A hybrid computational approach to nonlinear estimation

A general theory of nonlinear observers is developed. It is broad enough to include all existing approaches both deterministic and stochastic. We show that all observers reduce to the solution in a viscosity sense of a partial differential equality of Hamilton-Jacobi-Bellman (HJB) type. Based on this, we have developed a hybrid algorithm for a nonlinear state observer that utilizes two levels of computation. On the higher level we solve the HJB equality by approximating it by a discrete time and space nonlinear program. At the lower level, we initiate local observers that resemble extended Kalman filters at the local minima of the HJB solution. We present numerical results of the hybrid method with the inclusion of a forgetting factor to speed up the convergence. The algorithm has been tested for the low-dimensional systems. The performance of the hybrid estimator is contrasted with that of an extended Kalman filter. We have found that given a badly chosen initial condition of a nonlinear system, an extended Kalman filter can get trapped in a region far from the true value while the hybrid approach achieves an accurate estimate.