A quadrature discretization method for solving optimal control problems

This paper presents a pseudospectral method for solving optimal control problems based on nonclassical orthogonal and weighted interpolating polynomials. Traditional pseudospectral methods expand the state and control trajectories using global Lagrange interpolating polynomials based on a specific class of orthogonal polynomials from the Jacobi family, such as Legendre or Chebyshev polynomials, which are orthogonal with respect to a specific weight function over a fixed interval. Although these methods have many advantages, the location of the grid points are more or less fixed. The method presented in this paper generalizes the existing methods and allows a much more flexible selection of grid points by the arbitrary selection of the orthogonal weight function and interval. Additional flexibility is introduced by using weighted interpolants in the expansion of the states and controls, rather than simple Lagrange polynomials. A method for directly estimating the costates from the Kuhn-Karush-Tucker multipliers, which are obtained from the solution of the direct optimal control problem, is derived. An additional method for obtaining estimates of the costates by postprocessing is also presented. It is shown through numerical studies that the commonly used Legendre Gauss Lobatto points do not necessarily give the most numerically efficient or accurate solutions. Significant savings in computation time are possible by appropriately selecting the orthogonal weight function and weighted interpolating polynomial.