Legendre-Gauss Research Articles

Costate approximation in optimal control using integral Gaussian quadrature orthogonal collocation methods

Two methods are presented for approximating the costate of optimal control problems in integral form using orthogonal collocation at Legendre–Gauss (LG) and Legendre–Gauss–Radau (LGR) points. It is shown that the derivative of the costate of the continuous-time optimal control problem is equal to the negative of the costate of the integral form of the continuous-time optimal control problem. Using this continuous-time relationship between the differential and integral costate, it is shown that the discrete approximations of the differential costate using LG and LGR collocation are related to the corresponding discrete approximations of the integral costate via integration matrices. The approach developed in this paper provides a way to approximate the costate of the original optimal control problem using the Lagrange multipliers of the integral form of the LG and LGR collocation methods. The methods are demonstrated on two examples where it is shown that both the differential and integral costate converge exponentially as a function of the number of LG or LGR points. Copyright © 2014 John Wiley & Sons, Ltd.

Pseudospectral methods for solving infinite-horizon optimal control problems

An important aspect of numerically approximating the solution of an infinite-horizon optimal control problem is the manner in which the horizon is treated. Generally, an infinite-horizon optimal control problem is approximated with a finite-horizon problem. In such cases, regardless of the finite duration of the approximation, the final time lies an infinite duration from the actual horizon at t = + ∞ . In this paper we describe two new direct pseudospectral methods using Legendre–Gauss (LG) and Legendre–Gauss–Radau (LGR) collocation for solving infinite-horizon optimal control problems numerically. A smooth, strictly monotonic transformation is used to map the infinite time domain t ∈ [ 0 , ∞ ) onto a half-open interval τ ∈ [ − 1 , 1 ) . The resulting problem on the finite interval is transcribed to a nonlinear programming problem using collocation. The proposed methods yield approximations to the state and the costate on the entire horizon, including approximations at t = + ∞ . These pseudospectral methods can be written equivalently in either a differential or an implicit integral form. In numerical experiments, the discrete solution exhibits exponential convergence as a function of the number of collocation points. It is shown that the map ϕ : [ − 1 , + 1 ) → [ 0 , + ∞ ) can be tuned to improve the quality of the discrete approximation.

A unified framework for the numerical solution of optimal control problems using pseudospectral methods

A unified framework is presented for the numerical solution of optimal control problems using collocation at Legendre–Gauss (LG), Legendre–Gauss–Radau (LGR), and Legendre–Gauss–Lobatto (LGL) points. It is shown that the LG and LGR differentiation matrices are rectangular and full rank whereas the LGL differentiation matrix is square and singular. Consequently, the LG and LGR schemes can be expressed equivalently in either differential or integral form, while the LGL differential and integral forms are not equivalent. Transformations are developed that relate the Lagrange multipliers of the discrete nonlinear programming problem to the costates of the continuous optimal control problem. The LG and LGR discrete costate systems are full rank while the LGL discrete costate system is rank-deficient. The LGL costate approximation is found to have an error that oscillates about the true solution and this error is shown by example to be due to the null space in the LGL discrete costate system. An example is considered to assess the accuracy and features of each collocation scheme.

Design of quadrature rules for Müntz and Müntz‐logarithmic polynomials using monomial transformation

AbstractA method for constructing the exact quadratures for Müntz and Müntz‐logarithmic polynomials is presented. The algorithm does permit to anticipate the precision (machine precision) of the numerical integration of Müntz‐logarithmic polynomials in terms of the number of Gauss–Legendre (GL) quadrature samples and monomial transformation order. To investigate in depth the properties of classical GL quadrature, we present new optimal asymptotic estimates for the remainder. In boundary element integrals this quadrature rule can be applied to evaluate singular functions with end‐point singularity, singular kernel as well as smooth functions. The method is numerically stable, efficient, easy to be implemented. The rule has been fully tested and several numerical examples are included. The proposed quadrature method is more efficient in run‐time evaluation than the existing methods for Müntz polynomials. Copyright © 2009 John Wiley & Sons, Ltd.

The RKGL method for the numerical solution of initial-value problems

We introduce the RKGL method for the numerical solution of initial-value problems of the form y ′ = f ( x , y ) , y ( a ) = α . The method is a straightforward modification of a classical explicit Runge–Kutta (RK) method, into which Gauss–Legendre (GL) quadrature has been incorporated. The idea is to enhance the efficiency of the method by reducing the number of times the derivative f ( x , y ) needs to be computed. The incorporation of GL quadrature serves to enhance the global order of the method by, relative to the underlying RK method. Indeed, the RKGL method has a global error of the form Ah r + 1 + Bh 2 m , where r is the order of the RK method and m is the number of nodes used in the GL component. In this paper we derive this error expression and show that RKGL is consistent, convergent and strongly stable.