Pseudospectral Optimal Control - Details

Details

These methods include forms of the collocation at Legendre–Gauss–Lobatto points, collocation at Chebyshev–Gauss–Lobatto points, Legendre–Gauss points (known as the Gauss Pseudospectral Method), and collocation at Legendre–Gauss–Radau points (known as the Radau Pseudospectral Method). It is also noted that versions of the Gauss and Radau pseudospectral methods have been developed for solving infinite-horizon optimal control problems. It is important to note that the Lobatto pseudospectral method has the property that the differentiation matrix is square and singular whereas the Gauss and Radau pseudospectral methods have the property that the differentiation matrices are non-square and full rank. This last property of the Gauss and Radau pseudospectral methods leads to the fact that either of these latter two methods can be written equivalently in either differential or implicit integral form.

In pseudospectral methods, integration is approximated by quadrature rules, which provide the best numerical integration result. For example, with just N nodes, a Legendre-Gauss quadrature integration achieves zero error for any polynomial integrand of degree less than or equal to . In the PS discretization of the ODE involved in optimal control problems, a simple but highly accurate differentiation matrix is used for the derivatives. Because a PS method enforces the system at the selected nodes, the state-control constraints can be discretized straightforwardly. All these mathematical advantages make pseudospectral methods a straightforward discretization tool for continuous optimal control problems. One interesting property of pseudospectral optimal control is that, if done correctly, it permits commutativity between discretization and dualization. Specifically, this commutativity exists if the Gauss pseudospectral method (GPM, which uses Legendre–Gauss points) or the Radau pseudospectral method (RPM, which uses Legendre–Gauss–Radau points) are used. For either the GPM or RPM, the KKT multipliers are related to the costates of the continuous problem in an algebraically simple manner. In the case of Gauss-Lobatto points, this commutativity is lost because the transformed adjoint system is singular in the discretized costate.