On polytopes in Hurwitz region

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Abstract

We consider the stability region of monic polynomials in the coefficient space. Starting from specially chosen stable points we define polytopes, whose edges are either stable or belong to the stability boundary. Therefore by the Edge Theorem the interiors are stable. Stable polytopes, in the reverse directions which are directed to infinity have been studied in the work (Dzhafarov et al., 2019). Some applications of the obtained results to the synthesis of fixed-order stabilizing controllers are given.

Introduction

Consider the following polynomial a(s)=a1+a2s++ansn1+sn.The polynomial a(s) corresponds to the n-dimensional vector a=(a1,a2,,an)TRn.

The polynomial (1) has n roots and if all roots s1,s2,,sn satisfy the condition Re(si)<0 (i=1,2,,n) then it is called Hurwitz stable [1], [2]. Throughout this paper stable will mean Hurwitz stable.

Define Hn={a=(a1,a2,,an)TRn:a(s) is stable}.

It is well known that HnR+n, where R+n={x=(x1,x2,,xn)TRn:xi>0,(i=1,2,,n)}.

This necessary condition is not sufficient for n3.

The convex hull of a finite number of points from Rn is called a polytope in Rn and these points are called the generators.

Related questions concerning outer and inner approximations, geometric and topological properties of stability regions for continuous and discrete time systems have been studied in many works (see [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22]).

In the work [3], stable polytopes with infinitely large edges are defined. In the present work we define stable polytopes in the reverse directions than the directions used in [3]. These directions are directed to the stability boundary. In Section 2, we define stable polytopes of this type. Section 3, contains an algorithm for determining the initial stable point and two examples on stabilization.

The main difference of the present work in comparison with [3] is that the polytopes from [3] are directed inside the stability region, have (n+1) vertices, among which n vertices are lying far from the initial stable point. The polytopes P (see (2)) presented in this work are directed to the reverse direction to the stability boundary. n vertices of these polytopes are on the stability boundary. It is not difficult to give an example of an affine subset of Rn which meets with P and does not meet Pt (see (3)) from [3].

From Routh–Hurwitz matrix criterion (see [1, p. 51]) it follows that:

  • -

    A third order polynomial b1+b2s+b3s2+s3 with bi>0 (i=1,2,3) is stable if and only if b2b3b1>0;

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    A fourth order polynomial b1+b2s+b3s2+b4s3+s4 with bi>0 (i=1,2,3,4) is stable if and only if b2b3b4b1b42b22>0.

The Edge Theorem [7] says that a polynomial polytope with invariant degree is stable if and only if all edges are stable.

Section snippets

Stable polytopes in Hn

In this section, we define new polytopes in Hn (Theorem 2, Theorem 3). The idea of this construction is similar to the construction used in [3]. For comparison, in the below (Theorem 1) we describe the infinite polytopes constructed in [3].

A second order polynomial s2+k1s+k2 is stable if and only if k1>0 and k2>0.

Let n be even, m=n2, and α1,α2,,αm be different positive numbers satisfying the conditions αi1, (i=1,2,,m). Consider the following nth order stable polynomial a0(s)=(s2+α1s+α1)(s2+

Some applications

Here we consider some applications of the obtained results to the stabilization problems. It is well known that in many cases this problem can be reduced to the problem of a stable member in the closed loop affine family p(s,c)=p0(s)+c1p1(s)++clpl(s), where p0(s) is an nth order monic polynomial and p1(s),,pl(s) are of lower orders. The problem consists of determining the values of the controller parameters c1,,cl for which p(s,c) is stable. The polynomials p0(s),p1(s),,pl(s) correspond to n

Conclusion

In [3] we have defined infinite Hurwitz stable polytopes starting from specially chosen stable points in the coefficient space. The edges of these polytopes are directed to infinity. In this work starting from the same stable points we define polytopes in the reverse directions than directions used in [3]. These polytopes are directed to the stability boundary, have (n+1) vertices, among which n vertices belong to the stability boundary. Some applications of the obtained results to the

CRediT authorship contribution statement

Vakif Dzhafarov: Supervision, Conceptualization, Writing - original draft. Özlem Esen: Methodology, Visualization, Validation. Taner Büyükköroğlu: Software, Writing - review & editing, Investigation.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The authors wish to thank the reviewers for their constructive comments improving this note.

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