Kharitonov-like results in the space of Markov parameters

The largest stability box in the space of Markov parameters is determined. This result relies of Markov's theorem of determinants which gives a stability test (for a family of polynomials) involving only two (vertex) polynomials. V.L. Kharitonov's theorem (1978), which works for boxes in polynomial coefficient space, requires four polynomials to be stable     

On the stability of uncertain polynomials with dependentcoefficients

A sufficient condition is given for reducing the conservatism of the stability bounds for a family of polynomials with dependent coefficients, including nonlinear coefficients. It is also proved that if a finite family of stable polynomials has the same even part, then the polynomial with the even part and the odd part formed by adding any positive multiple of the even parts and odd parts, respectively, of the given family is also stable. Similar results holds if the given family of polynomials has the same odd part. A numerical example with nonlinear coefficients is given to illustrate the technique, and it is observed that the stability bounds obtained are larger than those acquired by Kharitonov's theorem     

On damping ratio of polynomials with perturbed coefficients

C.B. Soh and C.S. Berger (1988) derived a sufficient condition for a family of interval polynomials to have a damping ratio of φ using Kharitonov's theorem for complex polynomials. This paper points out that the transformations used by Soh and Berger to obtain the sufficient conditions also guarantee a simplification of Kharitonov's theorem for complex polynomials. That is, the number of required polynomials to be Hurwitz is half the number specified by Soh and Berger     

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     

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。     

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     

区间多项式族左扇区稳定鲁棒性及不变惯性定理

本文考虑区间多项式关于左扇区的稳定鲁棒性，用值集排零方法推导出强哈氏定理和不变惯性定理，并给出了构造强哈氏多项式及棱边族的简明算法。     

A note on Kharitonov's theorem

Kharitonov's theorem on the asymptotic stability of an equilibrium position of a family of systems of linear differential equations (1978) is connected with an elementary property of Bezoutian matrices associated with positive coefficient polynomials     

An alternative proof of Kharitonov's theorem

An alternative proof is presented of Kharitonov's theorem for real polynomials. The proof shows that if an unstable root exists in the interval family, then another unstable root must also show up in what is called the Kharitonov plane, which is delimited by the four Kharitonov polynomials. This fact is proved by using a simple lemma dealing with convex combinations of polynomials. Then a well-known result is utilized to prove that when the four Kharitonov polynomials are stable, the Kharitonov plane must also be stable, and this contradiction proves the theorem     

Kharitonov's theorem extension to interval polynomials which candrop in degree: a Nyquist approach

This paper studies the Hurwitz stability of interval polynomials which can drop in degree from a Nyquist point of view. These families make it possible to deal with system families taking into account different dynamic behaviors in the modeling of a plant. The behavior of the interval polynomials is studied, and it is proven that Kharitonov's theorem can be extended     

A Kharitonov-like theorem for interval polynomial matrices

As an extension of Kharitonov's theorem, robust stability of interval polynomial matrices is studied. Here a polynomial matrix is said to be stable if its determinant has all roots with negative real parts. The present paper shows that the robust stability of interval polynomial matrices is equivalent to that of the subclasses where each row (column) has only one element that involves Kharitonov edge polynomials and all the other elements take on one of the four Kharitonov vertex polynomials.     

On robust Hurwitz polynomials

In this note, Kharitonov's theorem on robust Hurwitz polynomials is simplified for low-order polynomials. Specifically, forn = 3, 4, and 5, the number of polynomials required to check robust stability is one, two, and three, respectively, instead of four. Furthermore, it is shown that forn geq 6, the number of polynomials for robust stability checking is necessarily four, thus further simplification is not possible. The same simplifications arise in robust Schur polynomials by using the bilinear transformation. Applications of these simplifications to two-dimensional polynomials as well as to robustness for single parameters are indicated.     

Robust stability analysis of polynomials with linearly dependentcoefficient perturbations

A computational tractable procedure for robust pole location analysis of uncertain linear time-invariant dynamical systems, whose characteristic polynomial coefficients depend linearly on parameter perturbations, is proposed. It is shown that, in the case of linearly dependent coefficient perturbations, the stability test with respect to any unconnected domain of the complex plane can be carried out, and the largest stability domain in parameter space can be computed by using only a quick test on a particular set of polynomials named vertex polynomials. The procedure requires only one sweeping function and simple geometrical considerations at each sweeping step. This leads to a very short execution time, as is shown in an example. A unification with Kharitonov's theory and edge theorem is also provided     

Robustness analysis of interval matrices based on Kharitonov'stheorem

This paper deals with the robustness analysis problem of interval matrices. Expansion of det (A+B) is used to get characteristic polynomials of two corresponding systems. Kharitonov's theorem is applied to test Hurwitz properties of the two polynomials. A new sufficient condition, for all eigenvalues of the interval matrix to lie in a damping-cone on complex plan, is derived. Illustrative examples are given     

A generalization of Kharitonov's four-polynomial concept for robuststability problems with linearly dependent coefficientperturbations

Kharitonov's four-polynomial concept is generalized to the case of linearly dependent coefficient perturbations and more general zero location regions. To this end, a specially constructed scalar function of a scalar variable is instrumental to the robustness analysis. The present work is motivated by two fundamental limitations of Kharitonov's theorem, namely: (1) the theorem only applies to polynomials with independent coefficient perturbations and (2) it only applies to zeros in the left-hand plane     

A root distribution criterion for interval polynomials

The problem of finding the conditions under which an interval polynomial has a given number of roots in the open left-half plane and the other roots in the open right-half plane, irrespective of the values of its coefficients, is considered. A simple criterion is provided to test interval polynomials for the root distribution invariance, viewed as an extension of Kharitonov's theorem. The goal is to provide an alternative theorem and then give an efficient means of checking the root distribution invariance     

一类多项式族的D鲁棒稳定性分析

本文提出了一种用于多项式族D鲁棒稳定性分析的新方法,与现有的同类方法相比,本文方法更易于理解和应用。这种方法适用于复平面上任何具有连续边界的D域,它是Kharitonov定理的扩展。     

Stability of a polytope of matrices: counterexamples

While there have been significant breakthroughs for the stability of a polytope of polynomials since V.L. Kharitonov's (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     

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.     

Damping ratio of polynomials with perturbed coefficients

Let a family of polynomials be P(s)=t ₀Sⁿ+t₁s ^n-1 . . .+t_n where O<α _j⩽t_j⩽β. Recently, C.B. Soh and C.S. Berger have shown that a necessary and sufficient condition for this equation to have a damping ratio of φ is that the 2ⁿ⁺¹ polynomials in it which have t_k=α_k or t_k=β_k have a damping ratio of φ. The authors derive a more powerful result requiring only eight polynomials to be Hurwitz for the equation to have a damping ratio of φ using Kharitonov's theorem for complex polynomials