H-infinity control is one of the robust control theories, a controller design method aimed at guaranteeing control performance against disturbances and model uncertainties in a system. In particular, it aims to minimize the system’s response to the worst-case disturbance.
H-infinity Norm
In H-infinity control, the H-infinity norm is used to evaluate system performance. The H-infinity norm represents the maximum gain of the system’s frequency response, meaning the maximum effect of disturbances on the output.
Given the closed-loop transfer function \(T_{zw}\) from disturbance \(w\) to the controlled output \(z\), its H-infinity norm is defined as:
\[ ||T*{zw}||*\infty = \sup*{\omega} ||T*{zw}(j\omega)|| \]This represents the maximum value (peak value) of the transfer function gain across all frequencies \(\omega\).
From an energy gain perspective, it is also defined as:
\[ ||T*{zw}||*\infty = \sup\_{w \neq 0} \frac{||z||\_2}{||w||\_2} \]Here, \(||\cdot||_2\) represents the L2 norm (energy) of the signal.
\[ ||z||\_2^2 = \int_0^\infty z^T(t)z(t)dt \]\[ ||w||\_2^2 = \int_0^\infty w^T(t)w(t)dt \]The goal of the H-infinity control problem is to design a controller \(K\) that keeps the H-infinity norm \(||T_{zw}||_\infty\) below a positive constant \(\gamma\). That is, finding a controller that satisfies:
\[ ||T*{zw}||*\infty < \gamma \]This \(\gamma\) is called the performance index or gain, and a smaller value indicates better disturbance rejection performance.
Formulation of the H-infinity Control Problem
The H-infinity control problem is often formulated as the following optimization problem using state-space representation:
\[ \min*K \sup*{w \neq 0} \frac{||z||\_2}{||w||\_2} \]This aims to minimize the energy gain from disturbance \(w\) to output \(z\) by designing the controller \(K\).
From a game-theoretic perspective, it can also be expressed as:
\[ \min_u \max_w \int_0^\infty (z^T(t)z(t) - \gamma^2 w^T(t)w(t)) dt \]This is interpreted as a minimax problem where the control input \(u\) is chosen to minimize \(z^T(t)z(t)\), while the worst-case disturbance \(w\) maximizes \(w^T(t)w(t)\).
H-infinity control is applied in various fields requiring high robustness, including aerospace, robotics, and process control.