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

In this paper, we propose a computational approach to solve a model-based optimal control problem. Our aim is to obtain the optimal so- lution 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 sys- tem 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 prob- lem 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 ap- proaches 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.