On Uniform Solvability of Parameter-Dependent Lyapunov Inequalities and Applications to Various Problems
On Uniform Solvability of Parameter-Dependent Lyapunov Inequalities and Applications to Various Problems
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DOI：
10.1137/040619417
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发表时间：
2006-09
期刊：
影响因子：
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通讯作者：
P. Krishnamurthy;F. Khorrami
中科院分区：
文献类型：
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作者：
P. Krishnamurthy;F. Khorrami
我们考虑的问题，找到一个共同的二次李雅普诺夫函数，以证明一个不一定有界的家庭的矩阵，其中包括设计自由的稳定性。一般来说，这可以被看作是选择一个家庭的控制器（或观测器）增益，使家庭的闭环系统矩阵承认一个共同的二次李雅普诺夫函数的问题。我们给出了矩阵族的各种结构的几个充分必要条件。具体来说，我们认为矩阵家庭，可以通过相似性变换从一个较低的Hessenberg结构。家庭的矩阵包含一个子集的对角矩阵，这是不变的设计自由度下，也被认为是因为它们发生在许多应用。明确给出了李雅普诺夫不等式一致可解的条件，并涉及矩阵中项的相对大小不等式。各种激励应用所获得的结果观测器和控制器的设计时变，切换，非线性系统的突出。
We consider the problem of finding a common quadratic Lyapunov function to demonstrate the stability of a not necessarily bounded family of matrices which incorporate design freedoms. Generically, this can be viewed as the problem of picking a family of controller (or observer) gains so that the family of closed-loop system matrices admits a common quadratic Lyapunov function. We provide several necessary and sufficient conditions for various structures of matrix families. Specifically, we consider matrix families which can be obtained through similarity transformations from a lower Hessenberg structure. Families of matrices containing a subset of diagonal matrices, which is invariant under the design freedoms, are also considered since they occur in many applications. The conditions for uniform solvability of the Lyapunov inequalities are explicitly given and involve inequalities regarding relative magnitudes of terms in the matrices. Various motivating applications of the obtained results to observer and controller designs for time-varying, switched, and nonlinear systems are highlighted.