New transcription method based on orthogonal collocation

[franckgaga]

Similar as implemented here: https://www.do-mpc.com/en/latest/theory_orthogonal_collocation.html

[franckgaga]

Hello @baggepinnen !

I'm starting to implement a new OrthogonalCollocation transcription method and I have a question. In the section here, how do we compute the $a_{j,k}$ coefficients ? I took a look at Biegler (2010) and they provide the values for the polynomial roots (that is, the $\tau$ values), but not the time derivative of the Lagrange polynomial at the roots. What did you do in DyadControlSystems, you used ForwardDiff.jl for this ? edit: or hard-coded values based on analytical derivatives ?

[baggepinnen]

You can see my implementation here
https://github.com/baggepinnen/SeeToDee.jl/blob/main/src/SeeToDee.jl#L452C2-L467C4

[franckgaga]

Thanks for the reference @baggepinnen, it helps a lot.

edit: For posterity, to answer my initial question, most implementations (incl. SimpleColloc above) rely on analytical derivatives to compute the $a_{j,k}$ coefficients, since (1) it's fairly easy to compute using differentiation of polynomials and (2) it leads to a matrix multiplication with the collocation points in the equality constraints, which is cheap.

I have another question: it seems that continuity constraint is not explicitly added in SimpleColloc implementation. Is it because you only support schemes that always include a point at $\tau=1$ ? If so, I guess your implementation would need additional logic to support e.g. Gauss-Legendre scheme (that does not include the point at $\tau=1$) ?

[baggepinnen]

Yeah, I only support two schemes in SeeToDee, those both include the end point