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     

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 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     

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.     

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     

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     

A dual result to Kharitonov's theorem

A dual problem to Kharitonov's problem, involving a diamond instead of a rectangle, is considered. The results show that a family of polynomials with coefficients varying in the diamond is strictly Hurwitz if and only if eight one-dimensional exposed edges of the diamond are strictly Hurwitz     

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     

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.     

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     

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     

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.     

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。     

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     

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     

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     

Necessary and sufficient condition for the existence of Kharitonov-like theorems

Kharitonov has shown that a family of interval polynomials is Hurwitz if and only if the vertex polynomials are Hurwitz. A necessary and sufficient condition to check the existence of Kharitonov-like theorems for other regions of interest in the complex plane is presented.     

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     

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     

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