Coprime Factorizations of Multivariate Rational Matrices

Coprime factorization is a well-known issue in one-dimensional systems theory, having many applications in realization theory, balancing, controller synthesis, etc. Generalization to systems in more than one independent variable is a delicate matter: First, several nonequivalent coprimeness notions for multivariate polynomial matrices have been discussed in the literature: zero, minor, and factor coprimeness. Here we adopt a generalized version of factor primeness that appears to be most suitable for multidimensional systems: a matrix is prime iff it is a minimal annihilator. After reformulating the sheer concept of a factorization, it is shown that every rational matrix possesses left and right coprime factorizations that can be found by means of computer algebraic methods. Several properties of coprime factorizations are given in terms of certain determinantal ideals. Date received: September 10, 1998. Date revised: February 25, 1999.     

Testing controllability and constructibility in discrete linear systems

A simple criterion for complete controllability and constructibility of discrete-time linear constant systems is given in terms of left and right coprimeness of certain polynomial matrices.     

New coprimeness over multivariable polynomial matrices and its application to control delay systems

This paper presents a new notion of coprimeness over multivariable polynomial matrices, where a single variable is given priority over the remaining variables. From a characterization through a set of common zeros of the minors, it is clarified that the presented coprimeness is equivalent to weakly zero coprimeness in the particular variable. An application of the presented coprimeness to control systems with non-commensurate delays and finite spectrum assignment is also presented. Because the presented coprimeness is stronger than minor coprimeness, non-commensurate delays are difficult to deal with in algebraic control theory. The “rational ratio condition” between delays, which can reduce non-commensurate delays to commensurate delays, proves to be both powerful and practical concept in algebraic control theory for delay systems.     

Proper stable Bezout factorizations and feedback control of linear time-delay systems†

This paper deals with the existence and construction of proper stable Bezout factorizations of transfer function matrices of linear time-invariant systems with commensurate time delays. Existence of factorizations is characterized in terms of spectral controllability (or spectral observability)of the co-canonical (or canonical) realization of the transfer function matrix. An explicit procedure for computing proper stable Bezout factorizations is given in terms of a specialized ring of pure and distributed time delays. This procedure is utilized to construct finite-dimensional stabilizing compensators and to construct feedback systems which assign the characteristic polynomial of the closed-loop system.     

Structured system models Part 3. An application to feedback compensator design

The controllability and observability indices are studied and applied to the feedback compensator design. The compensator design method uses polynomial matrices as system models. As the main result, a new algorithm is introduced for the construction of a first candidate for the feedback compensator. A new algorithm is also given for constructing a state-space model from polynomial matrix models. Such a realization is needed if there is originally only a polynomial matrix model for the system.     

不确定多通道奇异大系统分散鲁棒H∞控制

研究不确定多通道奇异系统的鲁棒分散H_∞控制问题,假定不确定性是时不变、范数有界,且存在于系统和控制输入矩阵中．主要考虑分散H_∞输出反馈控制问题．推导出了使不确定多通道奇异系统能鲁棒稳定且满足一定的性能指标的充分必要条件,没有等式约束的非线性矩阵不等式条件,采用两步同伦法迭代来求解非线性矩阵不等式(NMI),首先,通过逐步对控制器的系数矩阵加上结构限制,计算出当确定性不存在时的标称系统的分散H_∞控制器．然后,逐步改变标称系统分散控制器的系数,计算出不确定性参数存在时的分散鲁棒控制器．在每一阶段,每一次迭代过程中,通过交替固定NMI的一个变量,使NMI转变为线性矩阵不等式(LMI)．数值例子说明了本文提出的方法的有效性．