Stability of Polytopic Polynomial Matrices

A unified framework for deterministic and probabilistic D-stability analysis of uncertain polynomial matrices

Many problems in systems and control theory can be formulated in terms of robust D-stability analysis, which aims at verifying if all the eigenvalues of an uncertain matrix lie in a given region D of the complex plane. Robust D-stability analysis is an NP-hard problem and many polynomial-time algorithms providing either sufficient or necessary conditions for an uncertain matrix to be robustly D-stable have been developed in the past decades. Despite the vast literature on the subject, most of the contributions consider specific families of uncertain matrices, mainly with interval or polytopic …

关于带有全称量词的多项式矩阵的正则定理

Positivstellensätze for polynomial matrices with universal quantifiers

本文研究了带有全称量词的多项式矩阵不等式约束下的正则定理，提出矩阵值正则定理及稀疏正则定理，扩展了经典多项式正则定理的应用范围。

Characteristic Polynomials of Symmetric Matrices over the Univariate Polynomial Ring

Viewing a bivariate polynomial f in R[x,t] as a family of univariate polynomials in t parametrized by real numbers x, we call f real rooted if this family consists of monic polynomials with only real roots. If f is the characteristic polynomial of a symmetric matrix with entries in R[x], it is obviously real rooted. In this article the converse is established, namely that every real rooted bivariate polynomial is the characteristic polynomial of a symmetric matrix over the univariate real polynomial ring. As a byproduct we present a purely algebraic proof of the Helton-Vinnikov Theorem which s…

具有受限子行列式的整数规划中的多项式矩阵

Polynomial Matrices in Integer Programming With Restricted Subdeterminants

研究与参数约束矩阵相关的离散优化问题，针对约束矩阵元素为单变量多项式且子行列式在特定集\(S\)中的情况，对识别和优化两个关键问题，通过特定集合\(S\)给出肯定回答，所涉矩阵有独立研究价值。

通过矩形块Hankel矩阵推广的Hurwitz稳定性准则用于非monic矩阵多项式

A generalized Hurwitz stability criterion via rectangular block Hankel matrices for nonmonic matrix polynomials

本文提出了一种基于矩形块Hankel矩阵的非monic矩阵多项式Hurwitz稳定性准则，通过列降阶方法扩展了传统稳定性判据。

The Probability of Primeness for Specially Structured Polynomial Matrices over Finite Fields with Applications to Linear Systems and Convolutional Codes

We calculate the probability that random polynomial matrices over a finite field with certain structures are right prime or left prime, respectively. In particular, we give an asymptotic formula for the probability that finitely many nonsingular polynomial matrices are mutually left coprime. These results are used to estimate the number of reachable and observable linear systems as well as the number of non-catastrophic convolutional codes. Moreover, we are able to achieve an asymptotic formula for the probability that a parallel connected linear system is reachable.