A. I. G. Vardulakis

Computation of the inverse of a polynomial matrix and evaluation of its Laurent expansion

Abstract An algorithm is given which constitutes a generalization of the algorithm for the inversion of a pencil sE-A due to Mertzios (1984) in the case of general systems described in differential operator form. Recursive formulae are obtained for the calculation of both the coefficient matrices of the adjoint of the polynomial matrix, as well as for the coefficients of the characteristic polynomial of the polynomial matrix. A simple method is also presented that allows the evaluation of the Laurent expansion for the inverse of a polynomial matrix of any order. Specifically, an ARMA model is given for the computation of the coefficient matrices of the Laurent expansion in terms of the coefficient matrices of the polynomial matrix, without inverting the latter. The Laurent expansion of a polynomial matrix is used for the analysis and synthesis of the polynomial matrix descriptions that constitute a further generalization of the singular (or generalized) systems.

On infinite zeros

Abstract The infinite zero structure of completely controllable and observable linear multi-variable systems which give rise to square, non-singular, strictly proper transfer function matrices is investigated via the polynomial matrix approach to the solution of the decoupling problem. In the process various connections between infinite zeros, their degrees and geometric and polynomial matrix ideas are demonstrated. The asymptotic behaviour of closed loop eigenvectors under high gain output feedback is also examined.

Internal stabilization and decoupling in linear multivariable systems by unity output feedback compensation

We consider the problem of internally stabilizing and simultaneously diagonally decoupling a linear multivariable system by unity output feedback compensation. A sufficient condition is derived for the existence of a cascade proper compensator C(s) such that when employed in a unity feedback loop involving the proper transfer function matrix P o of a free of unstable hidden modes system Sigma(P_{o}) , will not only internally stabilize the feedback closed-loop system Sigma(P_{o}, C) but will also give rise to a closed-loop transfer function matrix H_{yr}^{diag} , which is nonsingular, diagonal, and has desired poles. Based on this analysis, an algorithmic procedure for the computation of such a compensator is presented.

Pole-shifting using output feedback

A simple test is given in analytic (matrix) terms which provides both necessary and sufficient conditions for arbitrary assignment of all of the system poles using only constant output feedback. A formula to determine the required output compensator is presented, and the whole procedure is illustrated by an example.

Zero placement and the ‘ squaring down ’ problem: a polynomial matrix approach

Certain results in the theory of polynomial matrices, free R [s]-modules and minimal bases of rational vector spaces are used in order to investigate the ‘ squaring down ’ and zero placement problem.

Structure, Smith-MacMillan form and coprime MFDs of a rational matrix inside a region P =ω∪{∞}

Abstract The structure of the Smith-MacMillan form of a rational matrix T(s) inside a region P=ω∪{∞} (where ω is asymmetric with respect to the real axis subset of the finite complex plane C) is determined. Algorithmic procedures based on elementary row and column operations over the euclidean ring R P(s) consisting of all rational functions with no poles in P are given. Coprimeness in P of a pair of rational matrices is studied in detail. These results lead to constructive procedures for determining the coprime in P matrix fraction descriptions of T(a).

Relations between strict equivalence invariants and structure at infinity of matrix pencils

The structure at infinity of matrix pencils is investigated via their Smith-MacMillan forms at infinity [1]. Relations between some strict equivalence invariants and pole-zero structure at infinity are established.

On the stable exact model matching problem

Abstract The stable exact model matching problem (SEMMP) is investigated. We state and prove a number of equivalent necessary and sufficient conditions for the existence of proper solutions to the exact model matching problem that are also Ω-stable, i.e. have no poles inside a symmetric ‘forbidden’ subset Ω of the finite complex plane C . These results can be viewed as the counterpart of the results in [3] and [9] for the case of the ring of proper and Ω-stable rational functions.

A sufficient condition for n specified eigenvalues to be assigned under constant output feedback

A sufficient condition for n ( n -dimension of the state vector) specified eigenvalues to be assigned under the use of constant output feedback is derived. The form of the output feedback constant gain matrix which results in the above eigenvalue assignment is also established.

Infinite elementary divisors of polynomials matrices and impulsive solutions of linear homogeneous matric differential equations

Impulsive solutions of linear homogeneous matrix differential equations are re-examined in the light of the theory of Jordan chains that correspond to infinite elementary divisors of the associated polynomial matrix. Infinite elementary divisors of general polynomial matrices are defined and their relation to the pole-zero structure of polynomial matrices at infinity is examined. It is shown that impulsive solutions are due to Jordan chains of a “dual” polynomial matrix that correspond to infinite elementary divisors that are associated with the orders of “zeros at infinity” of the original matrix.