Contractivity Analysis and Control Design for Lur'e Systems: Lipschitz, Incrementally Sector Bounded, and Monotone Nonlinearities
Ryotaro Shima, Alexander Davydov, Francesco Bullo
TL;DR
The paper tackles contractivity verification and controller design for Lur'e systems with nonlinearities that are Lipschitz, incrementally sector bound, or monotone, in both continuous- and discrete-time settings. It derives state-independent LMIs that are necessary and sufficient for strong infinitesimal contraction with respect to a $P$-weighted norm and rate $\eta$, and provides LMIs that yield controller gains via $K$ and $K_\Psi$ from feasible $(W,Z)$. It extends Lipschitz results to the incremental sector-bounded and monotone cases, showing monotonicity as a special case under symmetry, and presents discrete-time analogs of the contraction conditions. A numerical controller-design example demonstrates feasibility and contraction of the closed-loop system, highlighting practical applicability to tube-MPC and PI-like schemes in nonlinear control.
Abstract
In this paper, we study the contractivity of Lur'e dynamical systems whose nonlinearity is either Lipschitz, incrementally sector bounded, or monotone. We consider both the discrete- and continuous-time settings. In each case, we provide state-independent linear matrix inequalities (LMIs) which are necessary and sufficient for contractivity. Additionally, we provide LMIs for the design of controller gains such that the closed-loop system is contracting. Finally, we provide a numerical example for control design.