Abstract
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 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.
Highlights
This note gives LMI conditions that guarantee the absolute stability of the feedback interconnection between a discrete-time linear time-invariant (LTI) system and a memoryless, time-invariant, sectorbounded and slope-restricted nonlinearity
We provide the mathematical definition of the absolute stability problem
It has become standard to state such conditions in terms of LMIs where possible [6]. In particular these can 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 sector bound
Summary
This note gives LMI conditions that guarantee the absolute stability of the feedback interconnection between a discrete-time linear time-invariant (LTI) system and a memoryless, time-invariant, sectorbounded and slope-restricted nonlinearity. We provide the mathematical definition of the absolute stability problem . We are concerned with the strictly proper discrete-time LTI system xk+1 = Axk + Buk, x0 ∈ Rnx , yk = Cxk (1). Where A is Schur with xk ∈ Rnx and with uk, yk ∈ Rp; A ∈ Rnx×nx , B ∈ Rnx×p, and C ∈ Rp×nx. Throughout the paper, we will assume that (1) is a minimal representation. It is in negative feedback with a memoryless, time-invariant (static) nonlinearity φ : Rp → Rp with the relation
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