Minimum fuel control of a second-order linear process with a constraint on time-to-run

The optimal control as a junction of the instantaneous state, i.e., the optimal “feedback” or “closed-loop” control, is derived for the controlled second-order linear process with constant coefficients x + 2bx + c²x = u for so-called minimum-fuel or minimum-effort operation (i.e., such tliat the time integral of the magnitude of the control u is minimized), subject to an amplitude limitation on the control u ≤ L. The objective is to force the phase state from an arbitrary instantaneous value (x, x) to the origin within an arbitrarily prescribed time-to-run T. The solution is obtained for the nonoscillatory cases (b² ≥ c² ≥ 0) when L is finite, and for arbitrary real b and c when L is infinite; i.e., when the control is not amplitude-limited. The form of the optimal control is shown to be “bang-off-bang” with the most general initial conditions; i.e., during successive time intervals, u is constant at one limit, identically zero, and constant at the limit of opposite polarity. Explicit expressions for the switching surfaces in state space (T, x, x) at which u changes value and, hence, of the optimal feedback control u (T, x, x), are given, both with and without amplitude limitation. Without such (L = ∞) the optimal control is impulsive and the areas of the impulses in terms of the current slate are obtained by a limiting procedure.