The Legendre–Gauss–Radau Pseudospectral Method

This section develops the Legendre–Gauss–Radau (LGR) pseudospectral method, derives both its differential and integral forms, compares it with the LG method, and explains why LGR is often preferred for practical optimal control because it provides a control variable at the initial time.

Review of the LG Method¶

For LG collocation,

\tau_1>-1,\qquad \tau_N<1.

(1)

The state approximation is

X(\tau)=\sum_{i=0}^{N}X_iL_i(\tau),

(2)

leading to

D_0X_0+D_{1:N}X_{1:N}=F_{1:N},

(3)

and

X_{N+1}=X_0+\mathbf w^TF_{1:N}.

(4)

LGR Collocation Points¶

Unlike LG, the LGR points include the left endpoint,

\tau_1=-1,

(5)

while

\tau_N<1.

(6)

Introduce one additional non-collocated point

\tau_{N+1}=1.

(7)

State Approximation¶

Approximate the state by

X(\tau)= \sum_{i=1}^{N+1}X_iL_i(\tau),

(8)

with

L_i(\tau)= \prod_{\substack{j=1\\j\neq i}}^{N+1} \frac{\tau-\tau_j}{\tau_i-\tau_j}.

(9)

The approximation uses $N+1$ support points and therefore has degree $N$ .

Derivative Approximation¶

Differentiating,

\dot X(\tau)= \sum_{i=1}^{N+1}X_iL_i'(\tau).

(10)

The derivative has degree $N-1$ and is uniquely determined by $N$ conditions.

Collocation¶

Enforce the dynamics at the $N$ LGR points,

\dot X(\tau_j)=f(X_j,U_j), \qquad j=1,\ldots,N.

(11)

Define

D_{ji}=L_i'(\tau_j),

(12)

giving

\sum_{i=1}^{N+1}D_{ji}X_i=F_j.

(13)

Partition

D= \begin{bmatrix} D_1&D_{2:N+1} \end{bmatrix},

(14)

to obtain

\boxed{ D_1X_1+D_{2:N+1}X_{2:N+1}=F_{1:N}. }

(15)

Integral Form¶

The continuous dynamics satisfy

X(\tau)=X(-1)+ \int_{-1}^{\tau}f(s)\,ds.

(16)

Evaluating at

\tau_j,\qquad j=2,\ldots,N+1,

(17)

and approximating each integral by LGR quadrature gives

X_j= X_1+ \sum_{i=1}^{N}\alpha_{ji}F_i.

(18)

Collecting all equations,

\boxed{ X_{2:N+1} = \mathbf1X_1+ A_{2:N+1}F_{1:N}. }

(19)

Equivalence¶

D_{2:N+1}

(20)

is nonsingular,

X_{2:N+1} = -D_{2:N+1}^{-1}D_1X_1 + D_{2:N+1}^{-1}F_{1:N}.

(21)

The key identities are

-D_{2:N+1}^{-1}D_1=\mathbf1,

(22)

and

D_{2:N+1}^{-1}=A_{2:N+1}.

(23)

Hence the differential and integral formulations are mathematically identical.

Terminal State¶

The final row yields

X_{N+1} = X_1+ \sum_{i=1}^{N}w_iF_i,

(24)

which is exactly the LGR Gaussian quadrature formula.

Comparison with LG¶

Property	LG	LGR
Left endpoint	No	Yes
Right endpoint	No	No
Extra state point	Initial	Final
Control at initial time	No	Yes
Quadrature exactness	$2N-1$	$2N-2$
Integration method	Yes	Yes

Control Approximation¶

LG provides controls only at interior Gauss points,

U_1,\ldots,U_N,

(25)

so there is no control at $t_0$ .

LGR provides

U_1,\ldots,U_N,

(26)

where $U_1$ corresponds to

\tau=-1,

(27)

allowing the optimal control to be applied immediately.

General Time Interval¶

For

t\in[t_0,t_f],

(28)

the mapping

t=\frac{t_f-t_0}{2}\tau+\frac{t_f+t_0}{2}

(29)

gives

\frac{dx}{d\tau} = \frac{t_f-t_0}{2}f(x,u,t),

(30)

and

J= \Phi+ \frac{t_f-t_0}{2} \int_{-1}^{1}L(x,u,\tau)\,d\tau.

(31)

Only the scaling factor changes; the transcription remains unchanged.

Advantages¶

True Gaussian quadrature method.
Differential and integral forms are equivalent.
Initial control value is available.
Terminal state is recovered naturally.
Sparse NLP structure.

Summary¶

LGR includes the left endpoint.
The state polynomial has degree $N$ .
Dynamics are collocated at $N$ Radau points.
The differentiation matrix has a full-rank square submatrix.
Its inverse is the Radau integration matrix.
The terminal state follows directly from Radau quadrature.
LGR sacrifices one degree of quadrature exactness relative to LG but gains an initial control value.