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LQ control of descriptor systems by cancelling structure at infinity

机译：通过取消无穷大结构来对描述符系统进行LQ控制

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This paper presents results concerning LQ control of arbitrary descriptor systems and a numerical algorithm for achieving it. The algorithm consists of two steps. The first step is, using the Silverman algorithm (orthogonal version), transformation of the problem of LQ control of descriptor system, into a problem of LQ control of a strictly proper system (with singular or non-singular new matrix Sigma(22)). If new Sigma(22) is singular, we perform the second step, i.e. solving the LQ singular control problem using again the Silverman algorithm. Regarding the existing results on the LQ singular control problem, we have new results: a counter example that this problem may not have a stable solution, most general solvability conditions (with or without stability), and a new numerical algorithm based on using orthogonal matrices. When stabilizability conditions are satisfied, the algorithm places simultaneously stable poles in the closed-loop system. We show that the traditional Riccati equation associated to this problem has not a solution (although the problem has), and introduce a generalization of Riccati equation, together with a method for its solving, as well as a generalization of spectral factorization of Popov function. Examples that cannot be solved by the existing methods are presented.
机译：本文介绍了有关任意描述符系统的LQ控制的结果以及实现该问题的数值算法。该算法包括两个步骤。第一步是使用Silverman算法（正交版本），将描述符系统的LQ控制问题转化为严格正确的系统的LQ控制问题（具有奇异新矩阵Sigma（22）） 。如果新的Sigma（22）是奇异的，我们执行第二步，即再次使用Silverman算法解决LQ奇异控制问题。关于LQ奇异控制问题的现有结果，我们得到了新结果：一个反例，说明该问题可能没有稳定的解决方案，大多数一般的可溶性条件（有或没有稳定），以及基于正交矩阵的新数值算法。当满足稳定性条件时，该算法会同时在闭环系统中放置稳定极点。我们证明了与此问题相关的传统Riccati方程并没有解决方案（尽管有问题），并介绍了Riccati方程的推广及其求解方法，以及Popov函数的谱分解。给出了现有方法无法解决的示例。

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