Control Allocation Methods

The mathematics of distributing control demands across redundant, constrained actuators - pseudo-inverses, weighting, and optimization.

The allocation equation

Suppose a vector of virtual control demands v must be produced by actuator commands u through a control-effectiveness matrix B, so that Bu = v. When there are more actuators than demands (B is wide), infinitely many u satisfy the equation, and allocation chooses among them. When actuators are limited, an exact solution may not exist and allocation finds the best feasible one.

Pseudo-inverse allocation

The simplest choice is the Moore-Penrose pseudo-inverse: u = B^+ v, which gives the minimum-norm command that achieves the demand. Weighting the norm (u = W^-1 B^T (B W^-1 B^T)^-1 v) lets designers favor cheaper or faster actuators. This is fast and closed-form but ignores hard limits.

Constrained optimization

When limits bind, allocation becomes a constrained problem solved each cycle: minimize a cost on command size and on the error v - Bu, subject to lower and upper bounds and rate limits on u. Quadratic programming solves this efficiently and predictably, degrading to the closest achievable demand when the exact one is infeasible.

Prioritized allocation

Not all demands are equal. Prioritized (cascaded) allocation satisfies high-priority objectives first - vertical stability, protection - then uses remaining actuator freedom for lower-priority goals. This guarantees that under saturation the system sheds the least important objectives rather than failing everything at once.

Why it belongs in real time

Because actuator authority and limits change through a discharge, allocation must run live inside the control cycle, not be fixed offline. The method must be deterministic and bounded in time - a solved quadratic program with a worst-case iteration count - so it fits the hard-real-time budget of the control stack.