Consider a control system, $\dot{x}=Ax+Bu\\ y=Cx$.

Define the transfer function $G(s)=C(sI-A)^{-1}B$.

Then it is claimed that the following definitions of induced norm are equivalent.

$\|G\|_{\infty}=\sup_{s\in \mathbb{C}}\sigma_{max}G(s)\\ \|G\|_{\infty}=\sup_{\omega\in\mathbb{R}}\sigma_{max}G(i\omega)$.

I am looking for a proof for this claim. Basically it says that it suffices to maximize the leading singular value over purely imaginary $s$ rather than searching over all $s\in \mathbb{C}$.

  • $\begingroup$ Wouldn't maximizing over $\mathbb{C}$ always lead to infinity if the minimal realization of the system has at least one pole? $\endgroup$ – fibonatic May 11 at 11:22

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