A global exponential observer based on numerical differentiation

Abstract

The extended Kalman filter is known to have excellent filtering characteristics, but its convergence is guaranteed only if it is initialized close enough to the true state value. Numerical differentiation based observers, on the other hand, may be designed to be globally convergent to a neighborhood of the true state value, but when the measurements are corrupted by significant uncertain signals their state estimates are delayed. In a previous work we proposed an observer design scheme for nonlinear systems which combines these two techniques to yield a globally exponentially converging observer. The necessity to implement an extended Kalman filter has two shortcomings. One is it does not apply when we face a system with an observability condition weaker than that the linearized dynamics about the estimated trajectory is completely uniformly observable. The other limitation is the computation burden. In this paper we propose an alternative which is still a global exponential observer. While the computation burden may not have been significantly reduced, definitely, it applies to a wider class of nonlinear systems. Noteworthy we prove that this global exponential observer is also bounded which is a feature the nonlinear observers in the literature rarely possess. The observer scheme is illustrated through a bioprocess example whose linearization is not completely uniformly observable.