Pseudospectral optimal control

Definition

Consider a trajectory optimization problem on $[t_0, t_f]$ with dynamics $x' = f(x, u, t)$, cost $\phi(x(t_f)) + \int_{t_0}^{t_f} L(x, u, t) \, dt$, and boundary conditions. Map the interval to $[-1, 1]$ by $t = \frac{t_f - t_0}{2} \tau + \frac{t_f + t_0}{2}$. Let $N \ge 2$ and let $-1 = \tau_0 < \tau_1 < \cdots < \tau_N = 1$ be the Legendre–Gauss–Lobatto nodes, where $\tau_1, \dots, \tau_{N-1}$ are the zeros of $P_N'$ and $P_N$ is the Legendre polynomial of degree $N$. Let $w_0, \dots, w_N$ be the corresponding quadrature weights, and let $\ell_0, \dots, \ell_N$ be the Lagrange basis polynomials of the nodes. Let $D$ be the $(N + 1) \times (N + 1)$ matrix with $D_{kj} = \ell_j'(\tau_k)$. The Legendre pseudospectral method replaces the problem by the nonlinear program with unknowns $x_0, \dots, x_N \in \R^n$ and $u_0, \dots, u_N \in \R^m$:

1. minimise $\phi(x_N) + \frac{t_f - t_0}{2} \sum_{k=0}^N w_k L(x_k, u_k, t(\tau_k))$;
2. subject to $\sum_{j=0}^N D_{kj} x_j = \frac{t_f - t_0}{2} f(x_k, u_k, t(\tau_k))$ for $k = 0, \dots, N$;
3. and to the boundary conditions on $x_0$ and $x_N$ and the path constraints at the nodes.

The state is then approximated by $x(\tau) \approx \sum_j x_j \ell_j(\tau)$. Pseudospectral optimal control is the family of such methods, with Legendre, Chebyshev or other Gaussian nodes.