A new computational approach for optimal control problems with multiple time-delay

Abstract

The control parameterization method used together with the time-scaling transformation is an effective approach to approximating optimal control problems into optimal parameter selection problems when no time delays are involved. The approximate problems can then be solved by gradient-based optimization algorithms. However, the time-scaling transformation, which works well for optimizing variable switching times of the approximate piecewise constant/linear control functions obtained after the application of the control parameterization method, is not applicable to optimal control problems with time delays. In this paper, we consider a class of nonlinear optimal control problems with multiple time delays subject to canonical equality and inequality constraints. Our aim is to develop a novel transformation procedure that converts a given time-delay system into an equivalent system – defined on a new time horizon – in which the control switching times are fixed, but the dynamic system contains multiple variable time delays expressed in terms of the durations between the switching times for each of the approximate control functions in the original time horizon. On this basis, we show that an optimal control policy for the equivalent system can be obtained efficiently using gradient-based optimization techniques. This optimal control policy can then be used to determine the optimal switching times and optimal control variables for the original system. We conclude the paper by solving two example problems.