Convergence of full-order observers for the slow states of a singularly perturbed system (Part I: Theory)

Abstract

Estimation of physical variables of nonlinear systems with two-time scales is a hard task to address. Whilst nonlinear systems exhibiting a singularly perturbed structure are common in engineering applications, current observer design results apply only to a specific class of plants and observers. We consider a broader class of plants and observers to generalise existing results on observer design for slow states of nonlinear singularly perturbed systems. Under reasonable assumptions, it is shown that the estimation error can be made semi-globally practically asymptotically stable in the singular perturbation parameter. This subsequently leads to appropriate conditions for the observer design for slow variables that guarantee satisfactory estimation error performance in the full system.

Convergence of full-order observers for the slow states of a singularly perturbed system (Part II: Applications)

“…In this section, we demonstrate that our results in [1] cover the results reported in [2]. We show a stronger result under stronger assumptions than results in [2].…”

“…Even when we restrict our attention to the same class of systems considered in [2], our results are more general; furthermore, our results cover a much larger class of plants and observers than [2]. In this manuscript, we verify that [1] covers another class of systems and an observer that cannot be covered by [2]. We have also checked that our results apply for plants in which the reduced (slow) system is such that nonlinear observers in [4] - [10] can be used to estimate the slow variables.…”

“…Here, we demonstrate the results in [1] naturally cover two classes of plants and two nonlinear full-order observers. We validate that our framework applies for the class of systems and the nonlinear observer covered by current results in [2].…”

“…We also consider the Luenberger-type nonlinear observer analysed in [2], see [3]. We show that our assumptions and results in [1] hold for that class of plants and that nonlinear observer. In section III, we consider a class of plants with fast linear dynamics and with reduced (slow) models that fit the framework of the circle criterion observer [4].…”

“…In our companion paper [1], we study the performance of nonlinear observers used to estimate the slow states of singularly perturbed systems. We analyse the robustness, with respect to singular perturbations, of an observer designed using the reduced (slow) system without considering the fast states and implemented on the original plant.…”