A gradient algorithm for optimal control problems with model-reality differences

Sie Long Kek; Mohd Ismail Abd Aziz; Kok Lay Teo

doi:10.3934/naco.2015.5.251

Article Contents

2015, Volume 5, Issue 3: 251-266. Doi: 10.3934/naco.2015.5.251

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A gradient algorithm for optimal control problems with model-reality differences

1.
Department of Mathematics, Universiti Tun Hussein Onn Malaysia, 86400 Parit Raja
2.
Department of Mathematics, Universiti Teknologi Malaysia, 81310 UTM, Skudai
3.
Department of Mathematics and Statistics, Curtin University, Perth, W.A. 6845

Received: May 2014

Revised: March 2015

Published: August 2015

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Abstract

In this paper, we propose a computational approach to solve a model-based optimal control problem. Our aim is to obtain the optimal solution of the nonlinear optimal control problem. Since the structures of both problems are different, only solving the model-based optimal control problem will not give the optimal solution of the nonlinear optimal control problem. In our approach, the adjusted parameters are added into the model used so as the differences between the real plant and the model can be measured. On this basis, an expanded optimal control problem is introduced, where system optimization and parameter estimation are integrated interactively. The Hamiltonian function, which adjoins the cost function, the state equation and the additional constraints, is defined. By applying the calculus of variation, a set of the necessary optimality conditions, which defines modified model-based optimal control problem, parameter estimation problem and computation of modifiers, is then derived. To obtain the optimal solution, the modified model-based optimal control problem is converted in a nonlinear programming problem through the canonical formulation, where the gradient formulation can be made. During the iterative procedure, the control sequences are generated as the admissible control law of the model used, together with the corresponding state sequences. Consequently, the optimal solution is updated repeatedly by the adjusted parameters. At the end of iteration, the converged solution approaches to the correct optimal solution of the original optimal control problem in spite of model-reality differences. For illustration, two examples are studied and the results show the efficiency of the approach proposed.

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Mathematics Subject Classification: Primary: 93C05, 93C10; Secondary: 93B40.

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