control//robust control//H-infinity control
H-infinity control is a robust control synthesis method that finds the controller minimizing the worst-case amplification, over all frequencies, from disturbances and model uncertainty to the errors you care about, shaped by frequency weights the designer chooses; it is used where requirements are multivariable, demanding and must be certified with numbers, mostly in aerospace and in some precision mechatronics such as hard-disk servos. The name comes from the \(H_\infty\) norm, the peak gain of a transfer matrix across frequency. Where the LQR minimizes an average (a quadratic cost), H-infinity minimizes the worst case.
H-infinity control is a robust control synthesis method that finds the controller minimizing the worst-case amplification, over all frequencies, from disturbances and model uncertainty to the errors you care about, shaped by frequency weights the designer chooses; it is used where requirements are multivariable, demanding and must be certified with numbers, mostly in aerospace and in some precision mechatronics such as hard-disk servos. The name comes from the H∞H_\inftyH∞ norm, the peak gain of a transfer matrix across frequency. Where the LQR minimizes an average (a quadratic cost), H-infinity minimizes the worst case.
The usual set-up stacks the closed-loop maps that matter, each multiplied by a weight that says how small it must be at each frequency, and asks for
∥[W1SW2T]∥∞<1,\left\lVert\begin{bmatrix}W_1S\\ W_2T\end{bmatrix}\right\rVert_\infty<1,[W1SW2T]∞<1,
where SSS is the sensitivity function (how much disturbance reaches the output), T=1−ST=1-ST=1−S the complementary sensitivity (how much reference and sensor noise get through), W1W_1W1 large at low frequency to demand disturbance rejection and W2W_2W2 large at high frequency where the model is unreliable. If the inequality holds, the loop is stable for every plant within the uncertainty that W2W_2W2 describes. Since S+T=1S+T=1S+T=1, both cannot be small at the same frequency, and the weights are where that conflict is settled.
All the work is in choosing the weights. The synthesis itself is a solver call (python-control's hinfsyn and mixsyn); a poor weight gives a controller that is optimal for the wrong question.
It produces high-order controllers, often as many states as the plant plus all the weights, which are then reduced (model reduction) before they go on the flight computer.
It is a game. H-infinity control is a zero-sum game against a nature that chooses the worst disturbance, and the controller is the minimax strategy (zero-sum game).
Mu-synthesis extends it to structured uncertainty (a real parameter here, an unmodelled lag there), where treating all uncertainty as one lump would be too conservative.
Its guarantee, like every robust guarantee, covers only the uncertainty described. Maturity: respected and niche, deployed mostly in aerospace; Monte Carlo testing of a simpler design remains far more common.