Transcriptions: Pseudospectral Methods

Pseudospectral methods approximate the time-dependent variables of an optimal control problem using global polynomials, typically expressed in terms of Legendre or Chebyshev basis functions. The differential-algebraic equations of the problem are collocated at orthogonal collocation points, which are chosen to minimise the interpolation error. For smooth problems, pseudospectral methods exhibit spectral convergence: the approximation error decreases exponentially as the number of collocation points increases, so that highly accurate solutions can be obtained with comparatively coarse grids.

PSOPT implements pseudospectral collocation based on the following families of points:

• Legendre–Gauss–Lobatto (LGL) points, which include both endpoints of the interval.

• Chebyshev–Gauss–Lobatto (CGL) points, associated with Chebyshev polynomials.

• Legendre–Gauss–Radau (LGR) points, which include one endpoint of the interval.

• Legendre–Gauss (LG) points, which are interior to the interval.

The Radau and Gauss schemes have well-known advantages for costate estimation, as their associated Karush–Kuhn–Tucker multipliers can be mapped to accurate approximations of the costate (adjoint) variables of the continuous problem. PSOPT computes these costate estimates, which allows solutions to be checked against the necessary conditions of optimality.

Pseudospectral discretisations can be applied globally over each phase, or over multiple mesh intervals within a phase. PSOPT includes hp-adaptive mesh refinement for its pseudospectral discretisations: the refinement algorithm estimates the discretisation error and automatically decides, for each mesh interval, whether to increase the polynomial degree (p-refinement) or to subdivide the interval (h-refinement), so that the solution meets a specified accuracy. This combination is particularly effective for problems whose solutions are smooth over most of the horizon but contain localised regions of rapid change.