Full‐block multipliers for repeated, slope‐restricted scalar nonlinearities

Abstract

Summary This paper provides a comprehensive treatment of full‐block multipliers within the integral quadratic constraints framework for stability analysis of feedback systems containing repeated, slope‐restricted scalar nonlinearities. We develop a novel stability result that offers more flexibility in its application because it allows for the inclusion of general Popov and Yakubovich criteria in combination with the well‐established Circle and Zames‐Falb stability tests within integral quadratic constraint theory. A particular focus lies on the formulation of stability criteria in terms of full‐block multipliers, some of which are new, and thus typically involve less conservatism than current methods. Furthermore, a new asymptotically exact parametrization of full‐block Zames‐Falb multipliers is given that allows to exploit the complete potential of this stability test. Copyright © 2017 John Wiley & Sons, Ltd.

Convex Searches for Discrete-Time Zames–Falb Multipliers

“…In [21], [4], it has been shown that their relative performances vary with different examples. It must be highlighted that results in the basis functions can be significantly improved by manually selecting the parameters of the basis [14], [15]. Similarly, manual tuning of delta functions can be useful for time-delay systems [22].…”

Stability analysis of piecewise affine systems with multi-model predictive control

“…A unified framework has been proposed in (Jönsson and Rantzer, 2000) for the search of multipliers. Recently, Fetzer and Scherer (2017a) proposed a comprehensive analysis for the case of slope restricted nonlinearities in discrete time is presented, showing that the stability test in the literature are related.Consequently, IQCs have been widely used to perform stability and robustness analyses of dynamic systems (D'Amato et al, 2001;Heath and Li, 2010;Fetzer and Scherer, 2017b) in the frequency domain as well as in the time-domain employing dissipativity theory (Brogliato et al, 2007).…”

Conic sector analysis using integral quadratic constraints

“…Notable achievements are the small gain theorem , which underpins the robust control literature and the passivity theorem and related results . In addition, other researchers have considered the case when one of the systems in the feedback interconnection was linear and this has yielded a number of absolute stability results such as the popular Circle and Popov criteria, and also the theory associated with Zames‐Falb multipliers …”

“…3,4 In addition, other researchers have considered the case when one of the systems in the feedback interconnection was linear and this has yielded a number of absolute stability results such as the popular Circle and Popov criteria, and also the theory associated with Zames-Falb multipliers. [5][6][7][8] The circle criterion originally arose in the two-part paper of Zames,9,10 where the stability of systems of the form depicted in Figure 1 was studied. In most applications of the Circle Criterion, the troublesome nonlinearity, Δ, is considered to be static with a graph belonging to a particular sector.…”