Constructing convex directions for stable polynomials

New constructions of convex directions for Hurwitz stable polynomials are obtained. The technique is based on interpretations of the phase-derivative conditions in terms of the sensitivity of the root-locus associated with the even and odd parts of a polynomial     

Robust strictly positive real synthesis for polynomial families of arbitrary order

1IntroductionTheproblemofestimatingtheparametersofmultiplesinusoidsinnoisehasre-ceivedconsiderableattentioninthepastthirtyyears,andalotofalgorithmshavebeenestablishedtosolvetheproblem.Amongallofthealgorithms,themaximumlikelihood(ML)estimatorisaprominentone[1],andseveralalgorithms,suchasANP[2]andIMP[3,4]arerelatedtoML.ThedrawbackoftheMLestimatorisitshighcomputationalcomplexity,sothealternatingprojection(AP)algorithm[5]wasdevelopedtomakeitsrealtimerealizationpossible.Butthefundamentaldefic…     

Robust stability of a family of disc polynomials

In his well-known theorem, V. L. Kharitonov established that Hurwitz stability of a set f1 of interval polynomials with complex coefficients (polynomials where each coefficient varies in an arbitrary but prescribed rectangle of the complex plane) is equivalent to the Hurwitz stability of only eight polynomials in this set. In this paper we consider an alternative but equally meaningful model of uncertainty by introducing a set fD of disc polynomials, characterized by the fact that each coefficient of a typical element P(s) in fD can be any complex number in an arbitrary but fixed disc of the complex plane. Our result shows that the entire set is Hurwitz stable if and only if the ‘center’ polynomial is stable, and the H _∞-norms of two specific stable rational functions are less than one. Our result can be readily extended to deal with the Schur stability problem and the resulting condition is equally simple.     

Stabilisation of matrix polynomials

A state feedback is proposed to analyse the stability of a matrix polynomial in closed loop. First, it is shown that a matrix polynomial is stable if and only if a state space realisation of a ladder form of certain transfer matrix is stable. Following the ideas of the Routh–Hurwitz stability procedure for scalar polynomials, certain continued-fraction expansions of polynomial matrices are carrying out by unimodular matrices to achieve the Euclid’s division algorithm which leads to an extension of the well-known Routh–Hurwitz stability criteria but this time in terms of matrix coefficients. After that, stability of the closed-loop matrix polynomial is guaranteed based on a Corollary of a Lyapunov Theorem. The sufficient stability conditions are: (i) The matrices of one column of the presented array must be symmetric and positive definite and (ii) the matrices of the cascade realisation must satisfy a commutative condition. These stability conditions are also necessary for matrix polynomial of second order. The results are illustrated through examples.     

Stability of polynomials with conic uncertainty

In this paper we describe a conic approach to the stability theory of uncertain polynomials. We present necessary and sufficient conditions for a conic setp ₀+K of polynomials to be Hurwitz stable (K is a convex cone of polynomials of degree n and degp ₀=n). As analytical tools we derive an edge theorem and Rantzer-type conditions for marginal stability (semistability). The results are applied to prove an extremal-ray result for conic sets whose cone of directions is given by an interval polynomial.The second author would like to thank the Deutsche Forschungsgemeinschaft (DFG) for its support during the writing of this paper.     

Comments on "A necessary condition for Hurwitz polynomials"

A condition proven in the above paper to be a necessary condition for Hurwitz polynomials is not new. It is part of a necessary and sufficient condition for a polynomial to be Hurwitz.     

Necessary conditions for Hadamard factorizations of Hurwitz polynomials

A property of Hurwitz polynomials is related with the Hadamard product. Garloff and Wagner proved that Hadamard products of Hurwitz polynomials are Hurwitz polynomials, and Garloff and Shrinivasan shown that there are Hurwitz polynomials of degree 4 which do not have a Hadamard factorization into two Hurwitz polynomials of the same degree 4. In this paper, we give necessary conditions for an even-degree Hurwitz polynomial to have a Hadamard factorization into two even-degree Hurwitz polynomials; such conditions are given in terms of the coefficients of the given polynomial alone. Furthermore, we show that if an odd-degree Hurwitz polynomial has a Hadamard factorization then a system of nonlinear inequalities has at least one solution.     

