H-infinity - definition
H-infinity - definition
H-infinity (denoted H∞) is a mathematical and engineering concept used in control theory, signals and systems theory, referring to a class of norms and methods for designing fault-tolerant systems. In the context of linear systems analysis, H∞ defines the norm of a system's transfer function as the maximum gain of the input signal relative to the output signal in the frequency domain. Formally, the H∞ norm is equal to the supremum of the largest value of the singular transmittance along the complex axis.
This concept is used in the design of resilient control systems (robust control), where the aim is to ensure stability and performance despite uncertainties in model parameters, disturbances or non-linearities. Unlike classical methods based on the minimisation of the mean-square error, the H∞ approach focuses on the worst-case scenario - the maximum disturbance energy the system can pass through - making it a more conservative but also safer tool for critical applications.
From a systems theory point of view, the H∞ space contains all stable and proper norm-limited complex functions defined on a positive-frequency complex half-plane (for continuous systems) or inside the unit Wheels (for discrete systems). Analysis and synthesis in this space is based on Tools such as linear matrix inequalities (LMIs), Riccati transformations and the Bounded Real Lemma theorem.
The H∞ standard and associated design algorithms are widely used in advanced engineering fields such as flight control, adaptive systems, mechatronics or high-precision electronics, where it is crucial to maintain high quality control regardless of varying operating conditions and disturbance patterns.
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