Use of variable stage-lengths for constrained optimal control problems

The use of variable stage-lengths allows a relatively small number of time stages to be used for establishing: an accurate optimal control policy for systems exhibiting sudden switches in the control policy. Such problems where there are also state inequality constraints and where the final time is specified, are considered here. To handle the state inequality constraints, auxiliary state variables are introduced, and to satisfy the specified final time restriction, a quadratic penalty function with a shift term is used. Both of these restrictions are incorporated into a penalty function which is included in the augmented performance index. The approach is illustrated and tested with the optimal control of a tubular reactor, where the maximum temperature is specified. It was found that an optimal control policy consisting of only 10 stages of variable length gives the same performance index as was obtained with 96 stages of equal length. The use of 15 stages gives marginally better results. In each case, the state constraint is satisfied throughout the given time interval, and the computation time by iterative dynamic programming is reasonable.