Incremental quadratic stability

Abstract

The concept of incremental quadraticstability ($\delta$QS) is very useful in treating systems with persistently acting inputs.To illustrate, if a time-invariant $\delta$QS system is subject to a constant input or $T$-periodic input then,all its trajectories exponentially converge to a unique constant or $T$-periodictrajectory, respectively.By considering the relationship of $\delta$QS to the usual concept ofquadratic stability, we obtain a useful necessary and sufficientcondition for $\delta$QS.A main contribution of the paper is to consider nonlinear/uncertain systems whosestate dependent nonlinear/uncertain terms satisfy an incremental quadratic constraint which is characterized by a bunch of symmetric matriceswe call incremental multiplier matrices.We obtain linear matrix inequalities whose feasibility guarantee $\delta$QS of these systems.Frequency domain characterizations of $\delta$QS are then obtained from these conditions.By characterizing incremental multiplier matrices for many common classes of nonlinearities, wedemonstrate the usefulness of our results.