B.R. Barmish

A generalization of Kharitonov's four-polynomial concept for robust stability problems with linearly dependent coefficient perturbations

From a systems-theoretic point of view, Kharitonovs seminal theorem on stability of interval polynomials suffers from two fundamental limitations: First, the theorem only applies to polynomials with independent coefficient perturbations. Note that uncertainty in the physical parameters of a linear system typically results in dependent perturbations in the coefficients of the characteristic polynomial. Secondly, Kharitonovs Theorem only applies to zeros in the left half plane¿more general zero location regions are not accommodated. In view of this motivation, the main result of this paper is a generalization of Kharitonovs four polynomial concept to the case of linearly dependent coefficient perturbations and more general zero location regions.

Invariance of the strict Hurwitz property for polynomials with perturbed coefficients

Given a strictly Hurwitz polynomial f(¿) = ¿n+ an-1¿n-1+an-2¿n-2+...+a1¿+a0, it is of interest to know how much the coefficients ai can be perturbed while simultaneously preserving the strict Hurwitz property. For systems with n ¿ 4, maximal intervals of the ai are given in a recent paper by Guiver and Bose [1]. In this note, a theorem of Kharitonov is exploited to obtain a general result for polynomials of any degree.

The uniform distribution: rigorous justification for its use in robustness analysis

Consider a control system which is operated with admissible values of uncertain parameters which exceed the bounds specified by classical robustness theory. In this case it is important to quantify the tradeoffs between risk of performance degradation and increased tolerance of uncertainty. If a large increase in the uncertainty bound can be established, an acceptably small risk may often be justified. Since robustness problem formulations do not include statistical descriptions of the uncertainty, the question arises whether it is possible to provide such assurances in a “distribution-free” manner. In other words, if ℱ denotes a class of possible probability distributions for the uncertaintyq, we seek some worst-casef* ε ℱ having the following property: The probability of performance satisfaction underf* is smaller than the probability under any otherf ε ℱ. Said another way,f* provides the best possible guarantee. This new framework is illustrated on robust stability problems associated with Kharitonovs theorem and the Edge Theorem. The main results are straightforward to describe: Letp(s, q) denote the uncertain polynomial under consideration and takeP(ω) to be a frequency-dependent convex target set (in the complex plane) for the uncertain valuesp(jω, q). Consistent with value set analysis,P(ω) is assumed to be symmetric with respect to the nominalp(jω, 0). The uncertain parametersqi are taken to be zero-mean independent random variables with known support interval. For each uncertainty, the class ℱ is assumed to consist of density functions which are symmetric and nonincreasing on each side of zero. Then, for fixed frequencyω, the first theorem indicates that the probability thatp(jω, q) is inP(ω) is minimized by the uniform distribution forq. The second theorem, a generalization of the first, indicates that the same result holds uniformly with respect to frequency. Then probabilistic guarantees for robust stability are given in the third theorem. It turns out that in many cases, classical robustness margins can be far exceeded while keeping the risk of instability surprisingly small. Finally, for a much more general class of uncertainty structures, this paper also establishes the fact thatf* can be estimated by a truncated uniform distribution.

Extreme point results for robust stabilization of interval plants with first-order compensators

It has been shown previously that a first-order compensator robustly stabilizes an internal plant family if and only if it stabilizes all of the extreme plants. These extreme plants are obtained by considering all possible combinations for the extreme values of the numerator and denominator coefficients. In this work, the authors prove a stronger result, namely, that it is necessary and sufficient to stabilize only sixteen of the extreme plants. These sixteen plants are generated using the Kharitonov polynomials associated with the numerator and denominator. Furthermore, when additional information about the compensator is specified (sign of the gain and signs and relative magnitudes of the pole and zero), then, in some cases, it is necessary and sufficient to stabilize eight critical plants, while, in other cases, it is necessary and sufficient to stabilize twelve critical plants. >

New tools for robustness analysis

The author provides a personal perspective on research concerning Kharitonovs theorem of robustness analysis, as well as extensions and generalizations of this theorem. He discusses only a selected subset of the published literature which he feels provides the uninitiated reader with a coherent path through the rapidly expanding diversity of results. After a statement of Kharitonovs theorem and its limitations, he considers such issues as polytope stability problems and the edge theorem, and more general coefficient dependencies on perturbations.<<ETX>>

Stability of a polytope of matrices: counterexamples

While there have been significant breakthroughs for the stability of a polytope of polynomials since V.L. Kharitonovs (1978) seminal result on interval polynomials, for a polytope of matrices, the stability problem is considered far from completely resolved. Counterexamples are provided for three conjectures that are directly motivated by the results in the polynomial case. These counterexamples illustrate the fundamental differences between polynomial-stability and matrix-stability problems and indicate that some obvious lines of attack on the matrix polytope stability problem will fail. >

Polytopes of polynomials with zeros in a prescribed set

In the publication by A.C. Bartlett, C.V. Holot, and H. Lin (Proc. Amer. Contr. Conf., Minneapolis, MN, 1987) a fundamental result is established on the zero locations of a family of polynomials. It is shown that the zeros of a polytope P of nth-order real polynomials are contained in a simply connected set D if and only if the zeros of all polynomials along the edges of P are contained in D. The present authors are motivated by the fact that the requirement of simple connectedness of D may be too restrictive and applications such as dominant pole assignment and filter design where the separation of zeros is required. They extend the edge criterion of Bartlett et al. to handle any set D whose complement D/sup c/ has the following property: every point D in D/sup c/ lies on some continuous path which remains within D/sup c/ and is unbounded. This requirement is typically verified by inspection and allows for a large class of disconnected sets. >

Robustness margin need not be a continuous function of the problem data

For systems with structured real perturbations, it is shown that the robustness margin for stablhty can be a discon- tinuous function of the problem data

An extreme point result of robust stability of a diamond of polynomials

It is shown that a family of polynomials with real coefficients lying in a diamond is Hurwitz if and only if eight distinguished extreme polynomials are Hurwitz. For the case of complex coefficients, it is shown by a counterexample that no analogous extreme point result holds. >

Robust stability of a class of polynomials with coefficients depending multilinearly on perturbations

Necessary and sufficient conditions are given for robust stability of a family of polynomials. Each polynomial is obtained by a multilinearity perturbation structure. Restrictions on the multilinearity are involved, but, in contrast to existing literature, these restrictions are derived from physical considerations stemming from analysis of a closed-loop interval feedback system. The main result indicates that all polynomials in the family of polynomials have their zeros in the strict left half-plane if and only if two requirements are satisfied at each frequency. The first requirement is the zero exclusion condition involving four Kharitonov rectangles. The second requirement is that a specially constructed theta 0-parameterized set of 16 intervals must cover the positive reals for each theta epsilon (0,2 pi ). >