Real-Time Solutions of Bang-Bang and Singular Optimal Control Problems
In many applications of optimal control some or all of the control variables appear linearly in the objective function and the dynamical equations. Therefore, the optimal solutions may exhibit both bang-bang and singular subarcs. Unfortunately, the theory for linear problems of that type is not as well developed as for regular problems, in particular with respect to second order sufficiency conditions. This results in serious problems in developing real-time capable methods to approximate optimal solutions in the presence of data perturbations. In this paper, two discretization methods are presented by which linear optimal control problems can be transcribed into nonlinear programming problems. Based on a stability and sensitivity analysis of the resulting nonlinear programming problems it is possible to compute sensitivity differentials for the discretized problems, by means of which near-optimal solutions can now be computed in real-time for linear problems, too. The performance of one of these methods is demonstrated for the optimal control of a batch reactor.
KeywordsControl Problem Optimal Control Problem Nonlinear Programming Problem Singular Control Switching Structure
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