Solution of optimal control problems on a high-speed hybrid computer

This paper presents a method for finding the solutions to minimum-time optimal control problems. The procedure is to implement Pontryagin's Maximum Principle on an iterative hybrid computer. The state and adjoint equations as well as the control law are simulated using conven tional analog components. The troublesome two-point boundary-value problem, which is always associated with Maximum Principle, is solved by iteration, using a digital parameter optimizer. Thus, a manual trial-and-error search for the proper initial values of the adjoint variables is un necessary. We show that, for a large class of systems, it is not necessary to generate the Hamiltonian, because the neces sary condition that it normally must satisfy is redundant. This allows many problems to be greatly simplified. We also present an optimizing routine that solves the boun dary-value problem. This permits the proposed method to be used on any hybrid computer that incorporates a general-purpose digital computer. The solutions to two problems show that the proposed method is feasible. Average convergence times range from less than one second to about 70 seconds. These vary with the initial conditions on the state variables. The examples were solved using ASTRAC II, a small (40 amplifier), high speed (up to 1000 solutions per second), iterative hybrid computer with only modest component accuracy (0.25 per cent). Although the discussion and examples are limited to a minimum-time performance index, the method is easily extended to cover other criteria.