Adaptive impulsive observers for nonlinear systems: Revisited

This paper revisits the design of adaptive impulsive observers (AIOs) for nonlinear systems. The dynamics of observer state of the proposed AIO is modelled by an impulsive differential equation, by which the observer state is updated in an impulsive fashion. The parameter estimation law is modelled by an impulse-free time-varying differential equation associated with the impulse time sequence for determining when the observer state is updated. Unlike the previous work, the convergence analysis of the estimation error system is performed by applying a time-varying Lyapunov function based method, in conjunction with the application of a generalized version of Barbalat’s Lemma. A sufficient condition for the existence of AIOs is also derived. For some special cases, it is shown that the sufficient condition can be formulated in terms of linear matrix inequalities (LMIs), and the observer matrices can be attained by solving a set of LMIs. Furthermore, with an additional persistence-of-excitation-type constraint, it is proved that the sufficient condition can guarantee the convergence of parameter estimation. Two examples of chaotic oscillators are provided to illustrate the design procedure of the proposed AIOs.