Antonis-Ioannis G. Vardulakis

On the solution and impulsive behaviour of polynomial matrix descriptions of free linear multivariable systems

In this note we examine the solution and the impulsive behaviour of autonomous linear multivariable systems whose pseudo-state beta(t) obeys a linear matrix differential equation A(rho)beta(t) = 0 where A(rho) is a polynomial matrix in the differential operator rho:=d/dt. We thus generalize to the general polynomial matrix case some results obtained by Verghese and colleagues which regard the impulsive behaviour of the generalized state vector x(t) of input free generalized state space systems.

Proper and stable, minimal MacMillan degrees bases of rational vector spaces

The structure of proper and stable bases of rational vector spaces is investigated. We prove that if t(s) is a rational vector space, then among the proper bases of 3(s) there is a subfamily of proper bases which are 1) stable, 2) have no zeros in C\bigcup \{\infty\} and therefore are column (row) reduced at infinity, and 3) their MacMillan degree is minimum among the MacMillan degrees of all other proper bases of 3(s) and it is given by the sum of the MacMillan degrees of their columns taken separately. It is shown that this notion is the counterpart of Forneys concept of a minimal polynomial basis for the family of proper and stable bases of 3(s).

On the computation and parametrization of proper denominator assigning compensators for strictly proper plants

Given a right coprime MFD of a strictly proper plant P(s) = NR(s)DR(s) −1 with DR(s) column proper as imple numerical algorithm is derived for the computation of all polynomial solutions (X L(s), YL(s)) of the polynomial matrix Diophantine equation X L(s)DR(s) + YL(s)NR(s) = DC(s) which give rise to the class Φ(P, DC) of proper compensators C(s) := X L(s) −1 YL(s) that when employed in a unity feedback loop, result in closed-loop systems S(P, C) with a desired denominator DC(s). The parametrization of the proper compensators C(s) ∈ Φ(P, DC) is obtained and the number of independent parameters in the parametrization is given. We consider linear, time invariant, multivariable systems which are assumed to be free of unstable hidden modes, whose input-output relation is described by a strictly proper transfer function matrix P(s) (the plant). In this note we describe a numerically efficient algorithm for the computation of the class of proper compensators C(s) which, when employed in the unity feedback loop of Fig. 1, gives rise to a closed-loop system S(P, C) with a specific closed-loop denominator DC(s) (Rosenbrock & Hayton, 1978; Kucera, 1970). In particular, given a right coprime MFD of a strictly proper plant P(s) = NR(s)DR(s) −1 with DR(s) column proper (column reduced) and an appropriately defined polynomial matrix DC(s) with desired zeros, we extend the (Wolovich, 1974) resultant theorem and a theorem by Callier & Desoer (1982), Callier (2001) and Kucera & Zagalak (1999) in order to obtain an algorithm for the computation of all polynomial solutions (X L(s), YL(s)) of the polynomial matrix Diophantine equation X L(s)DR(s) + YL(s)NR(s) = DC(s) (1) which give rise to the class Φ(P, DC) of proper compensators C(s) := X L(s) −1 YL(s) that result in closed-loop systems S(P, C) with DC(s) as their closed-loop denominator. The issues of the parametrization of the proper compensators C(s) ∈ Φ(P, DC) and the number of independent parameters in the parametrization are also resolved. This is done by investigating the properties of a generalized version of Wolovichs resultant to obtain a series of new results regarding its algebraic structure. Despite the fact that similar results for Sylvester-type resultants have been presented in Bitmead et al. (1978), the Wolovich resultant has not received the expected attention, except perhaps by Wolovich (1974) and Hayton (1980) where Wolovichs resultant is used as a tool for testing the coprimeness of polynomial matrices.

Denominator Assignment, Invariants and Canonical Forms Under Dynamic Feedback Compensation in Linear Multivariable Systems

A result originally reported in [3] for linear time invariant single input-single output systems and concerning an invariant and a canonical form of the transfer function matrix of the closed loop system under dynamic feedback compensation is generalized for LTI multivariable systems. This result leads to a characterization of the class of closed loop transfer function matrices which are obtainable under feedback through a proper dynamic compensator and gives rise to an algorithmic design procedure for the computation of a proper dynamic (feedback), internally stabilizing and denominator assigning compensator for non-minimum phase plants.

Relations between denominator assigning proper feedback compensators and pole assignment by state feedback

We examine relations between denominator assigning proper compensators in the feedback path of linear, time invariant (LTI) multivariable systems, described by square strictly proper transfer function matrices, and pole assignment by state variable feedback. Through these results we establish conditions for the existence and computation of such compensators.

Notions of equivalence for linear multivariable systems

The present paper is a survey on linear multivariable systems equivalences. We attempt a review of the most significant types of system equivalence having as a starting point matrix transformations preserving certain types of their spectral structure. From a system theoretic point of view, the need for a variety of forms of polynomial matrix equivalences, arises from the fact that different types of spectral invariants give rise to different types of dynamics of the underlying linear system. A historical perspective of the key results and their contributors is also given.

Denominator assignment, invariants and canonical forms under dynamic feedback compensation in linear multivariable systems

A result originally reported in [3] for linear time invariant single input-single output systems and concerning an invariant and a canonical form of the transfer function matrix of the closed loop system under dynamic feedback compensation is generalized for LTI multivariable systems. This result leads to a characterization of the class of closed loop transfer function matrices which are obtainable under feedback through a proper dynamic compensator and gives rise to an algorithmic design procedure for the computation of a proper dynamic (feedback), internally stabilizing and denominator assigning compensator for non-minimum phase plants.

Symbolic Computations on Rings of Rational Functions and Applications in Control Engineering

A collection of algorithms implemented in Mathematica 7.0, freely available over the internet, and capable to manipulate rational functions and solve related control problems using polynomial analysis and design methods is presented. The package provides all the necessary functionality and tools in order to use the theory of

Descriptor systems toolbox : a Mathematica-based package for descriptor systems

\it \Omega-

Technical Communique: A note on the action of constant pseudostate feedback on the internal properness of an ARMA model

stable functions, and is expected to provide the necessary framework for the development of several other algorithms that solve specific control problems.