Geometric structure of observers for linear feedback control laws

This paper is concerned with constructing observers for linear feedback control laws. Two types of observers (Kalman-type and Luenberger-type) are considered concurrently. Geometric theory of dynamic covers is developed for evaluating the minimal orders of observers. New lower and upper bounds are obtained for the minimal order of function observers possessing an arbitrarily prescribed set of poles. They are expressed simply in terms of observability indices of an augmented system and give a new light on the structural properties of observers. They also suggest the possibility of significant order reduction compared with observers estimating the whole state. A new geometric concept of generator, a natural generalization of cyclic generator, plays a key role in their derivation. A frequency domain characterization of observers is derived which reveals an interesting algebraic property of observers. It is used for devising a design algorithm in the frequency domain; in which the problem is reduced to pole assignment by dynamic compensator of a restricted type. Another design algorithm is presented in the time domain. Some illustrative examples are shown.