On the Robust Stability of 2D Schur Polynomials

In this note, the problem of the robust stability for a two-dimensional (two-variable) Schur polynomial which is the characteristic polynomial of a discrete-time linear time-invariant system is investigated. A new approach based on the Rouché theorem is adopted. The extension to the robust stability for multidimensional (multivariable) polynomials is also provided. Interesting sufficient conditions for such robust stability are derived. A two-dimensional example is included to support the theoretical result.     

Robust Stability of Polynomials: New Approach

The problem of the robust stability of a Hurwitz polynomial which is the characteristic polynomial of a discrete-time linear time-invariant system is investigated. A new approach based on the Rouché theorem of classical complex analysis is adopted. An interesting sufficient condition for robust stability is derived. Three examples are included to support the theoretical result.     

基于LMIs的一类不确定线性切换系统的鲁棒控制

基于LMIs处理方法,研究了一类不确定线性切换系统在任意切换下的鲁棒控制问题.利用矩阵Schur补引理构造线性矩阵不等式,得到该系统的鲁棒稳定性的充要条件,同时也给出了在状态反馈下的鲁棒稳定性充要条件和在输出反馈下的充分条件.最后用数值例子对所得结果加以验证,说明了文中结果的正确性.     

时滞系统的鲁棒性与边界定理

本文首先注意到滞后型拟多项式与中立型拟多项式零点分布的根本区别，并指出一些通常的事实：如特征根对时滞的连续依赖性，所有特征根位于开左半复平面与稳定性的等价关系等对于中立型系统都不成立，然后建立了中立型拟多项式胞鲁棒稳定性的边界定理．依此将可公度时滞中立型系统族的鲁棒稳定性化为相应拟多项式胞所有边的Ｈｕｒｗｉｔｚ稳定性与其中立型项构成的子胞所有边的Ｓｃｈｕｒ稳定性的判定．最后给出了检验中立型胞鲁棒稳定性的一个有效的图解法．     

Multiplication of a Schubert polynomial by a Schur polynomial

Schur polynomials are a special case of Schubert polynomials. In this paper, we give an algorithm to compute the product of a Schubert polynomial with a Schur polynomial on the basis of Schubert polynomials. This is a special case of the general problem of the multiplication of two Schubert polynomials, where the corresponding algorithm is still missing. The main tools for the given algorithm is a factorization property of a special class of Schubert polynomials and the transition formula for Schubert polynomials.     

On the Optimum Radius of Robust Stability for Schur Polynomials

A recent paper (Ref. 1) established a new approach to estimate the robust stability radius of a Schur polynomial. This note points out that the approach given in Ref. 1 is not correct and also gives a counterexample to the main result of Ref. 1.     

Robustness bounds of Hurwitz and Schur polynomials

In this paper, the problem of robustness bounds of Hurwitz and Schur polynomials is addressed. For weightedL ₂-norm perturbations of a Hurwitz polynomialp(s) or a Schur polynomialp(z), a new method is developed for calculating the maximal perturbation bound under which stability is preserved. We show that such a robustness bound is related to the minimum of a rational function. The new method is superior to the previous one developed by Soh, Berger, and Dabke in Ref. 1. Our approach also provides solutions for the perturbation polynomial p(s) or p(z) with minimal coefficient norm which causep(s)+p(s) orp(z)+p(z) to be unstable.     

Schur coefficients in several variables

In the paper we study weakly continuous Schur-class-valued maps and their associated Schur coefficient families, that we call functional Schur coefficients. A case of special interest is the family of the “slices” through the polytorus of an n-variable function in the unit ball of H^∞(Dn), which is shown to be a weakly continuous map from the polytorus into the Schur class. The continuity properties of its functional Schur coefficients are used to characterize the rational inner functions in the polydisk algebra. As a consequence we obtain extensions in several variables of the Schur-Cohn test on zeroes of polynomials. This provides in particular a necessary and sufficient condition of stability for multi-dimensional AR filters.     

Schur and Schubert polynomials as Thom polynomials—cohomology of moduli spaces

The theory of Schur and Schubert polynomials is revisited in this paper from the point of view of generalized Thom polynomials. When we apply a general method to compute Thom polynomials for this case we obtain a new definition for (double versions of) Schur and Schubert polynomials: they will be solutions of interpolation problems.     

Application of semi-definite programming to robust stability of delay systems

We study here robust stability of linear systems with several uncertain incommensurate delays, more precisely the property usually called delay-dependent stability. The main result of this paper consists in establishing that the latter is equivalent to the feasibility of some Linear Matrix Inequality (LMI), a convex optimization problem whose numerical solution is well documented.The method is based on two main techniques:

• use of Padé approximation to transform the system into some singularly perturbed finite-dimensional system, for which robust dichotomy has to be checked;

• recursive applications of Generalized Kalman–Yakubovich–Popov (KYP) lemma to characterise by an LMI the previous property.

