Robotics

H-infinity

1981ActivePublished: 4 May 2026Updated: 4 May 2026Published

Key innovation

Minimization of the H∞ gain (supremum norm of the transfer function in the frequency domain) as a robust controller design criterion against worst-case disturbance signals.

How it works

Formulate the problem as a generalized plant P(s) with outputs z (to be minimized) and y (measurements), and inputs w (disturbances) and u (control). Find K(s) minimizing ||F_l(P,K)||_∞ < γ by solving two Riccati equations (or LMIs). The resulting controller K is optimal for the worst-case w.

Problem solved

How to design a controller that guarantees bounded performance degradation across an entire class of disturbances — rather than just their nominal values — without explicit knowledge of the disturbance shape.

Components

Generalized Plant P(s)Full representation of the design problem.

Generalised plant combining the nominal model, performance weights W_p and uncertainty weights W_u.

Weighting FunctionsSpecifies the robustness–performance tradeoff.

Frequency-domain filters W_1(s) (perf.), W_2(s) (control effort), W_3(s) (robustness) shaping the open loops.

H∞ Optimizer (Riccati / LMI)Synteza regulatora.

Solver minimising ||T_{zw}||_∞ by solving Riccati equations or an LMI formulation.

Implementation

Implementation pitfalls

High controller orderMedium

H∞ synthesises controllers of order equal to the generalised plant (can be very high). High-order controllers are hard to implement in embedded systems.

Fix:Model order reduction (Schur balanced truncation, Hankel norm approximation).

Weight selection for W — iterative and non-intuitiveMedium

Selecting shaping weights is iterative and requires expertise; bad weights produce a conservative or unstable controller.

Fix:Loop shaping (McFarlane–Glover), systematic H∞ loop shaping iteration.