Rosenbrock System Research Articles

A Rosenbrock framework for tangential interpolation of port-Hamiltonian descriptor systems

ABSTRACT We present a new structure-preserving model order reduction (MOR) framework for large-scale port-Hamiltonian descriptor systems (pH-DAEs). Our method exploits the structural properties of the Rosenbrock system matrix for this system class and utilizes condensed forms which often arise in applications and reveal the solution behaviour of a system. Provided that the original system has such a form, our method produces reduced-order models (ROMs) of minimal dimension, which tangentially interpolate the original model’s transfer function and are guaranteed to be again in pH-DAE form. This allows the ROM to be safely coupled with other dynamical systems when modelling large system networks, which is useful, for instance, in electric circuit simulation.

Linearizations of matrix polynomials viewed as Rosenbrock's system matrices

A well known method to solve the Polynomial Eigenvalue Problem (PEP) is via linearization. That is, transforming the PEP into a generalized linear eigenvalue problem with the same spectral information and solving such linear problem with some of the eigenvalue algorithms available in the literature. Linearizations of matrix polynomials are usually defined using unimodular transformations. In this paper we establish a connection between the standard definition of linearization for matrix polynomials introduced by Gohberg, Lancaster and Rodman and the notion of polynomial system matrix introduced by Rosenbrock. This connection gives new techniques to show that a matrix pencil is a linearization of the corresponding matrix polynomial arising in a PEP.

Revisiting Minimal Realizations [Lecture Notes

Although realizations of transfer functions are a standard topic in textbooks about linear systems theory, this article focuses on several points that are not widely treated. First, it is shown that, for a given dynamics matrix A, the smallest number of sensors and actuators that can be used to form a minimal realization is determined by properties of the Jordan form of A. This leads to the notions of minimally sensed and minimally actuated systems. Next, the rank of a transfer function is related to properties of the Jordan form of A and the rank of the Rosenbrock system matrix.

Eigenstructure assignment in linear geometric control

The focus of this paper is the connection between two foundational areas of linear time-invariant (LTI) systems theory: geometric control and eigenstructure assignment. In particular, we study the properties of the null-spaces of the reachability matrix pencil and of the Rosenbrock system matrix, which have been extensively used as two computational building blocks for the calculation of pole placing state feedback matrices and pole placing friends of output-nulling subspaces. Our objective is to show that the subspaces in the chains of kernels obtained in the construction of these feedback matrices interact with each other in ways that are entirely independent from the choice of eigenvalues. So far, these chains of subspaces have only been studied in the case of stationarity. In this case, it is known that these chains converge to the classic Kalman reachable subspace ℛ for the reachability matrix pencil and to the largest reachability subspace ℛ ⋆ in the case of the Rosenbrock matrix, respectively. Here we are interested in showing that even before stationarity has been reached, the partial chains are linked to structural properties of the system, and are therefore independent of the closed-loop eigenvalues that we wish to assign. We further characterize these subspaces by investigating the notion of largest subspace on which it is possible to assign the closed-loop spectrum (possibly maintaining the output at zero) without resorting to non-trivial Jordan forms.

Generalized Fiedler pencils with repetition for rational matrix functions

We introduce generalized Fiedler pencil with repetition(GFPR) for an n x n rational matrix function G(?) relative to a realization of G(?). We show that a GFPR is a linearization of G(?) when the realization of G(?) is minimal and describe recovery of eigenvectors of G(?) from those of the GFPRs. In fact, we show that a GFPR allows operation-free recovery of eigenvectors of G(?). We describe construction of a symmetric GFPR when G(?) is symmetric. We also construct GFPR for the Rosenbrock system matrix S(?) associated with an linear time-invariant (LTI) state-space system and show that the GFPR are Rosenbrock linearizations of S(?). We also describe recovery of eigenvectors of S(?) from those of the GFPR for S(?). Finally, We analyze operation-free Symmetric/self-adjoint structure Fiedler pencils of system matrix S(?) and rational matrix G(?). We show that structure pencils are linearizations of G(?).

