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

The Department of Mechanical, Energetics, Management, and Transportation Engineering (DIME), University of Genoa, Via Opera Pia 15, 16145 Genoa, Italy
Mathematics 2020, 8(2), 202; https://doi.org/10.3390/math8020202
Received: 5 January 2020 / Revised: 29 January 2020 / Accepted: 1 February 2020 / Published: 6 February 2020
(This article belongs to the Section Engineering Mathematics)
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.
Keywords: Lyapunov function; input-to-state stability; Hamilton–Jacobi inequality Lyapunov function; input-to-state stability; Hamilton–Jacobi inequality
MDPI and ACS Style

Alessandri, A. Lyapunov Functions for State Observers of Dynamic Systems Using Hamilton–Jacobi Inequalities. Mathematics 2020, 8, 202.

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