H-Infinity Optimal Control
H-infinity control minimizes the worst-case gain from disturbances to regulated outputs, delivering guaranteed robustness against bounded uncertainty.
Worst-case thinking
The H-infinity norm of a stable transfer function is the peak of its largest singular value across frequency. For a scalar system it is the maximum magnitude of the Bode plot. Minimizing this norm minimizes the worst-case amplification of any bounded-energy disturbance, which is exactly the quantity that robustness against unstructured uncertainty depends on.
Where H2 control optimizes average energy, H-infinity optimizes the peak. This makes it the natural framework when disturbances are unknown but bounded, or when a plant is only known to within a norm-bounded perturbation.
The suboptimal problem
Rather than minimizing the norm directly, the standard approach fixes a level gamma and asks whether a controller exists making the closed-loop H-infinity norm below gamma. This suboptimal problem has a clean solution via two Riccati equations whose stabilizing solutions must exist, be positive semidefinite, and satisfy a spectral-radius coupling condition. A bisection search on gamma then approaches the optimum.
Each channel of the generalized plant is weighted. The mixed-sensitivity setup stacks weighted sensitivity S, control sensitivity KS, and complementary sensitivity T, so a single H-infinity bound enforces tracking, actuator limits, and noise rejection at once.
Interpretation and limits
The small-gain theorem underlies the guarantees: if the closed loop has H-infinity norm below 1/gamma against a perturbation of norm below gamma, stability is preserved. The cost is conservatism, because unstructured uncertainty is often larger than reality. Structured uncertainty is handled better by mu-synthesis.
H-infinity controllers match the plant order plus the weights and can be reduced afterward. For tightly coupled multivariable systems such as multi-coil magnetic control of a spherical tokamak like Hyperion, the framework provides a principled way to trade robustness against performance in simulation studies.
- Peak-gain objective gives guaranteed robust stability margins
- Solved via Riccati equations or LMIs at each gamma level
- Mixed sensitivity encodes competing specifications in one norm
- Conservative under structured uncertainty; use mu-synthesis there