SOS-Convex Lyapunov Functions with Applications to Nonlinear Switched Systems
SOS-Convex Lyapunov Functions with Applications to Nonlinear Switched Systems
复制标题
SOS-凸 Lyapunov 函数及其在非线性切换系统中的应用
DOI：
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
R. Jungers
中科院分区：
文献类型：
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作者：
Amir Ali Ahmadi;R. Jungers
我们引入了 sos-凸 Lyapunov 函数的概念来进行离散时间切换系统的稳定性分析。这些是具有凸性代数证明的多项式李亚普诺夫函数，并且可以通过半定规划有效地找到。我们证明了 sos-凸 Lyapunov 函数对于切换线性系统的稳定性分析是通用的（即必要和充分的）。另一方面，我们通过一个明确的例子表明，sos-凸 Lyapunov 函数的最小次数可以任意高于（非凸）多项式 Lyapunov 函数，其导出的 Minkowski 泛函也是有效的（非多项式）凸 Lyapunov 函数。在本文的第二部分中，我们证明，如果将切换系统定义为有限数量的非线性函数的凸包，那么非凸公共李雅普诺夫函数的存在并不是切换稳定性的充分条件，但凸公共李雅普诺夫函数的存在是。这显示了 sos-凸 Lyapunov 函数的计算机制的有用性，它可以直接应用于切换非线性系统，或其线性化，为非线性系统的凸包提供局部切换稳定性的证明。给出了一个例子，其中次数小于 14 的多项式无法提供任意切换下吸引区域的估计。
We introduce the concept of sos-convex Lyapunov functions for stability analysis of discrete time switched systems. These are polynomial Lyapunov functions that have an algebraic certificate of convexity, and can be efficiently found by semidefinite programming. We show that sos-convex Lyapunov functions are universal (i.e., necessary and sufficient) for stability analysis of switched linear systems. On the other hand, we show via an explicit example that the minimum degree of an sos-convex Lyapunov function can be arbitrarily higher than a (non-convex) polynomial Lyapunov function, whose induced Minkowski functional is also a valid (non-polynomial) convex Lyapunov function. In the second part of the paper, we show that if the switched system is defined as the convex hull of a finite number of nonlinear functions, then existence of a non-convex common Lyapunov function is not a sufficient condition for switched stability, but existence of a convex common Lyapunov function is. This shows the usefulness of the computational machinery of sos-convex Lyapunov functions which can be applied either directly to the switched nonlinear system, or to its linearization, to provide proof of local switched stability for the convex hull of the nonlinear system. An example is given where no polynomial of degree less than 14 can provide an estimate to the region of attraction under arbitrary switching.