Generalized Fiedler Pencils for Rational Matrix Functions

We introduce generalized Fiedler (GF) pencils for an $n\times n$ rational matrix function $G(\lambda)$ for computing eigenvalues and poles of $G(\lambda)$ and show that the GF pencils are linearizations of $G(\lambda).$ These GF pencils generalize the GF pencils of matrix polynomials to the case of rational matrix functions. We show that the GF pencils of $G(\lambda)$ provide a class of linearizations of $G(\lambda)$ which contains potential candidates for structure-preserving linearizations of $G(\lambda)$. Indeed, we describe construction of a self-adjoint GF pencil of $G(\lambda)$ when $G(\lambda)$ is self-adjoint. We further show that the GF pencils of $G(\lambda)$ allow operation-free recovery of eigenvectors of $G(\lambda),$ that is, the eigenvectors of $G(\lambda)$ can be recovered from those of the GF pencils without performing any arithmetic operations. We also introduce GF pencils for the Rosenbrock system polynomial $\mathcal{S}(\lambda)$ associated with a linear time-invariant (LTI) state-spac...

Recovery of eigenvectors of rational matrix functions from Fiedler-like linearizations

Linearization is a standard method often used when dealing with matrix polynomials. Recently, the concept of linearization has been extended to rational matrix functions and Fiedler-like matrix pencils for rational matrix functions have been constructed. A linearization L ( λ ) of a rational matrix function G ( λ ) does not necessarily guarantee a simple way of recovering eigenvectors of G ( λ ) from those of L ( λ ) . We show that Fiedler-like pencils of G ( λ ) allow an easy operation free recovery of eigenvectors of G ( λ ) , that is, eigenvectors of G ( λ ) are recovered from eigenvectors of Fiedler-like pencils of G ( λ ) without performing any arithmetic operations. We also consider Fiedler-like pencils of the Rosenbrock system polynomial S ( λ ) associated with an LTI system Σ in state-space form (SSF) and show that the Fiedler-like pencils allow operation free recovery of eigenvectors of S ( λ ) . The eigenvectors of S ( λ ) are the invariant zero directions of the LTI system Σ.

On the computation of the fundamental subspaces for descriptor systems

ABSTRACTIn this paper, we investigate several theoretical and computational aspects of fundamental subspaces for linear time-invariant descriptor systems, which appear in the solution of many control and estimation problems. Different types of reachability and controllability for descriptor systems are described and discussed. The Rosenbrock system matrix pencil is employed for the computation of supremal output-nulling subspaces and supremal output-nulling reachability subspaces for descriptor systems.

Linearizations for Rational Matrix Functions and Rosenbrock System Polynomials

Our aim in this paper is twofold: First, for solving rational eigenproblems we introduce linearizations of rational matrix functions and propose a framework for their constructions via minimal realizations. We also introduce Fiedler-like pencils for rational matrix functions and show that the Fiedler-like pencils are linearizations of the rational matrix functions. Second, for computing zeros of a linear time-invariant system $\Sigma$ in state-space form, we introduce a trimmed structured linearization, which we refer to as a Rosenbrock linearization, of the Rosenbrock system polynomial $\mathcal{S}(\lambda)$ associated with $\Sigma.$ Also we introduce Fiedler-like pencils for $\mathcal{S}(\lambda),$ describe an algorithm for their constructions, and show that the Fiedler-like pencils are Rosenbrock linearizations of the system polynomial $\mathcal{S}(\lambda).$

Robust Eigenstructure Assignment in Geometric Control Theory

In this paper we employ the Rosenbrock system matrix pencil for the computation of output-nulling subspaces of linear time-invariant systems which appear in the solution of a large number of control and estimation problems. We also consider the problem of finding friends of these output-nulling subspaces, i.e., the feedback matrices that render such subspaces invariant with respect to the closed-loop map and output-nulling with respect to the output map, and which at the same time deliver a robust closed-loop eigenstructure. We show that the methods presented in this paper offer considerably more robust eigenstructure assignment than the other commonly used methods and algorithms.

A new two-stage Rosenbrock scheme for differential-algebraic systems

In this work conditions for the two-stage Rosenbrock system parameters with third-order accuracy for the differential-algebraic systems of index 1 have been obtained. The coefficients of the system are complex numbers. A new system with L2 stability has been constructed with third-order accuracy for differential-algebraic systems and with fourth-order accuracy for differential systems. The convergence of this scheme is proved. This scheme was tested on standard stiff tests and compared with the already known schemes of the same class.

