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.     

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     

Some results on robust stability of discrete-time systems

The theorem of Kharitonov on the Hurwitz property of interval families of polynomials cannot be extended, in genera, to obtain sufficient conditions for the stability of families of characteristic polynomials of discrete-time systems. Necessary and sufficient conditions for this stability problem are given. Such conditions naturally give rise to a computationally efficient stability test which requires the solution of a one-parameter optimization problem and which can be considered as a counterpart to the Kharitonov test for continuous-time systems. At the same time, the method used to derive the stability conditions provides a procedure for solving another stability robustness problem, i.e. the estimation of the largest domain of stability with a rectangular box shape around given nominal values of the polynomial coefficients     

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.     

Stability of a family of polynomials with coefficients bounded in adiamond

The robust stability property is examined for family of nth-order real polynomials where the coefficients are bounded within a diamond in the (n+1)-dimensional space. It is shown that such a family of polynomials is Hurwitz if and only if four specially selected edge polynomials are Hurwitz     

Stability analysis of polynomials with coefficients in disks

The aim of this note is to report results on the stability of a class of polynomials from the small gain theorem point of view. The authors consider families of polynomials whose coefficients lie in closed circular disks around their nominal values. Various measures of variation of polynomial coefficients around their nominal value are considered and in each case necessary and sufficient conditions are presented for stability of the resulting family of polynomials. The stability region could be any closed region of the complex plane. Based on similar ideas of small gain, the authors also provide sufficient conditions for testing the stability of systems with commensurate time delays, and for two-dimensional type systems. These conditions become both necessary and sufficient in some special cases. All tests are easy to implement and require checking the stability of a matrix (or equivalently checking the stability of the central polynomial) and evaluation of a norm     

Hurwitz testing sets for parallel polytopes of polynomials

In considering robustness of linear systems with uncertain paramenters, one is lead to consider simultaneous stability of families of polynomials. Efficient Hurwitz stability tests for polytopes of polynomials have earlier been developed using evaluations on the imaginary axis. This paper gives a stability criterion for parallel polytopes in terms of Hurwitz stability of a number of corners and edges. The ‘testing set’ of edges and corners depends entirely on the edge directions of the polytope, hence the results are particularly applicable in simultaneous analysis of several polytopes with equal edge directions.It follows as a consequence, that Kharitonov's four polynomial test for independent coefficient uncertainties is replaced by a test of 2q polynomials, when the stability region is a sector Ω = { e^iv | > 0, rπ/q < | v | ≤ π } and r/q is a rational number.     

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.     

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.     

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.     

Robustness analysis: a case study

The stability test of polynomials whose coefficients depend multilinearly on interval parameters is considered. The authors describe and compare four brute-force solution approaches. These are eigenvalue calculation, zero exclusion from a specified value set, algebraic tests of real and complex Hurwitz roots, and the parameter space method. They are applied to a simple example with two parameters and third-order polynomial. An interesting feature of the example is that it can have an isolated unstable point. The example may be useful as a benchmark for future approaches to the multilinear problem. All four methods are shown to be feasible for the simple example, but they require effort     

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     

A Note on Robust Stability Analysis of Fractional Order Interval Systems by Minimum Argument Vertex and Edge Polynomials

By using power mapping (s=vm), stability analysis of fractional order polynomials was simplified to the stability analysis of expanded degree integer order polynomials in the first Riemann sheet. However, more investigation is needed for revealing properties of power mapping and demonstration of conformity of Hurwitz stability under power mapping of fractional order characteristic polynomials. Contributions of this study have two folds:Firstly, this paper demonstrates conservation of root argument and magnitude relations under power mapping of characteristic polynomials and thus substantiates validity of Hurwitz stability under power mapping of fractional order characteristic polynomials. This also ensures implications of edge theorem for fractional order interval systems. Secondly, in control engineering point of view, numerical robust stability analysis approaches based on the consideration of minimum argument roots of edge and vertex polynomials are presented. For the computer-aided design of fractional order interval control systems, the minimum argument root principle is applied for a finite set of edge and vertex polynomials, which are sampled from parametric uncertainty box. Several illustrative examples are presented to discuss effectiveness of these approaches.      

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.     

Robust stability of polynomials with multilinearly dependentcoefficient perturbations

Analysis of the robust stability of a polynomial with multilinearly dependent coefficient perturbations is presented in this note. Some sufficient conditions for forming a convex polygon with the value set of the polynomials with multilinearly dependent coefficient perturbations are obtained. A zero-exclusion algorithm is then given to determine the D-stability of such polynomials. The well known Kharitonov's theorem and the edge theorem for stability analysis can be included as special eases of the authors' conclusions     

Matrices, polynomials, and linear time-variant systems

Some recent developments in the applications of matrices to problems arising in linear systems theory are described. It is shown how companion form matrices can be used to provide a unified framework for dealing with the qualitative analysis of polynomials, including such problems as determination of greatest common divisors. Relationships to classical theorems involving bigradients and to controllability are discussed. When applied to the determinantal stability criteria of Hurwitz and others, the companion matrix approach results in minors of half the original orders. The problem of minimal realization of a transfer function matrix is dealt with in terms of polynomial matrices using methods due to Rosenbrock, and links with the results on scalar polynomials are demonstrated. Some applications of Lyapunov theory to systems in state-space form are briefly reviewed.     

Invariance of the strict Hurwitz property for bivariate polynomialsunder coefficient perturbations

A result is given that enables one to determine the interval within which the coefficients of a real bivariate polynomial might be allowed to vary, centered around their respective nominal values, so that the strict Hurwitz property remains invariant. These results are suitable for generalization to multivariate polynomials with complex coefficients. Applications can be found in branches of network and control theory concerned with robust stability analysis. Such analysis should be of interest because of the established role of multivariate realizability theory in systems research and also because of the increasing importance that is being attached to the design of multidimensional feedback control systems     

有限状态变系数离散系统的稳定性检验

提出了系统参数独立于系统节拍变化的有限状态变系数离散系统模型、稳定性检验定理及稳定性检验的快速算法.这类系统的稳定的充分必要条件是其传递函数的分母多项式为有限Schur多项式簇.提出了改进的变系数Schur检验表,使系统的稳定性检验过程简化,计算量大为减少,有限次运算即可完成.     

An elementary proof of Kharitonov's stability theorem withextensions

Gives an elementary proof of Kharitonov's theorem using simple complex plane geometry without invoking the Hermite-Bieler theorem. Kharitonov's theorem is a stability result for classes of polynomials defined by letting each coefficient vary independently in an arbitrary interval. The result states that the whole class is Hurwitz if and only if four special, well-defined polynomials are Hurwitz. The paper also gives elementary proofs of two previously known extensions: for polynomials of degree less than six, the requirement is reduced to fewer than four polynomials; and the theorem is generalized to polynomials with complex coefficients