The $\mathcal{H}_{\infty,p}$ norm as the differential $\mathcal{L}_{2,p}$ gain of a $p$-dominant system

TL;DR

This paper establishes that the differential $ ext{L}_{2,p}$ gain of a $p$-dominant linear system equals its $ ext{H}_{ ext{infinity},p}$ norm, enabling robust analysis similar to classical stability methods.

Contribution

It introduces a novel equivalence between the differential $ ext{L}_{2,p}$ gain and the $ ext{H}_{ ext{infinity},p}$ norm for $p$-dominant systems, facilitating robust dominance analysis.

Findings

The $ ext{H}_{ ext{infinity},p}$ norm is equal to the differential $ ext{L}_{2,p}$ gain.

This norm is defined via the supremum over a complex plane strip with specific pole placement.

The approach allows for robustness analysis of nonlinear uncertain systems using linear system techniques.

Abstract

The differential $\mathcal{L}_{2,p}$ gain of a linear, time-invariant, $p$-dominant system is shown to coincide with the $\mathcal{H}_{\infty,p}$ norm of its transfer function $G$, defined as the essential supremum of the absolute value of $G$ over a vertical strip in the complex plane such that $p$ poles of $G$ lie to right of the strip. The close analogy between the $\mathcal{H}_{\infty,p}$ norm and the classical $\mathcal{H}_{\infty}$ norm suggests that robust dominance of linear systems can be studied along the same lines as robust stability. This property can be exploited in the analysis and design of nonlinear uncertain systems that can be decomposed as the feedback interconnection of a linear, time-invariant system with bounded gain uncertainties or nonlinearities.