H-Infinity Control
H-infinity control minimizes the worst-case gain from disturbances to errors, giving a systematic route to robust multivariable design.
Minimizing the Worst Case
H-infinity control designs a controller that minimizes the H-infinity norm of a chosen transfer function, from disturbances and references to errors and control effort. The H-infinity norm is the peak gain over all frequencies, so minimizing it minimizes the worst-case amplification the system can produce. This worst-case focus is what makes the method inherently robust.
The generalized plant
H-infinity design casts the problem in a standard form: a generalized plant with two kinds of inputs (external signals and control) and two kinds of outputs (performance signals and measurements). The designer specifies performance by inserting frequency-dependent weighting functions that shape which errors matter at which frequencies. The synthesis then finds the controller minimizing the weighted worst-case gain.
Weighting functions and loop shaping
- A weight on the sensitivity function shapes disturbance rejection and tracking at low frequency.
- A weight on the complementary sensitivity shapes noise rejection and robustness at high frequency.
- A weight on control effort limits actuator activity.
- Choosing these weights is the heart of the design; the numerical solver handles the rest.
How it is solved
Given the generalized plant and weights, the H-infinity controller is computed by solving two coupled Riccati equations (or, in modern solvers, linear matrix inequalities). The result is a controller of comparable order to the weighted plant that guarantees the worst-case gain stays below the achieved bound.
Strengths and costs
H-infinity handles multivariable systems and robustness in one unified framework, with explicit frequency-domain performance specifications. Its costs are that the resulting controllers can be high-order and that translating design intent into good weighting functions takes experience. For structured uncertainty, mu-synthesis extends the method further.
H-infinity control is standard in aerospace and other high-value applications where robustness to model error is non-negotiable, and it is a natural fit for control loops that must run against an approximate model of a complex, hard-to-characterize plant.