State Estimation for Linear Time-Varying Observation Systems

Abstract

We consider the state observation and estimation problems for linear time-varying systems of ordinary differential equations. We show that the uniform observability is equivalent to the approximate observability, i.e., the possibility of using delta sequences to estimate the current state with arbitrary accuracy without differentiating the output function. We propose a method for constructing state estimators for linear time-varying systems based on the quasi-differentiability of the coefficients with respect to a specially constructed lower-triangular matrix. For uniformly observable systems with quasi-differentiable coefficients, we obtain conditions for the existence of an exponential observer and describe a constructive method for designing such observers.