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

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 Metho

On the Discretized Algorithm for Optimal Proportional Control Problems Constrained by Delay Differential Equation

This paper seeks to develop an algorithm for solving directly an optimal control problem whose solution is close to that of analytical solution. An optimal control problem with delay on the state variable was studied with the assumption that the control effort is proportional to the state of the dynamical system with a constant feedback gain, an estimate of the Riccati for large values of the final time. The performance index and delay constraint were discretized to transform the control problem into a large-scale nonlinear programming (NLP) problem using the augmented lagrangian method. The delay terms were consistently discretized over the entire delay interval to allow for its piecewise continuity at each grid point. The real, symmetric and positive-definite properties of the constructed control operator of the formulated unconstrained NLP were analyzed to guarantee its invertibility in the Broydon-Fletcher-Goldberg-Shanno (BFGS) based on Quasi-Newton algorithm. Numerical example was considered, tested and the results responded much more favourably to the analytical solution with linear convergence. Keywords: Simpson’s discretization method, proportional control constant, augmented Lagrangian, Quasi –Newton algorithm, BFGS update formula, delays on state variable, linear convergence

Extension of ADMMAlgorithmin Solving Optimal Control Model Governed by Partial Differential Equation

This paper presents an Algorithm for the numerical solution of the Optimal Control model constrained by Partial Differential Equation using the Alternating Direction Method of Multipliers (ADMM) accelerated with a parameter factor in the sense of Nesterov. The ADMM tool was applied to a partial differential equation-governed optimization problem of the one-dimensional heat equation type. The constraint and objective functions of the optimal control model were discretized using the Crank-Nicolson and Composite Simpson’s Methods respectively into a derived discrete convex optimization form amenable to the ADMM. The primal-dual residuals were derived to ascertain the rate of convergence of themodel for increasing iterates. An existing example was used to test the efficiency and degree of accuracy of the algorithm and the results were favorable when compared the existing method