Algebraic design techniques for reliable stabilization
Algebraic design techniques for reliable stabilization
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DOI：
10.1109/tac.1982.1103086
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发表时间：
1982-10
影响因子：
6.8
通讯作者：
M. Vidyasagar;N. Viswanadham
中科院分区：
文献类型：
--
作者：
M. Vidyasagar;N. Viswanadham
本文研究了反馈镇定中的两个问题。第一类是同时稳定问题，可表述如下。给定植物G_{0}， G_{1}，…， G_{l}，是否存在一个单一的补偿器C使它们全部稳定？第二种是稳定补偿器的稳定，或者更一般地说，是“最不稳定”补偿器。给定一个对象G，我们想知道是否存在一个稳定补偿器C使G稳定；如果不是，那么任何稳定补偿器必须具有的右半位极（根据其麦克米伦度计数）的最小数量是多少？我们证明这两个问题在以下意义上是等价的。同时稳定l + 1个植物的问题可以简化为使用稳定补偿器同时稳定l个植物的问题，进而可以表述为以下纯代数问题。给定2l个矩阵A_{1}，…， A_{1}, B_{1}，…， B_{l}，其中A_{i}， B_{i}对所有i都是右素数，那么是否存在一个矩阵M使得A_{i} + MB_{i}对所有i都是幺模的？相反，使用稳定补偿器同时稳定l个对象的问题可以表示为同时稳定l + 1个对象的问题。给定一个右互素数对（A， B），确定是否存在一个M使得A + BM是单模的问题，是欧几里德定义域上矩阵除法算法问题的一个特例。我们给出了这个问题的答案，我们相信这个结果可能会引起一些独立的兴趣。我们证明了给定两个n \ * m的植物G_{0}和G_{1}，只要n或m大于1，我们可以同时稳定它们。相比之下，两个单输入单输出装置g0和g1的同时稳定性不是一般的。
In this paper we study two problems in feedback stabilization. The first is the simultaneous stabilization problem, which can be stated as follows. Given plants G_{0}, G_{1},..., G_{l} , does there exist a single compensator C that stabilizes all of them? The second is that of stabilization by a stable compensator, or more generally, a "least unstable" compensator. Given a plant G , we would like to know whether or not there exists a stable compensator C that stabilizes G ; if not, what is the smallest number of right half-place poles (counted according to their McMillan degree) that any stabilizing compensator must have? We show that the two problems are equivalent in the following sense. The problem of simultaneously stabilizing l + 1 plants can be reduced to the problem of simultaneously stabilizing l plants using a stable compensator, which in turn can be stated as the following purely algebraic problem. Given 2l matrices A_{1}, ..., A_{l}, B_{1}, ..., B_{l} , where A_{i}, B_{i} are right-coprime for all i , does there exist a matrix M such that A_{i} + MB_{i} , is unimodular for all i? Conversely, the problem of simultaneously stabilizing l plants using a stable compensator can be formulated as one of simultaneously stabilizing l + 1 plants. The problem of determining whether or not there exists an M such that A + BM is unimodular, given a right-coprime pair ( A, B ), turns out to be a special case of a question concerning a matrix division algorithm in a proper Euclidean domain. We give an answer to this question, and we believe this result might be of some independent interest. We show that, given two n \times m plants G_{0} and G_{1} we can generically stabilize them simultaneously provided either n or m is greater than one. In contrast, simultaneous stabilizability, of two single-input-single-output plants, g 0 and g 1 , is not generic.