A bond graph representation of an optimal control problem: the example of the DC motor
Résumé
Optimization has become of great importance in present day Systems Engineering. Also, bond graph language proves to be a very efficient tool for modelling, analysing and designing mechatronic systems from an energy and dynamic point of view. The idea presented in this paper is to combine an optimization formulation in the optimal control context with bond graph language. The objective is to transpose the optimization problem into bond graph formalism so that its exploitation will solve this optimization problem. This paper, being a primary investigation, restricts itself to an optimal control problem formulation applied to the example of the classical DC motor. Developments are based on the use of Pontryagin's classical Maximum Principle where the cost function is expressed as the integral of a quadratic form of the state space vector and the control input to be determined. This type of expression may correspond to a certain energy loss-minimization. It is shown that the formulation of this optimal control problem in bond graph makes logical use of the adjoint system concept. Later variable mapping enables an augmented bond graph representation of the whole problem to be set up. Finally the bicausality assignment to this augmented bond graph representation furnishes the solution to the optimal control problem under consideration.
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