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.     

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.     

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.     

Some necessary conditions for Hurwitz stability

In this paper, by using Newton and Marclaurin inequalities we obtain some necessary conditions for a polynomial with positive coefficients to be Hurwitz stable, which generalize the recent results obtained in Borobia and Dormido (Linear Algebra Appl. 338 (2001) 67) and the proof of the result in this paper is much shorter than that of in Borobia and Dormido (Linear Algebra Appl. 338 (2001) 67).     

Argument conditions for Hurwitz and Schur polynomials from networktheory

The monotonicity conditions, recently given, for the arguments of Hurwitz polynomials, as well as some of their associated polynomials, are, very simply, derived using well-established results in network theory. Corresponding results for Schur polynomials may also be obtained     

Necessary conditions for Schur-stability of interval polynomials

New bounds on the coefficient diameter of real Schur-stable interval polynomials are given using techniques from complex analysis. They can be used to unmask interval polynomials at low computational cost as being non-Schur-stable.     

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.     

A new model of the harmonic control based on Hadamard product

In terms of Hadamard product, a new model is proposed for the control of connection coefficients of the state variables of the systems. The control law to stabilize the systems via the regulations of connection coefficients is obtained via a Hadamard product involved bilinear matrix inequalities. This new control model may be of significant applications in many fields, especially may be of some special sense in the emergency control such as isolation and obstruction control.     

Necessary and Sufficient Conditions for Consensus in Third Order Multi-Agent Systems

We deal with a consensus control problem for a group of third order agents which are networked by digraphs. Assuming that the control input of each agent is constructed based on weighted difference between its states and those of its neighbor agents, we aim to propose an algorithm on computing the weighting coefficients in the control input. The problem is reduced to designing Hurwitz polynomials with real or complex coefficients. We show that by using Hurwitz polynomials with complex coefficients, a necessary and sufficient condition can be obtained for designing the consensus algorithm. Since the condition is both necessary and sufficient, we provide a kind of parametrization for all the weighting coefficients achieving consensus. Moreover, the condition is a natural extension to second order consensus, and is reasonable and practical due to its comparatively decreased computation burden. The result is also extended to the case where communication delay exists in the control input.      

Necessary and sufficient conditions for the Hurwitz and Schurstability of interval matrices

Establishes a set of new sufficient conditions for the Hurwitz and Schur stability of interval matrices. The authors use these results to establish necessary and sufficient conditions for the Hurwitz and Schur stability of interval matrices. The authors relate the above results to the existence of quadratic Lyapunov functions for linear time-invariant systems with interval-valued coefficient matrices. Using the above results, the authors develop an algorithm to determine the Hurwitz and the Schur stability properties of interval matrices. The authors demonstrate the applicability of their results by means of two specific examples     

Computation of the stability radius of a Hurwitz polynomial with diamond-like uncertainties

This paper considers the stability radius problem of Hurwitz polynomials whose coefficients have Hölder 1-norm-bounded uncertainties. We show that the solution to this problem demands the computation of the minimum of a piece-wise real-rational function ρ(λ), called the stability radius function. It is then shown that the calculations of ρ(λ) at the intersection points where ρ(λ) changes its representation and at the stationary points where ρ′(λ)=0 can be reduced to two sets of eigenvalue problems of matrices of the form H_β⁻¹H_γ, where both H_β and H_γ are frequency-independent Hurwitz matrices. Using root locus technique, we analyze this function further and prove that, in some special cases, the minimum of this function can be achieved only at the intersection points. Extensions of the eigenvalue approach to cover other robust stability problems are also discussed.     

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.     

Hurwitz判据的CAI软件设计

利用 MATL AB软件矩阵运算功能强和容易使用的特点 ,设计了 Hurwitz判据的 CAI软件。该 CAI软件比用 C、BASIC等语言设计更简单。     

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.     

鲁棒Hurwitz多项式的频域解释

本文用频域方法分析了鲁棒Hurwitz多项式,提出了一个判别Hurwitz多项式的充分必要条件,进而给出了Kharitonov定理和Hermitc-Bicbler定理的频域解释。对次数较低的多项式族,Kharitonov多项式的数目可以减少这一结论,本文也进行了讨论。     

On certain properties of Hurwitz determinants for interval polynomials

Consider the stable interval polynomialsF _n (z)=z ⁿ +a ₁ z ^n?1 +...+a _n?1 z+a _n wherea _i are real numbers, satisfying the inequalities α _i ≤a _i ≤β _i ,i=1,2, ...,n. In this paper we prove that mind _n (a) is the same foraεD andaεD ₁, whereD=[α₁, β₁]×[α₂, β₂]×...×[α _n , β _n ],D={(γ₁, γ₂,...γ _n )∈D:γ₁=α₁∨γ₁=β₁,... γ _n =α _n ∨γ _n =β _n }d _n (a)=detH, aεD, H—Hurwitz matrix for the polynomialF _n (z).     

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.     

Necessary and sufficient conditions for the stability of polynomials with linear parameter dependencies

In this paper we show that Kharitonov's theorem will provide a necessary and sufficient condition for the stability of a particular family of polynomials where the coefficients are real and linearly dependent on a set of uncertain parameters. The main results are obtained by restricting the linear map from the parameter to the coefficient space to contain the Kharitonov polynomials. The results are applied to both continous-time and discrete-time polynomials and future extensions are discussed.     

Inner-outer factorizations of right-invertible real-rational matrices

In this paper, the problems of doing inner-outer factorizations of right-invertible real-rational matrices are studied. By using some new properties of inner matrices, the problems can be solved as the general cases. A state-space algorithm for doing this factorization is also proposed.