A new extension to Kharitonov's theorem

An extension to a well-known theorem due to Kharitonov is presented, Kharitonov's theorem gives a necessary and sufficient condition for all polynomials in a given family to be Hurwitz stable. In Kharitonov's theorem, the family of polynomials considered is obtained by allowing each of the polynomial coefficients to vary independently within an interval. Kharitonov's theorem shows that stability of this family of polynomials can be determined by looking at the stability of four specially constructed vertex polynomials. Kharitonov's theorem is extended to allow for more general families of polynomials and to allow a given margin of stability to be guaranteed for the family of polynomials     

Two necessary conditions for a complex polynomial to be strictlyHurwitz and their applications in robust stability analysis

The necessary conditions for a complex polynomial to be strictly Hurwitz are reviewed and rigorously proved. Both necessary conditions have been extended to cover nonmonic polynomials instead of monic polynomials. Also, based on these two results, some necessary conditions for an interval polynomial to be stable in terms of being strictly Hurwitz are obtained. They can be used to quickly determine the instability of a complex interval polynomial family. Finally, their application to the study of robust stability, in the case where coefficient perturbation intervals are functions of a single parameter, is briefly discussed     

Testing the robust Schur stability of a segment of complex polynomials

Given two Schur stable complex polynomials p 0 ( z ) and p 1 ( z ) of the same degree n, we present a procedure for testing if convex combinations of the form (1 - u ) p 0 ( z ) + u p 1 ( z ) are Schur stable for all u ] [0, 1]. The procedure consists in constructing a polynomial array, which corresponds to the process of extracting the greatest common divisor of two polynomials, and testing the absence of real zeros of a real u polynomial of degree 2 n for u ] (0, 1). Since the latter task can be finished by using the Sturm theorem, the proposed procedure for testing the robust Schur stability of a segment of complex polynomials is efficient in the sense that it accomplishes the test in a finite number of arithmetic operations. As the derivation given in this paper establishes a connection between our procedure and Bose's resultant method, and identifies an intrinsic simplification for the latter method, the presented procedure can be viewed as an efficient algorithmic implementation of Bose's resultant method for testing the robust Schur stability of complex segment polynomials.     

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.     

Robustness analysis of polynomials with polynomial parameterdependency using Bernstein expansion

This paper considers the robust stability verification of polynomials with coefficients depending polynomially on parameters varying in given intervals. Two algorithms are presented, both rely on the expansion of a multivariate polynomial into Bernstein polynomials. The first one is an improvement of the so-called Bernstein algorithm and checks the Hurwitz determinant for positivity over the parameter set. The second one is based on the analysis of the value set of the family of polynomials and profits from the convex hull property of the Bernstein polynomials. Numerical results to real-world control problems are presented showing the efficiency of both algorithms     

不变根分布的多项式的最大摄动界

已知不确定的特征多项式p(s,q),其系数依赖于参数向量q,一个富有意义的问题是: 可以允许q摄动多大而使摄动后的多项式仍保持标称多项式p(s,0)所具有的惯性(亦称根 分布)数?这就是所谓不变根分布的多项式的最大摄动界问题.本文将就仿射线性及仿射双 线性两情况给出上述问题的解答与算法.     

