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AN EFFICIENT COMPUTATIONAL APPROACH TO A CLASS OF MINMAX OPTIMAL CONTROL PROBLEMS WITH APPLICATIONS

Part of: Existence theories

Published online by Cambridge University Press:  20 May 2010

B. LI ,
K. L. TEO ,
G. H. ZHAO  and
G. R. DUAN
B. LI*
Affiliation:
Harbin Institute of Technology, PR China (email: g.r.duan@hit.edu.cn) Curtin University of Technology, Australia (email: bin.li@postgrad.curtin.edu.au, k.l.teo@curtin.edu.au)
K. L. TEO
Affiliation:
Harbin Institute of Technology, PR China (email: g.r.duan@hit.edu.cn) Curtin University of Technology, Australia (email: bin.li@postgrad.curtin.edu.au, k.l.teo@curtin.edu.au)
G. H. ZHAO
Affiliation:
Dalian University of Technology, PR China (email: ghzhao6961@hotmail.com)
G. R. DUAN
Affiliation:
Harbin Institute of Technology, PR China (email: g.r.duan@hit.edu.cn)
*
✉For correspondence; e-mail: bin.li@postgrad.curtin.edu.au
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Abstract

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In this paper, an efficient computation method is developed for solving a general class of minmax optimal control problems, where the minimum deviation from the violation of the continuous state inequality constraints is maximized. The constraint transcription method is used to construct a smooth approximate function for each of the continuous state inequality constraints. We then obtain an approximate optimal control problem with the integral of the summation of these smooth approximate functions as its cost function. A necessary condition and a sufficient condition are derived showing the relationship between the original problem and the smooth approximate problem. We then construct a violation function from the solution of the smooth approximate optimal control problem and the original continuous state inequality constraints in such a way that the optimal control of the minmax problem is equivalent to the largest root of the violation function, and hence can be solved by the bisection search method. The control parametrization and a time scaling transform are applied to these optimal control problems. We then consider two practical problems: the obstacle avoidance optimal control problem and the abort landing of an aircraft in a windshear downburst.

Keywords

minmax optimal controlcontinuous state inequality constraintsconstraint transcriptioncomputation methodcontrol parametrizationtime scaling transformroot findingobstacle avoidanceaircraft abort landingwindshear downburst

MSC classification

Secondary: 49J15: Optimal control problems involving ordinary differential equations
Type
Research Article
Information
The ANZIAM Journal , Volume 51 , Issue 2 , October 2009 , pp. 162 - 177
Copyright
Copyright © Australian Mathematical Society 2010

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