On polytopes in Hurwitz region
Introduction
Consider the following polynomial The polynomial corresponds to the -dimensional vector .
The polynomial (1) has roots and if all roots satisfy the condition then it is called Hurwitz stable [1], [2]. Throughout this paper stable will mean Hurwitz stable.
Define
It is well known that , where
This necessary condition is not sufficient for .
The convex hull of a finite number of points from is called a polytope in 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 vertices, among which vertices are lying far from the initial stable point. The polytopes (see (2)) presented in this work are directed to the reverse direction to the stability boundary. vertices of these polytopes are on the stability boundary. It is not difficult to give an example of an affine subset of which meets with and does not meet (see (3)) from [3].
From Routh–Hurwitz matrix criterion (see [1, p. 51]) it follows that:
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A third order polynomial with is stable if and only if ;
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A fourth order polynomial with is stable if and only if .
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
In this section, we define new polytopes in (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 is stable if and only if and .
Let be even, , and be different positive numbers satisfying the conditions , . Consider the following order stable polynomial
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 , where is an th order monic polynomial and are of lower orders. The problem consists of determining the values of the controller parameters for which is stable. The polynomials correspond to
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 vertices, among which 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.
References (23)
- et al.
Infinite polytopes in hurwitz stablity region
Automatica
(2019) - et al.
Stable polyhedra in parameter space
Automatica
(2003) - et al.
Algebraic conditions for stability of cones of polynomials
Systems Control Lett.
(2002) - et al.
Smooth trivial vector bundle structure of the space of Hurwitz polynomials
Automatica
(2009) - et al.
On Kharitonov’s theorem without invariant degree assumption
Automatica
(2000) New Tools for Robustness of Linear Systems
(1994)- et al.
Robust Control: The Parametric Approach
(1995) - et al.
Root locations of an entire polytope of polynomials: It suffices to check the edges
Math. Control Signals Systems
(1988) A necessary and sufficient condition for the stability of convex combinations of stable polynomials or matrices
Bull. Pol. Acad. Sci. Tech. Sci.
(1985)- et al.
Stabilisation of discrete-time systems via Schur stability region
Internat. J. Control
(2018)
Fixed order controller for Schur stability
Math. Comput. Appl.
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