A Second-order Gradient Method for Determining Optimal Trajectories of Non-linear Discrete-time Systems

ABSTRACT A second-order method of successively improving a control sequence for a non-linear discrete-time system is derived. One step convergence is obtained for linear systems with quadratic performance functions. Although the results are of interest in their own right, a second-order method for continuous-time systems is obtained by formally allowing the sampling interval to approach zero. The equations so obtained differ slightly, because of the method of derivation, from results already obtained using the calculus of variations approach. The difference, which is an advantage of the method described in this paper, is that one vector differential equation less has to be integrated. The approach used in the derivation is motivated by dynamic programming and facilitates the application of gradient methods to stochastic problems which will be the subject of a future paper.