Controllability and Optimum Control

Abstract

The intuitive notion of controllability of dynamic systems had been used for many years by control-engineers. In the case of linear stable systems, described by eqs.

$$\dot{x}{\text{ = Ax + Bu}}\,\,{\text{,}}$$

((1))

$${\text{y = Cx\,\,,}}$$

((2))

where

• x = n-dimensional state-vector.

• u = r-dimensional control-vector.

• y = p-dimensional output-vector.

• A, B, C - matrices of the dimensions n x n, n x r, p x n, respectively.

that notion has been formulated strictly by Kalman (see Ref. [2, 3]).

According to Kalman the system S, described by (1), (2) is controllable if and only if: given that the system is in state x_O at time t=0, then for some finite time T > 0 there is a control u(t), t ∈ [0, T] such that x(T) = 0.