Higher Order Derivatives of Lyapunov Functions for Stability of Systems with Inputs
Higher Order Derivatives of Lyapunov Functions for Stability of Systems with Inputs
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DOI：
10.1109/cdc40024.2019.9029302
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发表时间：
2019-12
期刊：
影响因子：
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通讯作者：
Shenyu Liu;D. Liberzon
中科院分区：
文献类型：
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作者：
Shenyu Liu;D. Liberzon
本文研究了通过检查李雅普诺夫函数的高阶导数来确定动力系统稳定性的另一种方法。系统可以是时不变的或时变的;在这两种情况下，我们都定义了有输入时的高阶导数。然后，我们证明了如果存在这些高阶导数的非负系数（除了0阶项的系数必须为正）的半负定线性组合，则系统是全局一致渐近稳定的.证明涉及到一阶微分关系的比较原理的重复应用。我们还表明，一个系统的输入，其辅助系统承认满足上述条件的李雅普诺夫函数是输入状态稳定的。
In this paper we study an alternative method for determining stability of dynamical systems by inspecting higher order derivatives of a Lyapunov function. The system can be time invariant or time varying; in both cases we define the higher order derivatives when there are inputs. We then claim and prove that if there exists a linear combination of those higher order derivatives with non-negative coefficients (except that the coefficient of the 0-th order term needs to be positive) which is negative semi-definite, then the system is globally uniformly asymptotically stable. The proof involves repeated applications of comparison principle for first order differential relations. We also show that a system with inputs whose auxiliary system admits a Lyapunov function satisfying the aforementioned conditions is input-to-state stable.