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Real solutions of polynomial equations

Kiritsis K. H.

Control and Cybernetics

2017

Vol. 46, no. 2

177--185

In this paper the problem of existence of solution of polynomial equations over the ﬁeld of real numbers is considered. In particular, the explicit necessary and suﬃcient conditions are established for the equation A(z)X + B(z)Y = C(z) in polynomial matrices to have a solution for X and Y over the ﬁeld of real numbers, with X being non-singular, for every polynomial matrix C(z) from a given class.

Computation of a canonical form for neutral delay-diﬀerential systems

Boudellioua M. S.

Control and Cybernetics

2015

Vol. 44, no. 3

357--368

In this paper, symbolic computation techniques are used to obtain a canonical form for polynomial matrices arising from linear delay-diﬀerential systems of the neutral type. The canonical form can be regarded as an extension of the companion form, often encountered in the theory of linear systems, described by ordinary diﬀerential equations. Using the Smith normal form, the exact connection between the original polynomial matrix and the reduced canonical form is set out. An example is given to illustrate the computational aspects involved.

On the realization theory of polynomial matrices and the algebraic structure of pure generalized state space systems

Vardulakis A. I. G., Karampetakis N. P., Antoniou E. N., Tictopoulou E.

International Journal of Applied Mathematics and Computer Science

2009

Vol. 19, no 1

77-88

We review the realization theory of polynomial (transfer function) matrices via 'pure' generalized state space system models. The concept of an irreducible-at-infinity generalized state space realization of a polynomial matrix is defined and the mechanism of the 'cancellations' of 'decoupling zeros at infinity' is closely examined. The difference between the concepts of irreducibility and minimality of generalized state space realizations of polynomial (transfer function) matrices is pointed out and the associated concepts of dynamic and non-dynamic variables appearing in generalized state space realizations are also examined.