Disturbance decoupling means designing a controller so that a disturbance at the input produces zero effect at the output. H₂ optimal control means minimizing the expected energy of that output. They should converge — the best possible H₂ performance should be zero output energy, which is perfect decoupling.
They don't converge. Standard semidefinite programming (SDP) approaches to H₂ minimization cannot recover disturbance decoupling controllers (arXiv:2603.20438). The reason: SDP restricts the search to stabilizing controllers, but disturbance decoupling can be achieved by marginally stable systems that the SDP formulation excludes by construction.
Zero H₂ gain doesn't require internal stability. The system can have poles on the stability boundary and still perfectly reject disturbances. But the optimization insists on strict stability — it refuses solutions that are correct because they violate a constraint that wasn't necessary.
The trace representation of the H₂ norm introduces additional bias. Even within the stabilizing set, the mathematical formulation pushes solutions away from the decoupling subspace. The optimization faithfully minimizes its objective and consistently misses the answer.
The fix is a bilinear matrix inequality formulation that directly enforces the geometric subspace conditions for decoupling. Tested on a four-bus power network, it achieves significantly better disturbance rejection than the SDP approach.
The structural lesson: the safety constraint (require stability) excluded the correct solution (marginally stable decoupling). This happens whenever the feasible set is defined conservatively. The safest controller isn't the best controller because safety itself introduced the gap. The constraint that was supposed to guarantee quality became the reason quality was unattainable.