Summary
We present observer designs for hybrid systems in two scenarios: one where the jump times (i.e., the times when discontinuities in solutions occur) are known, and the other where these are unknown. For the first case, we first consider systems with linear maps by proposing a Kalman-like observer, then those with nonlinear maps by coupling a high-gain observer with a jump-based one under Lyapunov analysis. While the high-gain observer reconstructs the instantaneously observable part of the state from the flow output, the other one estimates the rest from the jump output and a fictitious one that models its interaction with the first part at jumps. When the jump times are unknown, we propose an observer by gluing the time domain, relying on transforming the hybrid system into some continuous-time dynamics without jumps. Applications to a walking robot and the stick-slip friction phenomenon are presented.
