Controllability by means of affine feedback

This paper studies the constant coefficient (closed loop) differential equation x = (A + BK)x + Bv + f(t) in [to, t1] ?? Rn which arises from substitution of the feedback controller u = Kx + v into the (open loop) control equation x = Ax + Bu + f(t). The open loop system is assumed to be controllable. A classical result due to Kalman states that this assumption is equivalent to the condition that rank [B,AB,...An-1B] = n. Under mild restrictions on f(t) we prove that for any states x0, x1 in Rn there exists a constant matrix K and a constant vector v for which the closed loop system has a response satisfying the boundary condititions x(t0) = x0 and x(t1)= x1. An equivalence between open loop and closed loop controllability is thereby established. Ultimately based upon the implicit function theorem, the approach taken lays the groundwork for computing feedback controllers that do the steering.