The H_∞,p norm as the differential L_2,p gain of a p-dominant system

Abstract

The differential L2,p gain of a linear, timeinvariant, p-dominant system is shown to coincide with the H∞,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 H∞,p norm and the classical H∞ 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.

Dominant Mixed Feedback Design for Stable Oscillations

“…Combining the notion of p-gain and the small gain theorem below [34], we have a framework for robust control of dominant systems, as in classical robust stability.…”

Model reduction by balanced truncation of dominant Lure systems

“…The dominance properties of an interconnected system can be studied using small L 2,p -gain conditions (Forni and Sepulchre, 2019;Padoan et al, 2019b). Theorem 2.…”

A tunable mixed feedback oscillator

“…For a quantitative perspective, the robustness of p-dominance property of the mixed-feedback amplifier can be studied using small-gain results, adapted to dominance through the notion of p-gain. Not surprisingly, the feedback interconnection of a p-dominant system with 0-dominant uncertainties ∆ l , ∆ p , and ∆ n remains p-dominant if the product of their gains is sufficiently small (their product must be less than one, as in the classical stability [18], [24]).…”

“…This means that the stable / oscillatory regimes of the mixed feedback are robust. A quantitative analysis can be developep through convex optimization, based on linear matrix inequalities [18], [24].…”