A Less Conservative LMI Condition for Stability of Discrete-Time Systems With Slope-Restricted Nonlinearities
IEEE Transactions on Automatic Control
Many conditions have been found for the absolute stability of discrete-time Lur'e systems in the literature. It is advantageous to find convex searches via LMIs where possible. In this technical note, we construct two less conservative LMI conditions for discrete-time systems with slope-restricted nonlinearities. The first condition is derived via Lyapunov theory while the second is derived via the theory of integral quadratic constraints (IQCs) and noncausal Zames-Falb multipliers. Both
... pliers. Both conditions are related to the Jury-Lee criterion most appropriate for systems with such nonlinearities, and the second generalizes it. Numerical examples demonstrate a significant reduction in conservatism over competing approaches. Index Terms-Convex LMI, discrete-time, Lyapunov stability, multiplier theory. I. INTRODUCTION A. Motivation Most theories for absolute stability problems in the discretetime setting were historically developed for the SISO case via the frequency-domain , -, , . It has become standard to state such conditions in terms of LMIs (linear matrix inequalities) where possible  . In particular these can easily be applied to multiinput multi-output problems that might not otherwise be tractable. In this technical note we provide new LMI stability conditions for a discrete-time Lur'e system where there is a slope restriction and (possibly different) sector bound. This is appropriate for many practical control systems: in particular MIMO loops with actuator saturation where the slope restriction and sector bound are the same. If the control structure is otherwise linear then the results are immediately applicable. We are also motivated by optimizing anti-windup  and input-constrained model predictive control where the nonlinearity of the controller can be shown to satisfy such a slope restriction  . The implementation of such controllers is inherently in the discrete domain; hence, there is a clear motivation for understanding the discrete-time domain problem. The case where the slope restriction and sector bound are not the same has received recent interest for continuous systems  .