Computing effectively stabilizing controllers for a class of nD systems

In this paper, we study the internal stabilizability and internal stabilization problems for multidimensional (nD) systems. Within the fractional representation approach, a multidimensional system can be studied by means of matrices with entries in the integral domain of structurally stable rational fractions, namely the ring of rational functions which have no poles in the closed unit polydisc U = {z = (z1, . . . , zn) ∈ C | |z1| 6 1, . . . , |zn| 6 1} . It is known that the internal stabilizability of a multidimensional system can be investigated by studying a certain polynomial ideal I = 〈p1, . . . , pr〉 that can be explicitly described in terms of the transfer matrix of the plant. More precisely the system is stabilizable if and only if V (I) = {z ∈ C | p1(z) = · · · = pr(z) = 0} ∩ U n = ∅. In the present article, we consider the specific class of linear nD systems (which includes the class of 2D systems) for which the ideal I is zero-dimensional, i.e., the pi’s have only a finite number of common complex zeros. We propose effective symbolic-numeric algorithms for testing if V (I) ∩ U = ∅, as well as for computing, if it exists, a stable polynomial p ∈ I which allows the effective computation of a stabilizing controller. We finally illustrate our algorithms on an example.