Properties of the entire set of Hurwitz polynomials and stabilityanalysis of polynomial families

It is proved in this paper that all Hurwitz polynomials of order not less than n form two simply connected Borel cones in the polynomial parameter space. Based on this result, edge theorems for Hurwitz stability of general polyhedrons of polynomials and boundary theorems for Hurwitz stability of compact sets of polynomials are obtained. Both cases of families of polynomials with dependent and independent coefficients are considered. Different from the previous ones, our edge theorems and boundary theorems are applicable to both monic and nonmonic polynomial families and do not require the convexity or the connectivity of the set of polynomials. Moreover, our boundary theorem for families of polynomials with dependent coefficients does not require the coefficient dependency relation to be affine     

A note on Routh-Hurwitz determinants and integral square errors

Let p(s) be a polynomial, all of whose roots lie in the left half plane. The purpose of this note is to show that when the transfer function p(0)/p(s) tracks unit step input, the integral square error (ISE) of the response has a simple expression in terms of the Routh—Hurwitz determinants for p. This enables us to generalize a result of Hall : he pointed out that for a quadratic polynomial with fixed natural frequency, the minimum ISE is attained for a damping factor of ½. We find the corresponding result for polynomials of any degree.     

A generalisation of Jury’s conjecture to arbitrary dimensions and its proof

In 1986 E. I. Jury conjectured by analogy to the theory of digital filters that a two-dimensional analog filter is BIBO stable if its transfer function is of the form H = 1/P, where P is a very strict Hurwitz polynomial (VSHP). In this article we prove a generalisation of Jury’s conjecture to r-dimensional analog filters (r ≥ 2) with proper transfer function H = Q/P, where the denominator P is a robustly stable polynomial, i.e., a strict Hurwitz polynomial which retains this property under small variations of its coefficients. In the bivariate case these polynomials are the VSHPs. Financial support by the Austrian FWF via project 18974 is appreciated.     

Real and complex polynomial stability and stability domain construction via network realizability theory

Network realzability theory provides the basis for a unified approach to the stability of a polynomial or a family of polynomials. In this paper conditions are given, in terms of certain decompositions of a given polynomial, that are necessary and sufficient for the given polynomial to be Hurwitz. These conditions facilitate the construction of stability domains for a family of polynomials through the use of linear inequalities. This approach provides a simple interpretation of recent results for polynomials with real coefficients and also leads to the formulation of corresponding results for the case of polynomials with complex coefficients.     

Sufficient conditions for Hurwitz polynomials and their application to the stability of a polytope of polynomials

Several sufficient conditions for the Hurwitz property of polynomials are derived by combining the existing sufficient criteria for the Schur property with bilinear mapping. The conditions obtained are linear or piecewise linear inequalities with respect to the polynomial coefficients. Making the most of this feature, the results are applied to the Hurwitz stability test for a polytope of polynomials. It turns out that checking the sufficient conditions at every generating extreme polynomial suffices to guarantee the stability of any member of the polytope, yielding thus extreme point results on the Hurwitz stability of the polytope. This brings about considerable computational economy in such a test as a preliminary check before going to the exact method, the edge theorem and stability test of segment polynomials.     

Schur stability and stability domain construction

A discrete version of Foster's reactance theorem is developed and, subsequently, used to delineate necessary and sufficient conditions for a given polynomial with complex or real coefficients to be of the Schur type. These conditions, obtained from the decomposition of a polynomial into its circularly symmetric and anti-circularly symmetric components, facilitate the construction of stability domains for a family of polynomials through the use of linear inequalities. These results provide the complete discrete counterpart of recent results for a family of polynomials which are required to be tested for the Hurwitz property.     

多项式族左扇区稳定的最大摄动区间

已知多项式ｐ（ｓ，ｒ）＝ｘ∑ｉ＝０ａｉ（ｒ）ｓ＾ｉ。其中诸系数ａｉ（ｒ）（ｉ＝０，１，…，ｎ）为参量ｒ的多项式函数且ｐ（ｓ，０）是左扇区稳定的多项式，本文给出ｒ的最大摄动区间以使对这区间中的所有ｒ．多项式ｐ（ｓ，ｒ）都是左扇区稳定的，