State estimation strategy without jump detection for hybrid systems using gluing function

In this paper, we propose a new observer design approach for a class of hybrid systems with jumps of the state. The idea is to “glue” the jump set (a part of domain where the jumps take place) onto its image. Then, on the “glued” domain, the hybrid system becomes a continuous-time system without any jumps. Therefore, it would be possible to design an observer for the system, via a conventional observer design method for continuous-time systems. This new observer design mechanism does not require any detection of the jumps. To this end, we introduce the notion of gluing function and provide a condition ensuring that the system dynamics after gluing is well-defined and continuous. Finally, as a case study, we design an observer for a bouncing ball system.