Duality bounds for discrete-time Zames-Falb multipliers

TL;DR

The numerical results allow us to show, by construction, that the set of plants for which a suitable Zames-Falb multiplier exists is non-convex, and to discuss numerical examples where the limitations are stronger than others in the literature.

Abstract

We develop phase limitations for the discrete-time Zames-Falb multipliers based on the separation theorem for Banach spaces. By contrast with their continuous-time counterparts they lead to numerically efficient results that can be computed either in closed form or via a linear program. The closed-form phase limitations are tight in the sense that we can construct multipliers that meet them with equality. We discuss numerical examples where the limitations are stronger than others in the literature. The numerical results complement searches for multipliers in the literature; they allow us to show, by construction, that the set of plants for which a suitable Zames-Falb multiplier exists is non-convex.

Figures (11)

• Figure 1: Lurye systems
• Figure 2: Phase limitation of the class $\mathcal{M}$ for frequencies $\omega=\frac{\alpha}{\beta}\pi$ with $\beta\leq 50$.
• Figure 4: The maximum allowed phase for $G(e^{j\omega_1})$, i.e. $-\pi+\sigma$, to ensure that there is no suitable multiplier $M\in\mathcal{M}$ implies in turn a maximum phase for the class of multiplier to recover the positivity of $\hbox{Re}\{M(e^{j\omega_1})G(e^{j\omega_1})\}$. The limitation depends on the period $T$.
• Figure 5: The minimum allowed phase for $G(e^{j\omega_1})$, i.e. $\pi-\sigma$, to ensure that there is no suitable multiplier $M\in\mathcal{M}$ implies in turn a minimum phase for the class of multiplier to recover the positivity of $\hbox{Re}\{M(e^{j\omega_1})G(e^{j\omega_1})\}$.
• Figure 6: When $M\in\mathcal{M_{\text{odd}}}$, the limitation is independent of the period as when $T=\beta$, the constraint is activated by the opposite element. As a result, the limit for $\sigma$ is achieved when $2\sigma=\pi/\beta$ regardless of the period $T$.

Theorems & Definitions (43)

• Definition 1: Luenberger:1997Limaye:2016
• Definition 2: Space $S_{NBV}^{m\times m}$ Jonsson:1996
• Lemma 1: Luenberger:1997
• Lemma 2
• Lemma 3: Patrick:1995
• Definition 3: IQC Megretski:1997Kao:2012
• Remark 1
• Theorem 1: Megretski:1997
• Remark 2
• Definition 4: Discrete-time LTI Zames-Falb multipliers OShea:1967Willems:1968