Hurwitz stability of weighted diamond polynomials

In this note we consider real diamond polynomials of degree n whose roots are required to lie in the open left half plane. A minimal number of exposed edges whose stability guarantees the stability of the entire polynomial family is selected. A condition under which the stability of the vertices of the diamond guarantees the stability of the entire polynomial family is provided. Some illustrative examples are given.     

On the complete instability of interval polynomials

In this paper, we study “complete instability” of interval polynomials, which is the counterpart of classical robust stability. That is, the objective is to check if all polynomials in the family are unstable. If not, a subsequent goal is to find a stable polynomial. To this end, we first propose a randomized algorithm which is based on a (recursive) necessary condition for Hurwitz stability. The second contribution of this paper is to provide a probability-one estimate of the volume of stable polynomials. These results are based on a combination of deterministic and randomized methods. Finally, we present two numerical examples and simulations showing the efficiency of the proposed methodology for small and medium-size problems.     

Some remarks on the robust Schur stability analysis of interval polynomials

The Schur stability analysis of an interval polynomial family can be quickly performed through a unique, suitably defined extreme polynomial. The purpose of this article is to provide some improvements with respect to the actually existing methods based on this approach.     

On stability of a weighted diamond of real quasi-polynomials

This paper deals with the stability problem for a weighted diamond of real quasi-polynomials. We show that under certain conditions on the weights and coefficients in the exponents, the stability of the weighted diamond follows from the stability of eight one-parameter families (edges) of quasi-polynomials. In order to check the diamond for stability, it is sufficient to examine only eight one-parameter families of quasi-polynomials in contrast to the case of a rectangle of quasi-polynomials, which requires checking the stability of an exponential number of one-parameter families of quasi-polynomials     

On Hurwitz stability for families of polynomials

The robustness of a linear system in the view of parametric variations requires a stability analysis of a family of polynomials. If the parameters vary in a compact set $A$, then obtaining necessary and sufficient conditions to determine stability of the family ${\mathfrak{F}}_A$ is one of the most important tasks in the field of robust control. Three interesting classes of families arise when $A$ is a diamond, a box or a ball of dimension $n+1$. These families will be denoted by ${\mathfrak{F}}_{D_n}$, ${\mathfrak{F}}_{B_n}$, and ${\mathfrak{F}}_{S_n}$, respectively. In this article, a study is presented to contribute to the understanding of Hurwitz stability of families of polynomials ${\mathfrak{F}}_A$. As a result of this study and the use of classical results found in the literature, it is shown the existence of an extremal polynomial $f({\alpha }^{\ast },x)$ whose stability determines the stability of the entire family ${\mathfrak{F}}_A$. In this case $f({\alpha }^{\ast },x)$ comes from minimizing determinants and in some cases $f({\alpha }^{\ast },x)$ coincides with a Kharitonov's polynomial. Thus another extremal property of Kharitonov's polynomials has been found. To illustrate our approach, it is applied to families such as ${\mathfrak{F}}_{D_n}$, ${\mathfrak{F}}_{B_n}$, and ${\mathfrak{F}}_{S_n}$ with $n\le 5$. The study is also used to obtain the maximum robustness of the parameters of a polynomial. To exemplify the proposed results, first, a family ${\mathfrak{F}}_{D_n}$ is taken from the literature to compare and corroborate the effectiveness and the advantage of our perspective. Followed by two examples where the maximum robustness of the parameters of polynomials of degree 3 and 4 are obtained. Lastly, a family ${\mathfrak{F}}_{B_5}$ is proposed whose extreme polynomial is not necessarily a Kharitonov's polynomial. Finally, a family ${\mathfrak{F}}_{S_3}$ is used to exemplify that if the boundary of $A$ is given by a polynomial equation in several variables, the number of candidates to be an extremal polynomial is finite.     

On the stability radius of a Schur polynomial

The robust Schur stability of a polynomial with uncertain coefficients will be investigated. The stability hypersphere for such polynomials will be determined in terms of Tshebyshev Polynomials.     

On the strict- and wide-sense stability robustness of uncertain systems: application of a new frequency criterion

In this paper, we discuss a new frequency-domain graphical approach for robust stability of uncertain systems. This method is based on Mikhailov's criterion and it requires the plotting of only one Hodograph and evaluation of two boundary conditions. Moreover, this approach is extended to a more general robust stability sense, i.e., extended and wide-sense Hurwitz stability, which may be applied to a number of robustness problems such as positivity, nonnegativity, spectral analysis, and array processing.     

