Observer design for hybrid systems with partially affine forms and known jump times: Applications to walking robots

We propose novel observer designs for hybrid systems, with a certain nonlinear structure that is affine with respect to certain state components and with known jump times, based on decomposing the state into parts exhibiting different observability properties. We assume that the state of the hybrid system can be decomposed into two components. During flows, the first component is independent of the second one and is assumed to be instantaneously observable from the flow output. The second one is required to be either detectable or backward distinguishable via the combination of flows and jumps, from the jump output as well as a fictitious output describing how this second part impacts the first one at jumps. An observer is designed to estimate each component: a high-gain flow-based observer using the flow output estimates the first one and an LMI/KKL-based jump-based observer using an extended jump output estimates the second one. Global exponential convergence and stability of the estimation error in the original coordinates are proven using Lyapunov analysis. The proposed observers are exercised in the problem of estimating the state and uncertainties at impacts in a bipedal walking robot.