Computation of time-optimal controls using a second-variation descent search

A second-variation descent search is proposed for computing time-optimal control functions for linear time-invariant plants. Since the piecewise constant structure of the optimal control function parameterizes the relevant two-point boundary value problem, the terms required in the variational analysis are simply first and second partial derivatives of an appropriate scalar cost function with respect to the parameters. Closed form expressions for these derivatives can be obtained. As a result, the second-order terms are incorporated into the analysis with only a modest increase in computational effort over that required for the first-variation methods appearing in the literature [1]-[4]. The usual second-variation algorithm can be applied only if the matrix of second derivatives is positive definite at each iteration. Since this condition generally fails to hold during the early stages of the search, a modified version of the algorithm is developed, based on the selection of a positive definite model for the matrix of second derivatives. Performance of the modified algorithm is shown to be significantly better than that of the familiar Newton algorithm in regions of parameter space far from the solution point. Particularly striking results are obtained when all the eigenvalues of the plant are real.