Convex Searches for Discrete-Time Zames–Falb Multipliers

Abstract

In this paper we develop and analyse convex searches for Zames-Falb multipliers. We present two different approaches: Infinite Impulse Response (IIR) and Finite Impulse Response (FIR) multipliers. The set of FIR multipliers is complete in that any IIR multipliers can be phase-substituted by an arbitrarily large order FIR multiplier. We show that searches in discrete-time for FIR multipliers are effective even for large orders. As expected, the numerical results provide the best 2stability results in the literature for slope-restricted nonlinearities. Finally, we demonstrate that the discrete-time search can provide an effective method to find suitable continuous-time multipliers.

Discrete‐time systems with slope restricted nonlinearities: Zames–Falb multiplier analysis using external positivity

“…One may note that the discrete‐time results of Reference 19 however, may still appear to be conservative and, for some of the examples given in the article, the estimate of the maximum slope size is still some way short of that predicted by the Kalman conjecture. While it is not expected that the Kalman conjecture holds in all cases, one might expect better results in some of them.…”

“…In some ways this article is therefore a companion paper to Reference 21 where the same was done for continuous‐time systems, but the methods used and some of the obstacles overcome differ in the discrete‐time case. The numerical results obtained are of variable competitiveness: sometimes rivaling the state‐of‐the‐art, 19 but sometimes yielding results some way from this.…”

“…There are some notable papers on the analysis of discrete‐time Lurie systems with slope‐restrictions. These include approaches relying on Lyapunov methods, 14,15 generic IQC‐based approaches, 16,17 and various papers on discrete‐time Zames–Falb approaches 8,18,19 . The state‐of‐the‐art in discrete‐time Zames–Falb analysis is described cogently in the recent paper by Carrasco et al 19 In that article, the authors compare various techniques for analyzing discrete‐time slope‐restricted Lurie systems, and also propose a new approach based on finite impulse response (FIR) Zames–Falb multipliers.…”

“…These include approaches relying on Lyapunov methods, 14,15 generic IQC‐based approaches, 16,17 and various papers on discrete‐time Zames–Falb approaches 8,18,19 . The state‐of‐the‐art in discrete‐time Zames–Falb analysis is described cogently in the recent paper by Carrasco et al 19 In that article, the authors compare various techniques for analyzing discrete‐time slope‐restricted Lurie systems, and also propose a new approach based on finite impulse response (FIR) Zames–Falb multipliers. These FIR multipliers provide superior results over most other methods, including the Lyapunov approaches of References 14,15 and those corresponding to discrete‐time versions of References 12,20.…”

“…While it is not expected that the Kalman conjecture holds in all cases, one might expect better results in some of them. Thus, while undeniably 19 provides the least Conservative results for discrete‐time Lurie systems with slope‐restrictions, currently available, the work in this article is motivated by conservatism which may yet exist in some instances.…”