Minimum-Fuel Feedback Control Systems: Second-Order Case

The control u(t), |u(t)| ≤1, is determined for the control of a linear plant G(s) from an initial state to a terminal state such that the fuel F = ∫
_o
^tf
|u(t)|dt is minimum. The maximum principle is used to prove that the optimal u(t) is necessarily piecewise constant and that u(t) = + 1, 0, or -1. The optimal feedback control function is derived for the plants G(s) = 1/s
²
and G(s)= 1/(s+1)(s+2) using the concept of the iso-fuel curves. The phase plane is divided into three regions of operation such that to each region corresponds a value for the control u(t).