Lyapunov Functions for State Observers of Dynamic Systems Using Hamilton–Jacobi Inequalities

Angelo Alessandri

doi:10.3390/math8020202

Abstract

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.

Highlights

The construction of state observers for nonlinear systems is difficult in general when it is required to ensure the property of global stability for the estimation error
Based on this general framework, we focus on observers for polynomial nonlinear systems and the sum-of-squares paradigm to find such Lyapunov functions
We address the stability analysis of the error dynamics by using input-to-state stability (ISS) [1,2], where the disturbances are regarded as input, and the role of the state is played by the estimation error incurred by the observer

Summary

Introduction

The construction of state observers for nonlinear systems is difficult in general when it is required to ensure the property of global stability for the estimation error Such a proof is obtained by means of the Lyapunov method. The contribution of this paper lies in highlighting the motivations to rely on ISS to both analyze the stability and construct observers for nonlinear systems Toward this end, we deal with the problem in quite a general nonlinear setting, where the design of observers is reduced to the satisfaction of a Hamilton–Jacobi inequality [35,36]. R[ x ] p×q and Σ[ x ]m the ring of real matrix polynomials of size p × q and the set of the SOS polynomials of degree equal or less than m, respectively

Input-to-State Stability for State Observers

State Observers for Polynomial Systems

Numerical Results

Conclusions