Continuous state observability and mode reconstructability of switched nonlinear systems with unknown switching function

For switched nonlinear systems, the problems of continuous state observability and mode reconstructability are tackled, assuming the switching function is unknown. First, we give a condition under which the continuous state is observable and the active mode is reconstructible. Furthermore, we provide some relaxed conditions that guarantee the observability of the continuous state vector x(t); those conditions allow estimating x(t) even if the active mode cannot be reconstructed. The observability analysis is carried out by the study of an invariant manifold on which the outputs of two systems (fixing each pair of different modes) are equal. Moreover, by assuming that there is a known minimum dwell-time, we propose a way of reconstructing the state vector and the switching function, provided the respective observability conditions are fulfilled. The reconstruction procedure involves the use of finite-time homogeneous sliding mode differentiator. In a short section, we show how the obtained observability conditions match with the known structural ones for linear systems. Then, we see that the proposed state reconstruction scheme can also be applied to that particular class of systems. We point out that the original system is not required to be in any specific form, which may be an advantage of the estimation method proposed in this paper. By an academic example with simulations, we illustrate the analysis carried out along the paper and the respective reconstruction method.