Orthogonal Collocation and Gaussian Quadrature

This section introduces orthogonal collocation, polynomial state approximations, Gaussian quadrature, and Legendre–Gauss (LG), Legendre–Gauss–Radau (LGR), and Legendre–Gauss–Lobatto (LGL) collocation points.

From Euler Collocation to General Collocation¶

Euler collocation enforces

\frac{x_{k+1}-x_k}{h_k}=f(x_k,u_k,t_k),

(1)

or equivalently

x_{k+1}-x_k-h_k f(x_k,u_k,t_k)=0.

(2)

The left-hand side approximates the derivative while the right-hand side evaluates the dynamics at the same point. This is the fundamental idea of collocation.

Polynomial Approximation¶

Over each mesh interval,

x^{(k)}(t)\approx \sum_{j=0}^{m} c_j^{(k)}\,s_j^{(k)}(t),

(3)

where $s_j^{(k)}(t)$ are basis polynomials.

Differentiating,

\dot x^{(k)}(t)\approx \sum_{j=0}^{m} c_j^{(k)}\dot s_j^{(k)}(t).

(4)

The derivative polynomial has degree $m-1$ .

Collocation Conditions¶

Choose $m$ collocation points inside each mesh interval,

\tau_1,\tau_2,\ldots,\tau_m.

(5)

At every collocation point,

\dot x^{(k)}(\tau_i)= f\!\left(x^{(k)}(\tau_i),u^{(k)}(\tau_i),\tau_i\right), \qquad i=1,\ldots,m.

(6)

Because the derivative polynomial has degree $m-1$ , these provide $m$ equations. One additional condition is required.

Continuity Condition¶

The missing equation is supplied by continuity:

x^{(k)}(t_k)=x^{(k+1)}(t_k).

(7)

For the first interval,

x(t_0)=x_0.

(8)

Thus every interval has exactly the correct number of equations.

Euler as a Special Case¶

Euler forward corresponds to:

a linear state approximation,
a constant derivative approximation,
one collocation point per interval,
one continuity condition.

Quadrature¶

The cost integral

J=\int_{-1}^{1}f(\tau)\,d\tau

(9)

is approximated by

J\approx \sum_{i=1}^{N}w_i f(\tau_i),

(10)

where $w_i$ are quadrature weights satisfying

\sum_i w_i=2.

(11)

Choosing Optimal Points¶

Rather than equally spaced points, Gaussian quadrature chooses both the locations $\tau_i$ and weights $w_i$ to minimize integration error.

LGL Points¶

Endpoints are included:

\tau_1=-1,\qquad \tau_N=1.

(12)

Interior points are roots of

\frac{d}{d\tau}P_{N-1}(\tau).

(13)

LGL exactly integrates polynomials of degree

2N-3.

(14)

LGR Points¶

Only the left endpoint is fixed. The points are roots of

P_N(\tau)+P_{N-1}(\tau).

(15)

LGR exactly integrates polynomials of degree

2N-2.

(16)

LG Points¶

Neither endpoint is fixed. The points are roots of

P_N(\tau).

(17)

LG exactly integrates polynomials of degree

2N-1.

(18)

Accuracy Comparison¶

Method	Endpoints Included	Exact Polynomial Degree
LGL	Both	$2N-3$
LGR	One	$2N-2$
LG	None	$2N-1$

Relaxing endpoint constraints increases quadrature accuracy.

Orthogonal Collocation¶

Legendre polynomials satisfy

\int_{-1}^{1}P_i(\tau)P_j(\tau)\,d\tau=0, \qquad i\neq j.

(19)

Because of this orthogonality, collocation at Gaussian points yields highly accurate numerical integration and transcription schemes.

Connection to Optimal Control¶

Within each mesh interval:

Approximate the state by a polynomial.
Differentiate the polynomial.
Enforce the dynamics at Gaussian collocation points.
Use Gaussian quadrature to approximate the integral cost.
Enforce continuity between neighboring mesh intervals.

This produces a sparse nonlinear programming problem.

Summary¶

Euler collocation is the simplest orthogonal collocation method.
Polynomial approximations replace piecewise linear approximations.
Continuity supplies the additional equation needed in each interval.
LG, LGR, and LGL differ only in endpoint constraints.
Gaussian quadrature dramatically improves integration accuracy.
These ideas form the basis of pseudospectral optimal-control methods.

Connection. The polynomial constructions used in orthogonal collocation rest on Legendre orthogonality, Lagrange interpolation, and exactness properties of Gaussian quadrature.