Phase limitations of Zames-Falb multipliers
This work provides theoretical insights into the limitations of Zames-Falb multipliers for nonlinear stability analysis, relevant to control theorists working on the Kalman conjecture.
The authors analyze phase limitations of Zames-Falb multipliers in both continuous and discrete time, generalizing a known phase limitation and developing a new one for discrete-time multipliers. They demonstrate that for certain fourth-order plants, no suitable Zames-Falb multiplier exists and simulations show instability, and they use the discrete-time limitation to show there is no direct discrete-time counterpart of the off-axis circle criterion.
Phase limitations of both continuous-time and discrete-time Zames-Falb multipliers and their relation with the Kalman conjecture are analysed. A phase limitation for continuous-time multipliers given by Megretski is generalised and its applicability is clarified; its relation to the Kalman conjecture is illustrated with a classical example from the literature. It is demonstrated that there exist fourth-order plants where the existence of a suitable Zames-Falb multiplier can be discarded and for which simulations show unstable behavior. A novel phase-limitation for discrete-time Zames-Falb multipliers is developed. Its application is demonstrated with a second-order counterexample to the Kalman conjecture. Finally, the discrete-time limitation is used to show that there can be no direct counterpart of the off-axis circle criterion in the discrete-time domain.