A multi-observer approach for the state estimation of nonlinear systems

We present a prescriptive approach for the state estimation of nonlinear systems. We first assume that we know a local observer, i.e. a dynamical system whose state converges to the plant state when it is initialized nearby the plant initial condition. We sample the set where the plant initial condition is assumed to lie with a finite number of points; noting that this set can be arbitrarily large. A local observer is initialized at each of these sampled points to form a bank of observers called multi-observer. A supervisor is then constructed to select one of these observers at any time instant. The selected state estimate is guaranteed to converge to the state of the plant, provided the number of samples is sufficiently large and a detectability property holds, which is expressed in terms of Lyapunov-based conditions. An explicit lower bound on the required number of observers is given when an estimate of the basin of convergence of the local observer is available. We explain how to apply the approach to nonlinear systems with globally Lipschitz and differentiable nonlinearities. Simulations results are presented for a Wilson-Cowan oscillator.