Computing effectively stabilizing controllers for a class of $n$D systems

TL;DR

This paper develops symbolic-numeric algorithms to determine internal stabilizability and compute stabilizing controllers for a specific class of multidimensional systems with zero-dimensional polynomial ideals.

Contribution

It introduces effective algorithms for testing stabilizability and computing stabilizing controllers for nD systems with finite common zeros of their polynomial ideals.

Findings

Algorithms successfully tested on 2D and 3D systems.

Provided prototype implementation with performance metrics.

Confirmed effectiveness for systems with zero-dimensional polynomial ideals.

Abstract

In this paper, we study the internal stabilizability and internal stabilization problems for multidimensional (nD) systems. Within the fractional representation approach, a multidimen-sional 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 n = {z = (z 1 ,. .. , z n) $\in$ C n | |z 1 | 1,. .. , |z n | 1}. It is known that the internal stabilizability of a multidimensional system can be investigated by studying a certain polynomial ideal I = p 1 ,. .. , p r 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 $\in$ C n | p 1 (z) = $\times$ $\times$ $\times$ = p r (z) = 0} $\cap$ U n = $\emptyset$. 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 p i 's have only a finite number of common complex zeros. We propose effective symbolic-numeric algorithms for testing if V (I) $\cap$ U n = $\emptyset$, as well as for computing, if it exists, a stable polynomial p $\in$ I which allows the effective computation of a stabilizing controller. We illustrate our algorithms through an example and finally provide running times of prototype implementations for 2D and 3D systems.