Efficiently solve Optimal Control Allocation with Active-Set

Solves $$ \min_u\ || Bu - \nu ||{W\nu}^2 + \gamma^2 || u - u_d ||_{W_u}^2\ s.t.\quad \underbar{u}_i \leq u_i \leq \overline{u}_i \quad \forall i $$

where $W_\nu$, $W_u$ diagonal and positive definite. $u,\nu$ vectors and $\gamma>0$ scalar and small. $\underbar{u}_i,\overline{u}_i$ bound the decision variables $u_i$.

Features

• Provides setup routines (setupWLS.h) to turn the weighted least-squares control allocation problem above into a standard least-squares form $||Au-b||$, s.t. $\underbar{u}\leq u \leq\overline{u}$.
• Exploits sparsity from diagonal weighing matrices.
• QR and CHOL choices use efficiently updating QR and Cholesky factorisations.

Limitations

• Does not solve general active-set problems, since highly optimised for control allocation where diagonal matrix $W_u$ gives rise to sparsity.
• CG no reliable. Only available when compiling with with -DAS_INCLUDE_CG
• CHOL less numerically stable with single floats. Choose low cond_bound and see solveActiveSet.h