# American Institute of Mathematical Sciences

2013, 3(1): 175-201. doi: 10.3934/naco.2013.3.175

 1 Boeing Research and Technology Europe, Avenida Sur del Aeropuerto de Barajas, 38, Edificio 4 Planta, 428042 Madrid, Spain 2 School of Aeronautics and Astronautics, Purdue University, W. Lafayette, IN 47907, United States

Received  February 2012 Revised  December 2012 Published  January 2013

The concept of incremental quadratic stability ($\delta$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 $T$-periodic trajectory, respectively. By considering the relationship of $\delta$QS to the usual concept of quadratic stability, we obtain a useful necessary and sufficient condition for $\delta$QS. A main contribution of the paper is to consider nonlinear/uncertain systems whose state dependent nonlinear/uncertain terms satisfy an incremental quadratic constraint which is characterized by a bunch of symmetric matrices we 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, we demonstrate the usefulness of our results.
Citation: Luis D'Alto, Martin Corless. Incremental quadratic stability. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 175-201. doi: 10.3934/naco.2013.3.175
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