Approximate solutions to nonlinearly-constrained optimal control problems

The use of variational methods in optimal control problems involves solving a two-point boundary-value problem (for states and costates) and satisfying an optimality condition. For problems with quadratic integral cost that have linear state dynamics and unconstrained controls, the co-state equations are also linear. Adjoining control constraints to the objective function introduces non linearity to the costate equation, and iterative numerical methods are required to converge upon the optimal control trajectory. The nonlinear costate terms arise at times in which the control constraints are active. In the numerical methodology proposed in this paper, an approximately optimal solution is converged upon from a feasible sub optimal initial control trajectory. In each iteration the control trajectory moves toward the unconstrained optimum solution while remaining feasible. Importantly, the state and costate equations are linear and the method is applied to a multi input system designed to minimize the response of a vibration isolation system by adjusting only the damping characteristics of a variable damping device.