Optimal Bang-Bang Control With Quadratic Performance Index

[+] Author and Article Information
W. M. Wonham

Center for Control Theory, Research Institute for Advanced Studies, Baltimore, Md.

C. D. Johnson

Electrical Engineering Department, University of Alabama, Huntsville Center, Huntsville, Ala.

J. Basic Eng 86(1), 107-115 (Mar 01, 1964) (9 pages) doi:10.1115/1.3653092 History: Received July 19, 1963; Online November 03, 2011


The following optimal regulator problem is considered: Find the scalar control function u = u(t) which minimizes the performance index

  120Tx(t), Qx(t)〉dt,
subject to the conditions
 = Ax + u(t)f,|u(t)| ≦ 1x(0) = x0(x0 is unrestricted)x(T) = 0(T is free)
Q , A are constant n × n-matrices; f is a constant n-vector. It is shown that optimal control includes both a bang-bang mode and a linear mode, the latter arising from the “singular” solutions of the Pontriagin canonical equations. Conditions are given under which nth-order systems are equivalent, for control purposes, to systems of first or second order. One example of a second-order system is worked in detail and some results of an analog computer study are presented.

Copyright © 1964 by ASME
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