A variable gain impulsive observer for perturbed Lipschitz nonlinear systems with delayed discrete measurements

This paper addresses the impulsive observers design problem of perturbed Lipschitz nonlinear systems subject to noisy delayed discrete measurements. The transmission delay, state perturbation, and measurement noise simultaneously hinder the convergence of impulsive observer. To deal with the difficulty, the observation error system with delayed impulses is represented as an augmented system with switching delay-free impulses. By taking into account the switching structure of the impulse dynamics, the impulse observation gain matrix is modelled as a function of the time interval between two consecutive sampling instant and update instant. Such a structure of the observation gain matrix is able to adapt to the variation of sampling period and transmission delay. To obtain a tractable design condition, the idea of piecewise linear interpolation is applied to represent the impulse observation gain matrix as a piecewise linear form, which is determined by finite static impulse gains. A piecewise Lyapunov function which matches the piecewise linear structure of the observation gain is constructed to analyze the exponential input-to-state stability (EISS) property of the observation error system. The EISS gain quantifies the robustness performance of the designed impulsive observer. Sufficient conditions for the existence of the proposed variable gain impulsive observers are presented. Two numerical examples are provided to illustrate the effectiveness of the developed impulsive observation strategy.