On the fuel-optimal singular control of nonlinear second-order systems

Author

Athans, M. ; Canon, M.D.

Author_Institution

Massachusetts Institute of Technology, Cambridge, Massachusetts, USA

Volume

9

Issue

4

fYear

1964

fDate

10/1/1964 12:00:00 AM

Firstpage

360

Lastpage

370

Abstract

Given a body subject to quadratic drag forces so that the position $y(t)$ and the applied control thrust $u(t)$ are related by $\\ddot{y}(t)+adot{y}(t)|dot{y}(t)| = u(t), |u(t)| \\leq 1$ , the control $u(t)$ is found which forces the body to a desired position, and stops it there, and which minimizes the cost $J=\\int\\liminf {0} \\limsup {T}{k + |u(t)|}dt$ . The response time $T$ is not fixed, $k > 0$ , and $|u(t)|$ is proportional to the rate of flow of fuel. Repeated use of the necessary conditions provided by the Maximum Principle results in the optimum feedback system. It is shown that if $k\\leq 1$ , then singular controls exist and they are optimal; if $k > 1$ , then singular controls are not optimal. Techniques for the construction of the various switch curves are given, and extensions of the results to other nonlinear systems are discussed.

Keywords

Fuel-optimal control; Singular optimal control; Control systems; Costs; Delay; Feedback; Force control; Fuels; Nonlinear control systems; Nonlinear systems; Optimal control; Switches;

fLanguage

English

Journal_Title

Automatic Control, IEEE Transactions on

Publisher

ieee

ISSN

0018-9286

Type

jour

DOI

10.1109/TAC.1964.1105731

Filename

1105731