Invariant subspaces and invertibility properties for singular systems: The general case

Open-loop definitions and properties of several subspaces for general singular systems are characterized by means of a fully algebraic distributional framework. Simple recursive algorithms for producing these spaces as well as related duality aspects turn out to follow directly from these definitions. Next, we provide definitions and conditions for two notions of left (right) invertibility of a general singular system in terms of our distributions, subspaces, and Rosenbrock's system matrix, and we show which conditions represent the “gap” between our invertibility concepts. Finally, we prove that in many cases left (right) invertibility is equivalent to left (right) invertibility of the system matrix.

Generalized state-space system matrix equivalents of a Rosenbrock system matrix

Generalized state-space system matrix equivalents of a Rosenbrock system matrix

2-D System Structure and Applications

The structure of 2-D systems is investigated by using the polynomial models of a matrix fraction description and the more general model of a Rosenbrock system matrix.These results are then applied to the case of the unit memory process to characterise the decoupling structure of the system

Module Theoretic Zero Structures for System Matrices

The foundation for a coordinate-free theory of the poles of a linear dynamical system was laid in 1965 by the module-theoretic work of Kalman. However, although the theoretical and application importance of transfer function zeros for feedback system design was widely known for single input, single output systems by 1955, it remained for Rosenbrock in 1970 to propose the ideas of transmission zeros, input-decoupling zeros, and output-decoupling zeros for multi-input, multi-output systems, by means of matrix theoretic methods. A coordinate-free, module-theoretic treatment of transmission zeros for a multi-input, mufti-output transfer function was given in 1981 by Wyman and Sain. This paper extends these coordinate-free, module-theoretic studies to include systems which need not be controllable or observable. Interpretation of the Rosenbrock system matrix is given on three levels: rational, finitely generated free-modular, and torsion divisible. On the second level, an $\Omega $-Zero Module$Z_\Omega $ is defined and imbedded in a short exact sequence showing that the input-decoupling zero module is contained as a factor module in $Z_\Omega $. On the third level, a $\Gamma $-Zero Module$Z_\Gamma $ is defined and imbedded in a short exact sequence showing that the output-decoupling zero module is contained as a submodule in $Z_\Gamma $. Both structures are studied further in regard to transmission zero module information with emphasis on lumped zeros, and the cases of right and left invertible transfer functions are given in detail. Not surprisingly, these investigations can support considerable fine detail, which is in accord with the widely held belief that questions on the nature of multivariable zeros must be broadly based.

Input- and output-decoupling structural zeros of linear systems described by Rosenbrock's polynomial matrices

Abstract This paper introduces the notion of input- (output-) decoupling structural zeros for linear time-invariant systems, described by Rosenbrock's system polynomial matrices. This notion is based on the definition of polynomial system matrices having the same structure that is given in this paper. Namely the input- (output-) decoupling structural zeros are decoupling zeros that are present for every choice of the system parameters. The main results are derived using digraph theory. An example illustrates the procedure for detecting structural zeros.

The determination of structural properties of a linear multivariable system by operations of system similarity. 1. Strictly proper systems

Abstract Structural properties of a linear multivariable system have been defined as invariants under various transformation groups using the theory of singular pencils of matrices or using a geometric approach. In this paper it is shown that for a strictly proper system in state-space form, these structural properties can be identified by bringing the system matrix to a particular form under system similarity. This allows a structural analysis to be incorporated in existing computer-aided design methods for multivariable control systems. In relation to these structural properties, Rosenbrock's definition of system zeros is investigated and it is shown how this definition fits into the framework of pencil theory and of the geometric approach. This leads to a precise structural interpretation of the relation between Rosenbrock's system zeros and the invariant zeros defined in accordance with pencil theory. The structural analysis presented in this paper also allows us to identify the zeros at infinity and ...

Signal-flow-graph reduction, Mason's rule and the system matrix

A simple algorithm for the simplification of signal-flow graphs is given, and an interesting connection with Rosen-brock's system matrix is shown.