Optimal Filtering For Systems With Delayed Measurements

Linear Estimation is a key research topic in many fields such as control, communications, signal processing, and so on. In the 1960's, Kalman filtering was presented and has been a major tool of state estimation and control since then. However, the standard Kalman filtering can be only applicable to the normal systems without delays not to the delayed systems. Much interest has been attached to the case of time delays for the actual requirements. Some problems such as estimation for linear systems with delayed measurements and control for systems with input delay haven't been studied well up to the present. In this paper, we are concerned with the optimal estimation problem for linear systems systems with delayed measurements, and a new optimal approach, re-organization of innovation analysis, is then proposed. The main idea is to re-organize the delayed measurements into delay-free measurements from different systems, and the associated innovation sequence is given in according to the re-organized measurements, Kalman filtering formulations based on the new approach are thus given. With the new technique, the more complicated H_∞white noise estimator, robust time-delayed Kalman filter, and time-delayed information fusion filter are then given. The paper mainly includes the following parts:Kalman filtering for linear discrete-time systems with delayed measurements. Reorganization of innovation analysis for discrete-time systems is introduced, and the case of systems with instantaneous measurement and delayed measurements is studied, the optimal filters are then given. Furthermore, the more complicated technique, re-organization of innovation analysis, is derived for discrete-time systems with multiple delayed measurements, and optimal filters and the flow chart are also given in the paper. Such an approach is much more computationally attractive which is a major advantage over the traditional system augmentation, and the comparison of the computation costs between the proposed re-organization of innovation analysis and the traditional augmentation method is also given, and two examples have been given to show this point. A numerical example is given to show the efficiency of the proposed approach. The proposed approach is further extended to the robust Kalman filtering for discrete-time systems with single delayed measurement and the time-delayed information fusion filtering problem, then the robust Kalman filter and the time-delayed information fusion filter are given.Kalman filtering for linear continuous-time systems with delayed measurements. Reorganization of innovation analysis for continuous-time systems is presented, the case of systems with instantaneous measurement and delayed measurements is studied well, and the proposed approach is extended to the case of continuous-time systems with multiple delayed measurements, and optimal filters are given. Two numerical examples and the flowchart for the computation are given to show the process of the computation. Explicit solutions to the problem are given without resorting to traditional partial differential equations.Re-organization of innovation analysis can be used to deal with many difficult problems, one of which is H_∞white noise estimation. In the paper, Krein space is introduced, and the H_∞white noise estimation for linear systems (including both discrete-time case and continuous-time case) is considered with the help of re-organization of innovation analysis and projection formulation in Krein space, thus the estimators (mainly filters and fixed-lag smoothers) and the associated sufficient and essential conditions are given. It is also shown that the white noise H_∞white noise filtering is equivalent to the H₂ white noise filtering in Krein space, and the H_∞white noise fixed-lag smoothing is in fact equivalent to an H₂ estimation problem for measurement delayed system in Krein space. The problem is much valuable both in theory and in practice.