Improved Optimum Radius for Robust Stability of Schur Polynomials

An approach based on the Rouché theorem was introduced in the literature to compute the optimum radius for robust stability of Schur polynomials. Later an attempt was made to improve the result, but it was shown to be incorrect. The purpose of this note is to show that an improved optimum radius still can be obtained by modifying the proposed method. The result of this note can be easily extended to the multidimensional cases.     

Applications of optimization methods to robust stability of linear systems

We study the robust stability problem for a family of polynomials. We allow for all the coefficients of the polynomials to be affinely perturbed, where the size of the perturbation is measured by an arbitrary convex function. We apply optimization techniques, and in particular convex duality methods, to derive simple formulas for the stability radius, to find a minimal perturbation which destroys stability, and to obtain necessary and sufficient conditions for robust stability. Our framework is general enough to cover many applications. As special cases, we obtain many results recently reported in the literature.The work of the first author was partially supported by AFOSR Grant 91-008 and NSF Grant DMS-92-01297.     

Combinatorial Aspects of the K-Theory of Grassmannians

In this paper we study Grothendieck polynomials indexed by Grassmannian permutations, which are representatives for the classes corresponding to the structure sheaves of Schubert varieties in the K-theory of Grassmannians. These Grothendieck polynomials are nonhomogeneous symmetric polynomials whose lowest homogeneous component is a Schur polynomial. Our treatment, which is closely related to the theory of Schur functions, gives new information about these polynomials. Our main results are concerned with the transition matrices between Grothendieck polynomials indexed by Grassmannian permutations and Schur polynomials on the one hand and a Pieri formula for these Grothendieck polynomials on the other.     

参数不确定非线性组合大系统的鲁棒稳定性及参数鲁棒域的估计(英)

本文研究了参数不确定非线性组合大系统．首先利用现代微分几何理论将系统化为由线性子系统互联而成的组合大系统，然后给出了参数不确定非线性组合大系统鲁棒稳定性的若干判据．最后通过一个例子说明了本文的结论及参数鲁棒域的估计方法．     

不确定二维多项式族的稳定性检验集合

不确定二维多项式族,包括多胞形二维多项式族,区间二维多项式族,钻石形二维多项式族,其鲁棒稳定性可由其棱边多项式的稳定性确定.如果二维多项式族的不确定参数较多,被检验棱边多项式的数目可能出现所谓组合爆炸问题.为解决这一问题,提出一检验集合,该集合根据棱边二维多项式的凸方向来构造,仅对属于该检验集合棱边二维多项式进行稳定性检验,从而大大地减少被检验的棱边二维多项式数目.给出一例来说明这种新方法的可行性.     

A system-theoretic framework for a wide class of systems II: Input-to-output stability

In this work characterizations of the notion of Weighted Input-to-Output Stability (WIOS) for a wide class of systems with disturbances are given. Particularly, for systems with continuous dependence of the solution on the initial state and the input, the WIOS property is shown to be equivalent to robust forward completeness from the input and robust global asymptotic output stability for the corresponding input-free system.     

Some characteristic properties of the weighted particular Schur polynomial mean

This paper provides some characteristic properties of the weighted particular Schur polynomial mean of several variables. In addition, an elementary proof of an important inequality involving the weighted particular Schur polynomial mean is given. Various related results involving a family of the Schur polynomials, symmetric polynomials, and other associated polynomials, together with the potential for their applications, are also considered.     

一类不确定线性切换系统二次鲁棒稳定性的充分条件

本文研究了一类不确定线性切换系统的二次鲁棒稳定性问题.首先利用矩阵集的严格完备性设计切换律,导出了二次鲁棒稳定的充分条件.同时得到了在任意切换策略下,当矩阵集的所有矩阵为负定时保证切换系统二次鲁棒稳定性.在适当的假设下,这些条件可以表示为矩阵不等式.最后,用数值例子对所得结果加以阐明,说明了文中结果的正确性.     

Completion problems forj pq-inner functions. II

This paper is a continuation of [AFK]. The notations used there will be preserved. We will consider completion problems forj _pq-inner polynomials which have a prescribed structure, namely for such matrix polynomials which have a prescribed structure, namely for such matrix polynomials which can be considered as suitably normalized resolvent matrix of an appropriate non-degenerate matricial Schur problem.     

Minor summation formula and a proof of Stanley’s open problem

In the open problem session of the FPSAC’03, R.P. Stanley gave an open problem about a certain sum of the Schur functions. The purpose of this paper is to give a proof of this open problem. The proof consists of three steps. At the first step we express the sum by a Pfaffian as an application of our minor summation formula (Ishikawa and Wakayama in Linear Multilinear Algebra 39:285–305, 1995). In the second step we prove a Pfaffian analogue of a Cauchy type identity which generalizes Sundquist’s Pfaffian identities (J. Algebr. Comb. 5:135–148, 1996). Then we give a proof of Stanley’s open problem in Sect. 4. At the end of this paper we present certain corollaries obtained from this identity involving the Big Schur functions and some polynomials arising from the Macdonald polynomials, which generalize Stanley’s open problem.