes and Duncan, and applications to the topology of spaces of rational matrix-valued functions, following the work of Brockett, Byrnes, and Duncan. This includes, in particular, a topological proof of the matrix Hermite-Hurwitz theorem.

Spectral Theory of the Linear-Quadratic Optimal Control Problem: Analytic Factorization of Rational Matrix-Valued Functions

The inversion of the Toeplitz operator $T_\Phi $, associated with the operator-valued function $\Phi $ defined on the unit circle, is well known to involve a special factorization, called analytic factorization, of the function $\Phi $. New results and algorithms concerning this factorization are presented, in the special case where $\Phi $ is rational and matrix-valued.