From restricted isometry property to observability under sparse measurement

In this paper, we show that it is possible to detect the active mode and reconstruct its associated state for a class of hybrid linear systems under sparse measurements. The solution we bring to this problem is firstly to analyze the observability of systems under random sampling and secondly to synthesize an impulsive observer. Here, the first approach is based on the concept of compressive sampling which is well-known in signal processing theory. A synthesis of impulsive observers will be presented for some special cases. These observers allows to detect the active mode and rebuild the state under sparse measurement. Simulation results are provided in order to highlight the well-foundedness of the proposed approach.