Finite Nyquist and Finite Inclusions Theorems for disjoint stability regions

doi:10.1016/j.sysconle.2009.09.001

Systems & Control Letters

Volume 58, Issue 12, December 2009, Pages 804-809

https://doi.org/10.1016/j.sysconle.2009.09.001 Get rights and content

Abstract

Finite Nyquist Theorem is an important tool in stability analysis and design of linear systems. Currently, Finite Nyquist Theorem can treat only simply connected convex stability regions with some extensions to simply connected non-convex regions. In this paper, we consider the generalization of Finite Nyquist Theorem for the case of union of disjoint convex stability regions. Based on this result, the Finite Inclusions Theorem is also formulated for a union of disjoint convex stability regions.

Introduction

The Finite Nyquist Theorem and its generalization for uncertain systems, i.e. the Finite Inclusions Theorem, [1], are important tools in stability analysis and design of linear systems [2], [3], [4], [5], [6]. The above theorems allow us to consider only a finite subset of frequencies in order to establish stability. This subject remains a topic of continuous research: useful extensions to this method were devised for example in [7].

Recently, the question of stability, where stability domain is given as a set of disjoint convex regions, got much attention [8], [9], [10]. Below we state a few examples of control problems requiring stability analysis with disjoint stability regions:

Problem 1

Model reduction of unstable LTI SISO system. When the unstable poles of the reduced system should be kept close to unstable poles of the original system, the stability region for the reduced system becomes the union of:

•
Regular stability region (for example the unit circle)
•
Small regions around the unstable poles of the original system.

Problem 2

Handling of systems with separate dynamics, for example the systems for which one part of the system should react fast, while the other part should react slow. Separate dynamics naturally leads to stability region being a union of disjoint regions. Designing a controller for such a system, determining maximal stability radius and other problems can benefit from the proposed method.

Problem 3

Non-fragility analysis of a given system. More precisely, given some nominal controller, find the maximal deviation from the nominal controller, so that the poles of the closed loop transfer function will remain in some predefined vicinity of the poles of the closed loop transfer function with nominal controller. In this case, again, the stability region is the union of small regions around the poles of the closed loop transfer function.

The purpose of this paper is to generalize the results of Finite Nyquist Theorem, stated in [1] only for a simply connected convex region, to the case of union of disjoint convex stability regions. Namely, given a polynomial, our purpose is to find a finite number of checks to be performed on the phase of this polynomial in order to ensure that its zeros are located in given disjoint convex regions. Based on this result, the Finite Inclusions Theorem is also formulated for the union of disjoint convex stability regions.

Note that, as shown in Section 3, the natural naive extension of the theorems in [1] to the case of disjoint regions does not hold.

The structure of the paper is as follows. In Section 2 we provide the notations used in the paper. In Section 3 we state and prove the new Finite Nyquist Theorem for disjoint stability regions. Based on the results in Section 3, the Finite Inclusions Theorem is formulated in Section 4. In Section 5 we provide an example. We conclude in Section 6.

Section snippets

Definitions and notations

The following notations will be used throughout the paper.

Given the monic polynomial $p (s) = (s - z_{n}) (s - z_{n - 1}) \dots (s - z_{1})$ , its phase, $\arg (p (s))$ , is defined as $\arg (p (s)) = \arg (s - z_{n}) + \dots + \arg (s - z_{1})$ where $\arg (s - z_{i}) \in [0, 2 π]$ , $1 \leq i \leq n$ . The $\arg (p (s))$ is a continuous function of $s$ unless $s$ passes through some zero $z_{i}$ . Note, that $\arg (p (s))$ can take values in $[0, 2 π n]$ .

A given closed Jordan curve $Ψ \subset C$ separates the complex plane into two regions. By $r e g Ψ$ we denote one of these regions, namely, either interior or exterior

Finite Nyquist Theorem for disjoint stability region

The Finite Nyquist Theorem for simply connected single region [1] is cited below for convenience.

