Quasi-Newton Embedded Augmented Lagrangian Algorithm for Discretized Optimal Proportional Control Problems

Abstract

In developing a robust algorithm for solving a class of optimal  control problems in which the control effort  is proportional to the state of the dynamic system, a typical model was studied which generates a constant feedback gain , an estimate of the Riccati for large values of the final time. Involving the third Simpson’s Rule, a discretized unconstrained non-linear problem via the Augmented Lagrangian Method was obtained. This problem was consequently subjected to the Broydon-Fletcher-Goldberg-Shannon(BFGS) based  on Quasi-Newton algorithm. The positive- definiteness of the estimated quadratic control operator was analyzed to guarantee its invertibility in the BFGS. Numerical examples were considered, tested and the results responded much more favourably to the analytical solution with linear convergence.

Keywords: Proportional control, feedback gain, Augmented Lagrangian Method, Discretization, BFGS , Simpson’s Rule and Quasi-Newton Method