Learning Techniques and Neural Networks for the Solution of N-Stage Nonlinear Nonquadratic Optimal Control Problems

This paper deals with the problem of designing closed-loop feed-forward control strategies to drive the state of a dynamic system (in general, nonlinear) so as to track any desired trajectory joining the points of given compact sets, while minimizing a certain cost function (in general, nonquadratic). Due to the generality of the problem, conventional methods (e.g., dynamic programming, maximum principle, etc.) are difficult to apply. Then, an approximate solution is sought by constraining control strategies to take on the structure of multi-layer feed-forward neural networks. After discussing the approximation properties of neural control strategies, a particular neural architecture is presented, which is based on what has been called the “Linear-Structure Preserving Principle” (the LISP principle). The original functional problem is then reduced to a nonlinear programming one, and backpropagation is applied to derive the optimal values of the synaptic weights. Recursive equations to compute the gradient components are presented, which generalize the classical adjoint system equations of N-stage optimal control theory. Simulation results related to non-LQ problems show the effectiveness of the proposed method.