Let $p (s) = \sum_{j = 0}^{n} α_{j} s^{j}$ , where $n \in Z, n \geq 0$ , and $α_{j} \in C$ and let $Γ \subset C$ be a closed Jordan curve such that $i n t Γ$ is convex. Then, $p (s)$ is of degree $n$ and has all its zeros in $i n t Γ$ if and only if there exist $m \geq 1$ angles $θ_{k} \in R$ and a counterclockwise sequence of points $s_{k} \in Γ$ , $1 \leq k \leq m$ , such that $\forall_{1 \leq k < m} | θ_{k + 1} - θ_{k} | < π$ $| 2 π n + θ_{1} - θ_{m} | < π$ $\forall_{1 \leq k \leq m} p (s_{k}) \neq 0$ $\forall_{1 \leq k \leq m} arg p (s_{k}) = θ_{k} + 2 π v_{k}, v_{k} \in Z .$ The simple paraphrase of the above theorem is: if we can

Finite Inclusions Theorem

In this section we provide the Finite Inclusions Theorem for a union of disjoint convex stability regions, based on the results of the previous section.

Theorem 2

Let $p (s, q) = \sum_{j = 0}^{n} α_{j} (q) s^{j}$ , $q \in Q$ where $Q$ is an arbitrary set (of possibly complex parameters), $n \in Z, n > 0$ , and $α_{j} : Q \to C$ . Let $Γ^{(i)} \subset C, 1 \leq i \leq q$ be closed Jordan curves such that $i n t Γ^{(i)}$ are convex and $Γ^{(i)} \cap Γ^{(j)} = 0̸$ for $i \neq j$ . Only one of the $i n t Γ^{(i)}$ is allowed to be unbounded. Let $x^{(i)} \in Z$ be given such that $\sum_{i = 1}^{q} x^{(i)} = n$ .

Then, every member of the polynomial

Example

We consider a practical example requiring the Finite Inclusions Theorem for disjoint regions. The poles of the nominal plant for this example are taken from [9], where the lateral dynamics of an aircraft was considered: $λ_{i} = {- 0.5, - 2 \pm 2 i, - 3 \pm 2 i} \Rightarrow p (s) = s^{5} + 10.5 s^{4} + 50 s^{3} + 122.5 s^{2} + 154 s + 52 .$ Our aim is to analyze the robustness of the pole locations in the presence of an uncertainty. We consider an interval polynomial of a predefined form (with parameterizable gain $k$ ) around the nominal polynomial (13): $\begin{matrix} p (s, q \end{matrix}$

Conclusion

In this paper we have stated and proved the Finite Nyquist Theorem for a union of disjoint convex stability regions. We have shown that a finite number of phase checks is sufficient to guarantee that all roots of a polynomial lie inside the stability region, even when the stability region is a union of convex regions. Based on this theorem, we also formulated the Finite Inclusions Theorem for a union of disjoint convex stability regions.

References (10)

R.D. Kaminsky et al.
The finite inclusions theorem
IEEE Transactions on Automatic Control
(1995)
L. Bruyere et al.
Robust analysis for missile lateral acceleration control using finite inclusion theorem
AIAA Journal of Guidance, Control and Dynamics
(2005)
T.E. Djaferis
Robust Control Design: A Polynomial Approach
(1995)
T.E. Djaferis
The finite inclusions theorem: A tool for robust design
Kybernetika
(1998)
T.E. Djaferis et al.
System Theory: Modeling, Analysis and Control
(1999)

There are more references available in the full text version of this article.

Cited by (1)

Introduction to Linear Control Systems
2017, Introduction to Linear Control Systems

View full text

Systems & Control Letters

Finite Nyquist and Finite Inclusions Theorems for disjoint stability regions

Abstract

Introduction

Section snippets

Definitions and notations

Finite Nyquist Theorem for disjoint stability region

Finite Inclusions Theorem

Example

Conclusion

The finite inclusions theorem

IEEE Transactions on Automatic Control

Robust analysis for missile lateral acceleration control using finite inclusion theorem

AIAA Journal of Guidance, Control and Dynamics

Robust Control Design: A Polynomial Approach

The finite inclusions theorem: A tool for robust design

Kybernetika

System Theory: Modeling, Analysis and Control

Introduction to Linear Control Systems