An efficient algorithm for checking the robust stability of a polytope of polynomials

An efficient algorithm for checking the robust stability of a polytope of polynomials is proposed. This problem is equivalent to a zero exclusion condition at each frequency. It is shown that such a condition has to be checked at only afinite number of frequencies. We formulate this problem as aparametric linear program which can be solved by the Simplex procedure, with additional computations between steps consisting of polynomial evaluations and calculation of positive polynomial roots. Our algorithm requires a finite number of steps (corresponding to frequency checks) and in the important case when the polytope of parameters is a hypercube, this number is at most of orderO(m ³ n ²), wheren is the degree of the polynomials in the family andm is the number of parameters. Supported by NASA under Contract No. NCC2-477 and by a Charles Powell Foundation Grant.     

On Hadamard powers of polynomials

We introduce the concept of the Hadamard power of a polynomial formed by real powers of its coefficients. We show that the Hadamard power of a Hurwitz polynomial remains Hurwitz.     

Some counterexamples in robust stability theory

This note deals with the problem of determining if a linear system whose characteristics polynomial depends multilinearly on n independent uncertain real parameters Δ_i, I = 1,…,n, is robustly stable. It is shown by example that a polynomial in n variables may have a unique real root, and that this observation disposes of several natural conjectures in robust stability theory. In particular, we show that, in a certain sense, there are no ‘edge’ or ‘m-dimensional face’ Kharitonov-like theorems for the general multilinear case. The result holds even when restricted to that subset of multilinear functions which can be written in the form f(Δ₁,…, Δ_n) = det(I + diag(Δ₁,…,Δ_n)M) for some complex matrix M.     

On the stability of a general polynomial in the space of Markov parameters

Markov parameters and the associated stability criterion were first introduced for continuous-time real polynomials. Recently, robust stability of such polynomials was considered in Markov parameters space, where efficient robust stability tests were obtained based on the Markov theorem. This has motivated the authors to extend the above idea to more general types of polynomials, and develop Markov parameters and the associated stability criterion for complex continuous-time as well as real and complex discrete-time polynomials. Moreover, for each polynomial type, we present compact relations in order to recover the coefficients of a polynomial corresponding to a given set of Markov parameters. The stability results presented here may be useful for 1-D and 2-D filtering and control applications.     

On delay-dependent stability conditions

It is quite common in stability analysis of time-delay systems to make a special transformation of the system under investigation in order to obtain stability conditions which depend on the values of the delays. In this note we discuss additional conditions for stability and robust stability of the transformed system which do not appear when the original system is considered.     

Guardian maps and the generalized stability of parametrized families of matrices and polynomials

The generalized stability of families of real matrices and polynomials is considered. (Generalized stability is meant in the usual sense of confinement of matrix eigenvalues or polynomial zeros to a prescribed domain in the complex plane, and includes Hurwitz and Schur stability as special cases.) Guardian maps and semiguardian maps are introduced as a unifying tool for the study of this problem. These are scalar maps which vanish when their matrix or polynomial argument loses stability. Such maps are exhibited for a wide variety of cases of interest corresponding to generalized stability with respect to domains of the complex plane. In the case of one- and two-parameter families of matrices or polynomials, concise necessary and sufficient conditions for generalized stability are derived. For the general multiparameter case, the problem is transformed into one of checking that a given map is nonzero for the allowed parameter values. This research was supported in part by the National Science Foundation’s Engineering Research Centers Program, NSFD CDR 8803012, and was also supported by the NSF under Grants ECS-86-57561, DMC-84-51515, and by the Air Force Office of Scientific Research under Grant AFOSR-87-0073.     

On robust stability of linear systems

In this paper the robust stability of linear continuous and discrete systems in the frequency domain is considered giving necessary and sufficient conditions. Based on that, the method of Tsypkin and Polyak (1991) is proved in a simple way. Extensions to discrete systems in the z-domain and sampled-data systems in the delta-domain are done using two ways of parametrization.     

A note on some differences between real and complex stability radii

Real and complex stability radii measure the robustness of stable linear systems under real and complex parameter perturbations, respectively. In this note we point out some basic differences between them. In particular, we investigate their different behaviour with respect to system interconnections and changes in system data.     

On robust stability of multivariate interval plants

In this work we consider the stability property of the feedback connection of a multivariate interval plant with a fixed compensator. Copyright © 2003 John Wiley & Sons, Ltd.     

On delay-dependent stability conditions for time-varying systems

In this paper, some recent results on additional dynamics for transformed time-delay systems are extended to the case of time-varying systems. Special equations which describe these dynamics are derived. Additional restrictions on stability and robust stability imposed by the transformations are obtained.     

Sufficient algebraic conditions for stability of cones of polynomials

In this paper a sufficient condition for a cone of polynomials to be Hurwitz is established. Such condition is a matrix inequality, which gives a simple algebraic test for the stability of rays of polynomials. As an application to stable open-loop systems, a cone of gains c such that the function u=−kc^Tx is a stabilizing control feedback for all k>0 is shown to exist.