Robust Observer-Based Feedback Control for Lipschitz Singularly Perturbed Systems

Abstract

The observer-based feedback control for singularly perturbed systems (SPSs) with Lipschitz constraint is addressed. A sufficient condition, independent of the perturbation parameter, for a full-order observer is presented in terms of linear matrix inequality (LMI) such that observation error is exponentially stable for all sufficiently small perturbation parameters. Then, for observer-based feedback control, a proper controller is constructed to guarantee the input-to-state stability of the system with regard to the observation error. Considering the convergence of observation error, the stability of the system can be obtained based on the input-to-state stability property. It is shown that the proposed method is simple and easy to operate. Moreover, the upper bound of the small perturbation parameter for stability of systems can be explicitly estimated with a workable computation way. Finally, two numerical examples show the effectiveness of the proposed method.

Robustness analysis of nonlinear observers for the slow variables of singularly perturbed systems

“…An observer for spring-mass-damper systems with singularly perturbed structure is given in Saha and Valasek 6,7 ; however, those results apply to a very specific class of systems and observers. Other results on observer design for nonlinear singularly perturbed systems are available in Wang and Liu 20 , and Darrogheh et al 21 .…”

Globalstability of the error dynamics of an observer designed for the slow states of a singularly perturbed system

“…It is shown a local convergence of the estimation error when the observer is implemented on the full system. A Luenberger-type observer to estimate the slow and fast states of a particular class of plants is reported in [5]. The observer gains are presented in terms of an LMI condition which is independent of the perturbation parameter and that guarantees exponential stability of the error dynamics.…”

“…Another work on the topic is presented in [4] in which a sliding mode observer for the slow states is designed. Note that results in [3]- [5] deal with specific plants and specific observers.…”