Robustness analysis of nonlinear observers for the slow variables of singularly perturbed systems

Abstract

Estimation of unmeasured variables is a crucial objective in a broad range of applications. However, the estimation process turns into a challenging problem when the underlying model is nonlinear and even more so when additionally it exhibits multiple time scales. The existing results on estimation for systems with two time scales apply to a limited class of nonlinear plants and observers. We focus on analyzing nonlinear observers designed for the slow state variables of nonlinear singularly perturbed systems. Moreover, we consider the presence of bounded measurement noise in the system. We generalize current results by considering broader classes of plants and estimators to cover reduced-order, full-order, and higher-order observers. First, we show that the singularly perturbed system has bounded solutions under an appropriate set of assumptions on the corresponding boundary layer and reduced systems. We then exploit this property to prove that, under reasonable assumptions, the error dynamics of the observer designed for the reduced system are semiglobally input-to-state practically stable when the observer is implemented on the original plant. We also conclude (Formula presented.) stability results when the measurement noise belongs to (Formula presented.). In the absence of measurement noise, we state results on semiglobal practical asymptotical stability for the error dynamics. We illustrate the generality of our main results through three classes of systems with corresponding observers and one numerical example.