Controllability, Feedback and Pole Assignment

Abstract

Consider as usual the system

$$\dot x(t) = Ax(t) + Bu(t),t \ge 0$$

(0.1)

Suppose we are free to modify (0.1) by setting

$$u(t) = Fx(t) + v(t),t \ge 0$$

(0.2)

where υ(·) is a new external input, and F: X → Uis an arbitrary map. We refer to F as the state feedback. The obvious result of introducing state feedback is to change the pair (A, B) in (0.1) into the pair (A + BF, B). We shall explore the effect of such a transformation of pairs on controllability and on the spectrum of A + BF. Our main result is that if (A, B) is controllable then σ(A + BF) can be assigned arbitrarily by suitable choice of F, and this property in turn implies controllability.