Linear Algebra for Control Theory

Frontmatter

Recursive Modeling of Discrete-Time Time Series

Abstract

The approach to modeling inspired by the behavioral framework consists in treating all measurements on an equal footing, not distinguishing between inputs and outputs. Consequently, the initial search is for autonomous models. In the linear, time invariant case, the main result guarantees the existence of a minimal complexity autonomous generating model Θ*. This means that all other models can be explicitely constructed from Θ*. Among them in most cases, the so-called input-output controllable models are of interest. The main purpose of this paper is to show how these models can be constructed in a an easy-to-implement, recursive way.

A. C. Antoulas

Pole Placement, Internal Stabilization and Interpolation Conditions for Rational Matrix Functions: A Grassmannian Formulation

Abstract

The problem of pole placement via dynamic feedback and the bitangential interpolation problem are shown to be both particular instances of a general subspace interpolation problem formulated for rational curves from projective space into a Grass-mannian manifold. The problem of determining the minimal degree d for an interpolant in terms of the problem data is shown to be computable via intersection theory in projective space. Using the projective dimension theorem bounds for the minimal degree interpolant curve are given.

Joseph A. Ball, Joachim Rosenthal

Feedback Stabilizibility over Commutative Rings

Abstract

In this paper, we survey a part of feedback stabilization for systems over commutative rings. Since many excellent sources exist which describe the motivation behind the study of systems over rings, we shall not touch on that here. The interested reader is referred to [1], [8], [9], [12], [13], and [14].

J. W. Brewer, L. C. Klingler, Wiland Schmale

Output Feedback in Descriptor Systems

Abstract

A summary is given of conditions under which a descriptor, or generalized state-space, system can be regularized by output feedback. Theorems are presented showing that under these conditions proportional and derivative output feedback controls can be constructed such that the closed loop system is regular and has index at most one. This property ensures the solvability of the resulting system of dynamic-algebraic equations. A canonical form is given that allows the system properties as well as the feedback to be determined. The construction procedures used to establish the theory are based only on orthogonal matrix decompositions and can therefore be implemented in a numerically stable way. A computational algorithm for improving the ‘conditioning’ of the regularized closed loop system is described.

Angelika Bunse-Gerstner, Volker Mehrmann, Nancy K. Nichols

On Realization Theory for Generalized State-Space Systems over a Commutative Ring

Abstract

The problem of finding a state-space realization for a given rational matrix over a commutative ring is considered. To simplify the problem, we assume a certain factored structure for the denominator polynomials in the matrix. Our main results state that this class of matrices is a module which can be decomposed into two independent and isomorphic submodules, each realizable via existing results for strictly proper matrices. Any rational matrix with factored denominators can be realized through this decomposition.

J. Daniel Cobb

Problems and Results in a Geometric Approach to the Theory of Systems over Rings

Abstract

The use of geometric methods in the study of linear systems with coefficients in a ring presents several difficulties, but, as in the case of coefficients in a field, it appears capable of providing efficient solutions to a number of control and observation problems. In this paper we survey some recent results concerning a class of Disturbance Decoupling Problems and we extend the techniques developed in that case to a class of problems concerning the Estimation of Linear Functions of the State.

G. Conte, A. M. Perdon

Completion of a Matrix so that the Inverse has Minimum Norm. Application to the Regularization of Descriptor Control Problems

Abstract

We discuss the problem of minimizing the spectral norm of the inverse of a matrix, when a submatrix of the matrix can be chosen arbitrarily. In a recent paper by the authors [3], it was shown that the solution of this problem can be discussed in a similar way as the problem of minimizing the norm of the matrix in terms of matrix Riccati inequalities.

Here we review the results for the norm of the inverse and then apply these results to the robust regularization of descriptor control problems. We also describe a numerical method and give numerical examples.

L. Elsner, C. He, V. Mehrmann

On the Rutishauser’s Approach to Eigenvalue Problems

Abstract

We consider the quotient-difference algorithm of H. Rutishauser from the point of view of the realization theory of rational functions. QR-like algorithms of the linear algebra can be thought of as discrete-time dynamical systems on certain classes of matrices. We show that these algorithms induce linear dynamical systems on various manifolds of rational functions. Applications to coding theory are considered.

L. Faybusovich

The Block form of Linear Systems over Commutative Rings with Applications to Control

Abstract

This paper deals with a new approach to the study of linear time-invariant discrete-time systems whose coefficients belong to an arbitrary commutative ring. Such systems arise in the study of integer systems, systems depending on parameters, and multi-dimensional systems. The key idea is to consider the representation of systems over a ring in terms of a block-input/block-output form. By time-compressing the block representation, new results are derived on assignability by state feedback control including the construction of deadbeat controllers. A new type of state observer is then considered based on a block-output form for the update term in the state estimate. The results on state observers are combined with the time-compression approach to state feedback control to yield a new type of input/output regulator. In the last section of the paper the results are applied to the problem of state and output tracking of set points.

Edward W. Kamen

Diffeomorphisms between Sets of Linear Systems

Abstract

Diffeomorphisms are given between different subsets of linear systems of fixed McMillan degree. The sets considered are the set of all systems of fixed McMillan degree, the subset of stable systems, the subset of bounded real systems, the subset of positive real systems, the subset of stable systems with Hankel singular values bounded by one. State space techniques are used in the proofs.

R. Ober, P. A. Fuhrmann

Transfer Function Approach to Disturbance Decoupling Problem

Abstract

We give a necessary and sufficient condition for existence of a controller which decouples the disturbance signal. The condition is based on state space computations. If it is satisfied, we parametrize the set of all disturbance decoupling controllers.

Marek Rakowski

Some Numerical Challenges in Control Theory

Abstract

We discuss a number of novel issues in the interdisciplinary area of numerical linear algebra and control theory. Although we do not claim to be exhaustive we give a number of problems which we believe will play an important role in the near future. These are: sparse matrices, structured matrices, novel matrix decompositions and numerical shortcuts. Each of those is presented in relation to a particular (class of) control problems. These are respectively: large scale control systems, polynomial system models, control of periodic systems, and normalized coprime factorizations in robust control.

Paul Van Dooren

Erratum

Joseph A. Ball, Joachim Rosenthal