TY - JOUR

T1 - Duality Bounds for Discrete-Time Zames-Falb Multipliers

AU - Zhang, Jingfan

AU - Carrasco, Joaquin

AU - Heath, William Paul

PY - 2022/7/1

Y1 - 2022/7/1

N2 - This note presents phase conditions under which there is no suitable Zames– Falb multiplier for a given discrete-time system. Our conditions can be seen as the discrete-time counterpart of Jönsson’s duality conditions for Zames–Falb multipliers. By contrast with their continuous-time counterparts and other phase limitations in the literature, 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. The numerical results allow us to conclude that the current state-of-the-art in searches for Zames–Falb multipliers is not conservative. Moreover, they allow us to show, by construction, that the set of plants for which a suitable Zames– Falb multiplier exists is nonconvex.

AB - This note presents phase conditions under which there is no suitable Zames– Falb multiplier for a given discrete-time system. Our conditions can be seen as the discrete-time counterpart of Jönsson’s duality conditions for Zames–Falb multipliers. By contrast with their continuous-time counterparts and other phase limitations in the literature, 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. The numerical results allow us to conclude that the current state-of-the-art in searches for Zames–Falb multipliers is not conservative. Moreover, they allow us to show, by construction, that the set of plants for which a suitable Zames– Falb multiplier exists is nonconvex.

U2 - 10.1109/tac.2021.3095418

DO - 10.1109/tac.2021.3095418

M3 - Erthygl

VL - 67

SP - 3521

EP - 3528

JO - IEEE Transactions on Automatic Control

JF - IEEE Transactions on Automatic Control

SN - 2334-3303

IS - 7

ER -