Optimal deterministic observer design for linear continuous-time systems with unknown output disturbances in a ball in L2

Abstract

The problem of the existence and design of an optimal observer for linear systems with unknown disturbances additive to the output is considered. The disturbances are assumed to be arbitrary L2 functions constrained to a ball in L2. The problem of reconstructing the best estimate of the state with respect to the worst disturbances is reformulated in terms of finding an operator L2-Rn Satisfying some linear equalities and of minimal norm. The necessary and sufficient condition for the existence of such an operator is proved to be the well-known observability condition. The optimal observer is in the form of an integral operator. An explicit formula for its integrand is given. In addition, differential equations fulfilled by the optimal state estimate for the cases of a fixed or time-variable observation interval are derived.