Costate Estimation using Multiple-Interval Pseudospectral Methods

Abstract

A method is presented for costate estimation in nonlinear optimal control problems using multiple-interval collocation at Legendre-Gauss or Legendre-Gauss-Radau points. Transformations from the Lagrange multipliers of the nonlinear programming problem to the costate of the continuous-time optimal control problem are given. When the optimal costate is continuous, the transformed adjoint systems of the nonlinear programming problems are discrete representations of the continuous-time first-order optimality conditions. If, however, the optimal costate is discontinuous, then the transformed adjoint systems are not discrete representations of the continuous-time firstorder optimality conditions. In the case where the costate is discontinuous, the accuracy of the costate approximation depends on the locations of the mesh points. In particular, the accuracy of the costate approximation is found to be significantly higher when mesh points are located at discontinuities in the costate. Two numerical examples are studied and demonstrate the effectiveness of using the multiple-interval collocation approach for estimating costate in continuous-time nonlinear optimal control problems. = initial time ut = control on time domain t 2 t 0 ; t f u = control on time domain 2 1; 1 U j = control approximation at time point j w j = jth Legendre-Gauss or Legendre-Gauss-Radau quadrature weight X = state approximation on time domain 2 1; 1 xt = state on time domain t 2 t 0 ; t f x = state on time domain 2 1; 1 X j = state approximation at time point j = Lagrange multiplier associated with discretized inequality path constraint = Lagrange multiplier associated with inequality path constraint ij = Kronecker delta function = accuracy tolerance j = Lagrange multiplier of discretized dynamic constraint at j = continuous costate on the time domain 2 1; 1 j = costate approximation at time point j = Lagrange multiplier associated with quadrature constraint = jump in the costate at mesh point = transformed time domain 1; 1 = Mayer cost = boundary condition function = Lagrange multiplier associated with the discretized boundary condition = Lagrange multiplier associated with the boundary condition

Time-optimal reorientation of the rigid spacecraft using a pseudospectral method integrated homotopic approach

“…For problems where the solutions are infinitely smooth and well behaved, a pseudospectral method has a simple structure and converges at an exponential rate . Most importantly, when the optimal costate is continuous, the results presented in show that, for the GPM and RPM, the transformed adjoint systems of the NLP are discrete representations of the continuous‐time first‐order optimality conditions and accurate costate estimations can be obtained from the Karush–Kuhn–Tucker (KKT) multipliers of the NLP. However, for problems with either nonsmooth solutions or nonsmooth problem formulations, applying the pseudospectral methods directly to solve these problems may result in low convergence speed and accuracy.…”

“…In recent years, development has increased in pseudospectral methods, in which the collocation points are based on accurate quadrature rules and the basic functions are typical Chebyshev or Lagrange polynomials . The most well‐developed pseudospectral methods are the Lobatto pseudospectral method , Gauss pseudospectral method (GPM) , and the Radau pseudospectral method (RPM) . Using pseudospectral collocation, the differential equations are converted into algebraic equations, which are generally more convenient to solve.…”

An Introduction to Trajectory Optimization: How to Do Your Own Direct Collocation

“…The number of terms in the infinite series is related to the order of the polynomial: a higher-order polynomial approximation will be more accurate. There are many papers that cover the detailed mathematics of orthogonal polynomials [26,44,4,33,58,28] and their use in trajectory optimization [29,62,18,30,19,53,3,57,23,24,15,21]. Here we will focus on the practical implementation details and on gaining a qualitative understanding of how orthogonal collocation works.…”

“…Orthogonal collocation is similar to direct collocation, but it generally uses higher-order polynomials. The collocation points for these methods are located at the roots of an orthogonal polynomial, typically either Chebyshev or Legendre [15]. Increasing the accuracy of a solution is typically achieved by increasing either the number of trajectory segments or the order of the polynomial in each segment.…”

Direct Trajectory Optimization and Costate Estimation of State Inequality Path-Constrained Optimal Control Problems Using a Radau Pseudospectral Method

“…The approach is motivated by the results of Ref. 35 where it was shown that the accuracy of the pseudospectral costate estimate can be quite poor due to discontinuities that arise in the presence of active state-inequality path constraints. The method developed in this paper utilizes a modified version of the Radau pseudospectral method.…”