### Phase limitations of Zames-Falb multipliers

Authors: Shuai Wang, Joaquin Carrasco, William P Heath

Journal: IEEE Transactions on Automatic Control

Publication Date: 19 July, 2017

School of: Electrical and Electronic Engineering

### In Abstract

**New insight to the classical problem of absolute stability**

Feedback systems are everywhere. Many can be usefully modelled as the interconnection of two components: one linear and one nonlinear. The Zames-Falb multipliers are a classical analysis tool for understanding this feedback interconnection when the nonlinear component belongs to class of slope-restricted nonlinearities. Applications range from control systems with actuators whose range is limited to oscillating systems in biology. Briefly, for a given linear system, if a suitable Zames-Falb multiplier can be found (or even just shown to exist) then the feedback loop is guaranteed to be stable with any slope-restricted nonlinearity. In the jargon, the feedback system is absolutely stable.

To date, it has been standard to assume only simple conditions on the linear system: for example that it is stable and its inverse is stable. Researchers at the University of Manchester have demonstrated further limitations on the availability of Zames-Falb multipliers, which in turn translate into more sophisticated requirements for the feedback system to be absolutely stable. Were this not so, the Kalman conjecture (a famous fallacy in systems theory) would be true. Phase limitations are developed for the Zames-Falb multipliers in both continuous and discrete time. These are consistent with both classical and recently discovered counterexamples to the Kalman conjecture.

The phase-limitations for discrete-time systems are more restrictive than their counterparts for continuous-time systems. The restrictions have strong implications for the design of sampled-data control systems in the face of nonlinearities. The researchers are now exploring the implications, for systems with nonlinear drives. Their current focus is on aerospace, automotive and precision manufacturing applications.

- Feedback loops with slope-restricted nonlinearities have non-trivial dynamics. Engineers and mathematicians have studied them for decades because they have wide-ranging applications. Nevertheless, their properties are still not fully understood.
- Absolute stability: a feedback loop is said to be absolutely stable if its stability is guaranteed for a class of nonlinearities. The concept is crucial to our modern understanding of robust control.
- Multiplier: a multiplier is a mathematical device that can be used to guarantee the absolute stability of a feedback loop.
- Kalman Conjecture: the Kalman conjecture postulates that a feedback loop with a slope-restricted nonlinearity is absolutely stable if it can be shown to be stable for all linear gains within the class. Fourth-order counterexamples are well-known in the literature. Researchers at the University of Manchester recently showed that there are second-order counterexamples in the discrete-time domain.