Exploiting Sparsity in Direct Collocation Pseudospectral Methods for Solving Optimal Control Problems

Abstract

In a direct collocation pseudospectral method, a continuous-time optimal control problem is transcribed to a finitedimensional nonlinear programming problem. Solving this nonlinear programming problem as efficiently as possible requires that sparsity at both the first-and second-derivative levels be exploited. In this paper, a computationally efficient method is developed for computing the first and second derivatives of the nonlinear programming problem functions arising from a pseudospectral discretization of a continuous-time optimal control problem. Specifically, in this paper, expressions are derived for the objective function gradient, constraint Jacobian, and Lagrangian Hessian arising from the previously developed Radau pseudospectral method. It is shown that the computation of these derivative functions can be reduced to computing the first and second derivatives of the functions in the continuous-time optimal control problem. As a result, the method derived in this paper reduces significantly the amount of computation required to obtain the first and second derivatives required by a nonlinear programming problem solver. The approach derived in this paper is demonstrated on an example where it is found that significant computational benefits are obtained when compared against direct differentiation of the nonlinear programming problem functions. The approach developed in this paper improves the computational efficiency of solving nonlinear programming problems arising from pseudospectral discretizations of continuous-time optimal control problems.

Stochastic 4D trajectory optimization for aircraft conflict resolution

“…Especially the Radau pseudospectral method [26] [27] is employed to solve the optimal control problems. For solving the stochastic optimal control problem, the stochastic process including the statistical information (expected value and variance) is approximated by the theory of the gPC method mentioned in Sec.…”

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

“…All three types of Gaussian quadrature points are defined on the domain [ − 1,1] but differ in that the LG points include neither of the endpoints, the LGR points include one of the endpoints, and the LGL points include both of the endpoints. The use of global polynomials together with Gaussian quadrature collocation points is known to provide accurate approximations that converge exponentially fast for problems whose solutions are smooth . An advantage of these methods is that by computing the solution of the control problem accurately at a small number of carefully chosen points, one obtains an accurate global approximation.…”

“…The continuous‐time optimal control problem is then transcribed to a finite‐dimensional nonlinear programming problem (NLP), and the NLP is solved using well known software . Recently, a great deal of research has been done on the class of Gaussian quadrature orthogonal collocation methods . In a Gaussian quadrature orthogonal collocation method, the state is approximated using a basis of either Lagrange or Chebyshev polynomials, and the dynamics are collocated at points associated with a Gaussian quadrature.…”

“…In a Gaussian quadrature orthogonal collocation method, the state is approximated using a basis of either Lagrange or Chebyshev polynomials, and the dynamics are collocated at points associated with a Gaussian quadrature. The most common Gaussian quadrature collocation points are Legendre–Gauss (LG) , Legendre–Gauss–Radau (LGR) , and Legendre–Gauss–Lobatto . All three types of Gaussian quadrature points are defined on the domain [ − 1,1] but differ in that the LG points include neither of the endpoints, the LGR points include one of the endpoints, and the LGL points include both of the endpoints.…”

Utilizing the Algorithmic Differentiation Package ADiGator for Solving Optimal Control Problems Using Direct Collocation

“…In this paper we study the use of the ADiGator algorithmic differentiation package in order to compute the the first and second derivatives of the NLP arising from direct collocation methods. While the methods of this paper may be applied to multiple direct collocation schemes, we focus on an hp-adaptive Legendre-Gauss-Radau [8,9,10,11,12,13] scheme which has been coded in the commercial optimal control software GPOPS-II. This provides us with a good test environment due to the manner in which GPOPS-II requires only the derivatives of the control problem [5], together with the fact that it is implemented in the same language as the ADiGator tool.…”