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.