Minimum Fuel Control of a Second-Order Linear Process With a Constraint on Time-to-Run

[+] Author and Article Information
H. O. Ladd

Space and Information Systems Division, Raytheon Company, Bedford, Mass.

Bernard Friedland

Aerospace Research Center, General Precision, Inc., Little Falls, N. J.

J. Basic Eng 86(1), 160-168 (Mar 01, 1964) (9 pages) doi:10.1115/1.3653101 History: Received July 30, 1963; Online November 03, 2011


The optimal control as a function 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

ẍ + 2bẋ + c2x = u
for so-called minimum-fuel or minimum-effort operation (i.e., such that 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, ẋ) 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, ẋ) at which u changes value and, hence, of the optimal feedback control u (T, x, ẋ), 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 state are obtained by a limiting procedure.

Copyright © 1964 by ASME
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