mathcalHinfty

The H-infinity norm, often denoted as $\mathcal{H}_{\infty}$, is a mathematical concept used primarily in control theory and signal processing. It represents the worst-case gain from an input signal to an output signal for a linear time-invariant (LTI) system. In simpler terms, it quantifies the maximum amplification of any possible input disturbance by the system.

Mathematically, the $\mathcal{H}_{\infty}$ norm of a stable linear system is defined as the supremum of the ratio of the L2 norm of the output signal to the L2 norm of the input signal, over all possible non-zero input signals. This norm is particularly useful because it provides a measure of system robustness. A system with a small $\mathcal{H}_{\infty}$ norm is considered robust, meaning it is less sensitive to external disturbances and model uncertainties.

The problem of designing controllers that minimize this $\mathcal{H}_{\infty}$ norm is known as $\mathcal{H}_{\infty}$ control. This approach aims to design controllers that keep the worst-case gain of the closed-loop system below a certain acceptable level. This is achieved by solving certain matrix inequalities, often referred to as the Algebraic Riccati Equations or Inequalities.

$\mathcal{H}_{\infty}$ control is a powerful technique for designing controllers that guarantee a certain level of performance and stability robustness in the face of uncertainties and disturbances. It has found wide applications in various engineering fields, including aerospace, robotics, and process control.