Lyapunov Functions for State Observers of Dynamic Systems Using Hamilton–Jacobi Inequalities
Lyapunov Functions for State Observers of Dynamic Systems Using Hamilton–Jacobi Inequalities
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DOI：
10.3390/math8020202
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发表时间：
2020-02
影响因子：
1.1
通讯作者：
A. Alessandri
中科院分区：
文献类型：
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作者：
A. Alessandri
李雅普诺夫函数可以分析由常微分方程组描述的动力系统的稳定性，而不需要求解此类方程。对于非线性系统，设计一个Lyapunov函数通常不是一件容易的事情。本文给出了一种构造Lyapunov函数的方法来证明估计问题的稳定性。为此，我们鼓励采用输入-状态稳定性(ISS)来处理状态观测器在进行非线性连续时间系统状态估计时所涉及的估计误差。这种稳定性由满足Hamilton-Jacobi不等式的ISS Lyapunov函数来保证。基于这一一般框架，我们重点研究了多项式非线性系统的观测器以及寻找此类Lyapunov函数的平方和范型。
Lyapunov functions enable analyzing the stability of dynamic systems described by ordinary differential equations without finding the solution of such equations. For nonlinear systems, devising a Lyapunov function is not an easy task to solve in general. In this paper, we present an approach to the construction of Lyapunov funtions to prove stability in estimation problems. To this end, we motivate the adoption of input-to-state stability (ISS) to deal with the estimation error involved by state observers in performing state estimation for nonlinear continuous-time systems. Such stability properties are ensured by means of ISS Lyapunov functions that satisfy Hamilton–Jacobi inequalities. Based on this general framework, we focus on observers for polynomial nonlinear systems and the sum-of-squares paradigm to find such Lyapunov functions.