When is Naive Low-Pass Filtering of Noisy Measurements Counter-Productive for the Dynamics of Controlled Systems?

Abstract

In many practical applications, noisy measurements are low-pass filtered by (quasi-)continuous-time filters with linear, fixed-order lag behavior that are manually tuned according to the designer's gut feeling. However, if the time constants of these low-pass filters are neglected during control synthesis, oscillations may arise even in cases in which the non-filtered closed-loop dynamics are described by linear system models with purely real eigenvalues. In this paper, Lyapunov methods are applied to system models given in terms of stochastic differential equations to account for measurement and process noise and to predict the influence of noise on the filter and closed-loop system dynamics. In addition, an optimization approach for the parameterization of state observers is derived which is based on these Lyapunov techniques. It is implemented by a suitable problem formulation in terms of linear matrix inequalities. Illustrative simulation case studies, including a comparison with the well-known stationary Kalman Filter, conclude this paper.