Implementation of an Algorithm for the Direct Solution of Optimal Control Problems

Abstract

This paper presents the implementation of a numerical algorithm for the direct solution of optimal control and parameter identification problems. The problems may include differential equations that define the state, inequality constraints, and equality constraints at the initial and final times. The numerical method is based on transforming the infinite dimensional optimal control problem into a finite dimensional nonlinear programming problem. The transformation technique involves dividing the time interval of interest into a mesh that need not be uniform. In each subinterval of the mesh the control input is approximated using a piecewise polynomial. In particular, the control can be approximated using: (i) piecewise constant, (ii) piecewise linear, or (iii) piecewise cubic polynomials. The explicit Runge-Kutta method is used to obtain an approximate solution of the differential equations that define the state. With the approach used here the states do not appear in the nonlinear programming (NLP) problem. As a result the NLP problem is very compact relative to other numerical methods used to solve nonlinear optimal control problems. The NLP problem is solved using a sequential quadratic programming (SQP) technique. The SQP method is based on minimizing the L1 exact penalty function. Each major step of the SQP method solves a strictly convex quadratic programming problem. The paper also describes a simplified interface to the computer programs that implement the method. An example is presented to demonstrate the algorithm.