Poincare mapping for a time-delay impulsive observer

Abstract

This note deals with a hybrid static gain observer for a linear time-invariant continuous plant under an intrinsic pulse-modulated feedback. The pulse parameters representing the discrete state of the hybrid system are not available for direct measurement and has to be estimated through continuous measurements. It poses an unusual observation problem. A considerable number of papers is devoted to the observability of hybrid systems, where the discrete states of a system are usually assumed known, while observers for hybrid systems that are able to reconstruct discrete states from only continuous measurements are not so well covered in the literature. With the time delay taken into account, the pulse-modulated model acquires an infinite-dimensional continuous part. The closed-loop dynamics become therefore both hybrid and infinite-dimensional, and this combination is mathematically challenging and so far rarely treated. However, the cascade structure of the continuous part, together with the impulsive feedback, allow the application of the concept of finite-dimensional reducibility (FD-reducibility), in particular the fact, that the dynamics of an impulsive time-delay system with an FD-reducible continuous part coincide on certain time intervals with the dynamics of a delay-free impulsive system. This idea plays a key role in the present study. Nevertheless the observer proposed here explicitly involve a delay and based on the infinitedimensional original plant model, on the one hand it complicates the study of its properties, but on the other allows observation of the system for the entire time interval. An pointwise discrete mapping (known in the hybrid systems theory as Poincare mapping), describing the observer dynamics is derived, constituting the contribution of the paper. With the use of this pointwise mapping the mathematical modeling of the system can be performed and obtained the observer stability conditions for synchronous mode with respect to the periodic solution of the system. Refs 14. Figs 2.