Hinf - guidosassaroli/controlbasics GitHub Wiki

In control theory, $H_{inf}$ control offers a powerful framework for designing controllers that ensure robust performance in the presence of model uncertainties and external disturbances. Rather than optimizing for a specific nominal model, H${inf}$ control focuses on minimizing the worst-case effect of disturbances across all frequencies. This is achieved by designing a controller that minimizes the $H{inf}$-norm of the system’s transfer function from disturbance inputs to performance outputs, effectively limiting the maximum energy amplification the system can experience. As a result, the controlled system is better equipped to handle a wide range of operating conditions and unforeseen perturbations. This robustness makes $H_{inf}$ control particularly valuable in applications where performance and stability must be guaranteed despite uncertainties. The $H_{inf}$ approach is inherently optimization-based, framing the controller design as a problem of achieving the best possible attenuation of disturbances, rather than simply following a reference trajectory or minimizing a specific cost function.

If you have a system described by matrices A,B,C,D and a disturbance $w$, and output $z$, you want to design a controller $K$ so that the transfer function from $w$ to $z$, call it $T_{zw}$, satisfies:

$$ ||T_{zw}||_{inf} < y $$

for the smallest possible $y$.

Example

Work in progress.