Necessary and sufficient condition for finite horizon H∞ estimation of time delay systems

This paper is concerned with the problems of finite horizon H_∞ filtering, prediction and fixed-lag smoothing for linear continuous-time systems with multiple delays. By applying an innovation approach in Krein space, a necessary and sufficient condition for the existence of an H_∞ filter, predictor or smoother is derived. The estimator is given in terms of the solution of a partial differential equation with boundary conditions. The innovation approach in Krein space enables us to convert the very complicated deterministic estimation problem into a stochastic one to which a simple H₂ innovation analysis method can be adapted. The result of this paper demonstrates that the Krein space approach is powerful in solving otherwise very complicated H_∞ problems. Our result is in contrast with many recent sufficient conditions for H_∞ filtering of delay systems.