Costate Estimation by a Legendre Pseudospectral Method

Abstract

We present a Legendre pseudospectral method for directly estimating the costates of the Bolza problem encountered in optimal control theory. The method is based on calculating the state and control variables at the Legendre-Gauss-Lobatto (LGL) points. An Nth degree Lagrange polynomial approximation of these variables allows a conversion of the optimal control problem into a standard nonlinear programming (NLP) problem with the state and control values at the LGL points as optimization parameters. By applying the Karush-Kuhn-Tucker (KKT) theorem to the NLP problem, we show that the KKT multipliers satisfy a discrete analog of the costate dynamics including the transversality conditions. Indeed, we prove that the costates at the LGL points are equal to the KKT multipliers divided by the LGL weights. Hence, the direct solution by this method also automatically yields the costates by way of the Lagrange multipliers that can be extracted from an NLP solver. One important advantage of this technique is that it allows a very simple way to check the optimality of the direct solution. Numerical examples are included to demonstrate the method.

Galerkin Optimal Control

“…Of the direct methods, specifically the global orthogonal collocation methods (a.k.a. pseudospectral methods) have proven to solve difficult problems with great ac-curacy [4,8,34,35] after becoming an actively researched topic in the 1990s by Elnagar et al [1] and Fahroo et al [2].…”

“…, N . In terms of the approximating polynomials, the system of equations 2) and the (N + 1) × 1 right-hand-side (RHS) vector, c, is defined as…”

“…Particularly, direct collocation methods, such as pseudospectral (PS) methods, have received much attention [1][2][3][4][5][6][7][8]. PS methods produce accurate solutions on a wide variety of optimal control problems.…”

“…Two recent highlights are the successful use of the Legendre PS method for the first ever zero-propellant attitude maneuver of the International Space Station [4] and the first ever minimum-time rotational maneuver performed in orbit by a NASA space telescope called TRACE [8]. In the Legendre PS method [1,2,5,6,9,10], the problem is discretized at the Legendre-Gauss-Lobatto (LGL) points, Legendre-Gauss-Radau (LGR) points or Legendre-Gauss (LG) points. The states are approximated with globally interpolating Lagrange polynomials and the cost function is typically approximated using Gaussian quadrature rule.…”

A Unified Pseudospectral Framework for Nonlinear Controller and Observer Design

“…This is the approach used in DIDO [31], a MATLAB application package for solving dynamic optimization problems. In recent years, PS methods have been successfully applied to solve a wide variety of optimal control problems, [6], [26], [12], [32], [34], [15], [19], [23], [16], [27]. As a result of its success, PS methods are now part of OTIS [22], NASA's software package for solving trajectory optimization problems.…”