Hinfinityteori

Hinfinityteori is a field within control theory that focuses on robust control using H-infinity optimization. It aims to guarantee acceptable performance even in the face of model uncertainty and external disturbances by minimizing the worst-case gain from disturbances to controlled outputs. The central mathematical object is the H-infinity norm, defined for a linear time-invariant transfer matrix G(s) as the maximum singular value of G(jω) across all frequencies. This norm represents the largest possible amplification of disturbances by the system.

In a typical setting, a plant P with a state-space realization is given, and the goal is to design a controller K so that the closed-loop system is stable and the transfer from disturbance to a performance output has an H-infinity norm below a prescribed level γ. This formulation leads to a robust control problem that accounts for uncertainties in the plant and aims for guaranteed worst‑case performance.

Two main solution approaches are used. Riccati-based synthesis provides constructive controller formulas under certain assumptions and often yields fixed-structure controllers. Convex optimization via linear matrix inequalities (LMIs) recasts the same problem into a solvable convex program, allowing more general plant models and uncertainty descriptions. Variants include Hinfinity filtering (designing observers with H-infinity performance) and mixed H2/H∞ problems that balance energy-based criteria with worst-case guarantees.

Hinfinityteori has applications in aerospace, automotive, robotics, power systems, and process control. It originated in the robust control research of the 1980s and remains a foundational framework for ensuring stability and performance under uncertainty.