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

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.

Stability Analysis of Systems with Fuzzy PI Controllers Applied to Electric Drives

“…Figure 10 presents the primary rule base with 3 × 3 r Thus, if it is assumed that the initial conditions are e0 = P an Equation (3), the command is did = P. If the speed control system ble, the state path can follow the following rules in order: ( 9), ( 6), steady state. If the system has initial condition specific to rules (9) state trajectory can only be traversed from these initial positions. example for an 'aperiodic linguistic behavior' would be (2), ( 8), ( base contains the vague state trajectories which, due to the integ controller, lead the system, from any point of the rule base from w the steady state rule.…”

“…The analysis of the stability of fuzzy control systems is approached in the literature by using various methods of analysis: utilization of a small-signal state-space linearized model [6] based on Lyapunov's direct method [7,8], using the Toeplitz matrices and their singular values [9], analyzing input-to-state stability and imposing some conditions [10], and applying Lyapunov's functions and theorem [11,12].…”

State Observation for Lipschitz Nonlinear Dynamical Systems Based on Lyapunov Functions and Functionals

“…We focus on a Lipschitz system given by two cascaded Van der Pol oscillators with the first and third state variable as outputs [26], i.e., ẋ = A x + f (x) + Dw y = C x + Ew with x ∈ R 4 , w ∈ R 4 , y ∈ R 2 , and…”

“…We computed this gain by solving (22) with Yalmip [37]: 1 is a noise-free simulation, while truncated random Gaussian noises are considered in the second run of Figure 2, which exhibits a bounded estimation error, as foreseen because of the ISS property. Generally speaking, it is not difficult to construct examples of system and observer, for which ISS does not hold (see, e.g., [25,26]). From this point of view, the considered class of the Lipschitz nonlinear system is more easily tractable, owing to the linear structure, which allows to apply ISS and derive stability conditions that are given by LMIs.…”

“…The proposed observers may be designed by using linear matrix inequalities (LMIs) [24]. As compared with recent results on ISS for the stability analysis of state observers [25,26], the novel contribution concerns the investigation of Lyapunov functional instead of Lyapunov functions. This goal is pursued by resorting to the decomposition of the dynamics of the error, which holds for systems with Lipschitz nonlinearities.…”