Augmented Gradient Projection Calculations for Regulator Problems with Pointwise State and Control Constraints

Abstract

A new implementation of the augmented gradient projection (AGP) scheme is described for discrete-time approximations to continuous-time Bolza optimal control problems with pointwise bounds on control and state variables. In the conventional implementation, control and state vectors are the primal variables, and local forms of the control problem’s state equations are treated as equality constraints incorporated in an augmented Lagrangian with a penalty parameter c. In the new implementation, the original control vectors and new artificial control vectors are the primal variables, and an integrated form of the state equations replaces the usual local form in the augmented Lagrangian. The resulting relaxed nonlinear program for the augmented Lagrangian amounts to a Bolza problem with pure pointwise control constraints, hence the associated gradient and Newtonian direction vectors can be computed efficiently with adjoint equations and dynamic programming techniques. For unscaled AGP methods and prototype regulator problems with bound constraints on control and state vectors, numerical experiments indicate rapid deterioration in the convergence properties of the conventional implementation as the discrete-time mesh is refined with the penalty constant fixed. In contrast, the new implementation of the unscaled AGP scheme exhibits mesh-independent convergence behavior. The new formulation also offers certain additional computational advantages for control problems with separated control and state constraints.