Stability and performance analysis of saturated systems via partitioning of the virtual input space

A Generalized Piecewise Quadratic Lyapunov Function Approach to Estimating the Domain of Attraction of a Saturated System∗∗This work was supported in part by the National Natural Science Foundation of China under Grant Nos. 61221003 and 61273105.

“…A set of matrix inequality conditions with less conservativeness are established for guaranteeing the contractive invariance of a level set of this new Lyapunov function. Simulation results show that the estimate resulting from this new Lyapunov function is significantly larger than those in [3,9,12].…”

“…This Lyapunov function can be equivalently written as a quadratic function of the augmented state vector with a positive definite block diagonal matrix, and has been generalized in [3] to a piecewise quadratic Lyapunov function of the augmented state vector with a general positive definite matrix. Furthermore, by using this piecewise quadratic Lyapunov function and the partitioning of the virtual input space that enables some special properties of saturation functions to emerge, a larger estimate has been obtained in [12]. Because of the dependence on the state of the saturated input, the positive definiteness requirement of the matrix in the piecewise quadratic Lyapunov function is apparently conservative.…”

“…INTRODUCTION For a linear system under saturated linear feedback, whose open-loop poles are not all in the closed left-half plane, the analytic characterization of its domain of attraction, in general, seems impossible. An alternative approach is to estimate the domain of attraction with a contractively invariant set, which is a subset of the domain of attraction [1,2,3,5,7,8,9,11,12]. One such contractively invariant set is the polyhedral invariant set, which could result in a good approximation of the domain of attraction.…”

“…Recently, a class of quadratic functions of the augmented state vector that contains the system state and saturation/deadzone function of the input have been employed to estimate the domain of attraction of saturated linear systems [3,4,10,12]. An integral of the saturation/deadzone function of the state is added to a quadratic Lyapunov function to form a Lure-Postnikov type Lyapunov function [4,10].…”

Lyapunov stability for piecewise affine systems via cone-copositivity

“…Unfortunately the S-procedure is lossy in general. Several variants of this technique have been proposed in the more recent literature including sliding modes (Samadi & Rodrigues, 2011), attraction domain estimation (Li & Lin, 2015), and relaxed LMIs for discrete-time PWA systems (Hovd & Olaru, 2013).…”