Estimation and Feedback in Linear Time-Varying Systems: A Deterministic Theory

Abstract

The problems of (i) state-variable feedback, (ii) deterministic state estimation, and (iii) state estimation and feedback are considered for a finite-dimensional linear time-varying system. The following are the questions; they can be regarded as natural and logical extensions to a time-varying case of the pole assignment problem [2]–[6] for a time-invariant system. (i) What is the relation between controllability and reachability of an open loop system and the possibility of constructing a linear state-variable feedback law which realizes the prescribed stability and instability degrees of the closed loop system? (ii) What is the relation between observability and reconstructibility of the system to be estimated and the possibility of constructing a linear state estimator which realizes the prescribed stability and instability degrees of the estimate error? (iii) What is the relation between controllability, reachability, observability and reconstructibility of an open loop system and the possibility of constructing a linear feedback compensator composed of a state estimator and an amplifier which realizes the prescribed stability and instability degrees of the closed loop system? Here, the stability and instability degrees mean the indices of exponential functions which bound the (zero-input) response from above and below respectively. This paper gives some principal answers to the questions.