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Construction of Parameter-Dependent Lyapunov Functions for Robust Control

用于鲁棒控制的参数相关李亚普诺夫函数的构造

基本信息

批准号：
05650393
负责人：
MORI Takehiro
金额：
$ 1.41万
依托单位：
Kyoto Institute of Technology
依托单位国家：
日本
项目类别：
Grant-in-Aid for General Scientific Research (C)
财政年份：
1993
资助国家：
日本
起止时间：
1993 至 1994
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-05650393/
关键词：
Lyapunov function Parameter-dependence Robust stability Polytope of polynomials Quadratic form Discrete systems Common Lyapunov function エルミート行列

项目摘要

1) In the first year of the project term, investigations were on ways to construct Lyapunov functions for linear systems represented by polynomials with uncertain coefficient parameters. This leads to an expected result that there actually exist a parameter-dependent Lyapunov function for the given polytopic uncertain polynomial. The Lyapunov function is also a polytope of Lyapunov functions that correspond to the extreme polynomials of the polytope of polynomials. We see a one-to-one correspondence between parameters of Lyapunov functions and those of polynomials.2) With this result, we then pass to uncertain nonlinear systems, anticipating some parallel results with the linear case. Deliberation together with numerical experiences, however, led us to conclude that for nonlinear systems seach for parameter-dependent Lyapunov functions was harder than expected. We were thus obliged to reconsider our idea and began to look for the "opposite-end" problem, so-called common Lyapunov function problem. This is because common Lyapunov functions for several systems make us easy to handle uncertain nonlinear systems.3) In the final year, we set out to consider this problem for linear systems in the first place. We could successfully identify some classes of discerte-time linear systems that have common quadratic Lyapunov functions. This yields a sufficient condition for the This could also give a scope to explore results for uncertain nonlinear systems Lyapunof function approach.

1)在项目的第一年，研究了如何构建李雅普诺夫函数的线性系统表示的多项式与不确定系数参数。这导致了一个预期的结果，实际上存在一个参数依赖的李雅普诺夫函数给定的多面体不确定多项式。李雅普诺夫函数也是李雅普诺夫函数的多胞形，其对应于多项式多胞形的极值多项式。我们看到了李雅普诺夫函数的参数和多项式的参数之间的一一对应。2）有了这个结果，我们然后传递到不确定的非线性系统，预期一些平行的结果与线性情况。然而，考虑与数值经验，使我们得出结论，非线性系统的参数依赖的李雅普诺夫函数的搜索比预期的更难。因此，我们不得不重新考虑我们的想法，并开始寻找“相反的结束”的问题，所谓的共同李雅普诺夫函数问题。这是因为几个系统的共同李雅普诺夫函数使我们很容易处理不确定的非线性系统。3）在最后一年，我们首先考虑线性系统的这个问题。我们可以成功地识别几类具有共同的二次李雅普诺夫函数的离散时间线性系统。这也可以为不确定非线性系统的Lyapunof函数方法提供一个探索结果的范围。