Delay-dependent H_∞ dynamic observers for non-linear systems with multiple time-varying delays

Abstract

The design of dynamic [Formula: see text] observers (DO) for non-linear Lipschitz systems with multiple time-varying delays and disturbances is studied. Sufficient conditions for the existence of these observers are presented in the form of rank equality. Compared to previously published work, the system under consideration includes non-linearity, non-commensurable delay, and external disturbance. Through the use of the Wirtinger inequality and the extended reciprocally convex matrix inequality, new and less conservative delay-dependent conditions in terms of linear matrix inequalities (LMIs) are derived based on the Lyapunov–Krasovskii functional method. Solving these LMIs makes it possible to obtain DO that satisfies an [Formula: see text] performance index. Through two numerical examples in which the comparison with the proportional observer (PO) and the proportional–integral observer (PIO) shows the efficiency of the proposed DO synthesis condition. Furthermore, the results indicate that the DO developed in this paper is more resilient to parameter perturbations.

H∞ dynamic observer design for a class of Lipschitz nonlinear discrete‐time systems with time varying delays

“…Lemma 3 (33). For any X ∈ R n and Y ∈ R n and any scalar 𝜚 > 0, the following inequality is obtained:…”

“…Remark In fact, we can extract the PO, PIO and DO of the form () by the same method as in the continuous time case in Reference 33: If matrices $$$ J={J}_{\tau }=0 $$$, $$$ R={R}_{\tau }=0 $$$, $$$ S={S}_{\tau }=0 $$$, and $$$ P={P}_{\tau }=0 $$$, $$$ i=0,\dots, M $$$, the observer () becomes that of the PO: $$$$ {\displaystyle \begi...$$…”

“…Remark The majority of articles study HDO for continuous time systems with or without delay (see References 33 and 34). In some situations, discrete‐time systems can be more practical for defining a system and designing the controller, due to current advances in this field.…”

“…The first structure of this observer was proposed by Park in Reference 32. Since then, numerous studies have explored the application of this type of observer in the continuous‐time domain, as evidenced by various works in the literature such as those by Naami et al, 33 Gao et al, 34 Boutat‐Baddas et al, 35 and Osorio‐Gordillo et al 36 However, limited attention has been paid to its discrete‐time equivalent. Notably, Reference 37 studied this observer for linear systems, while Reference 38 focused on linear parameters‐varying (LPV) systems.…”