Mathematical Problems in Engineering
Volume 2003 (2003), Issue 1, Pages 25-45

Exponentially dissipative nonlinear dynamical systems: a nonlinear extension of strict positive realness

VijaySekhar Chellaboina1 and Wassim M. Haddad2

1Mechanical and Aerospace Engineering, University of Missouri-Columbia, Columbia 65211, MO, USA
2School of Aerospace Engineering, Georgia Institute of Technology, Atlanta 30332-0150, GA, USA

Received 10 February 2002

Copyright © 2003 VijaySekhar Chellaboina and Wassim M. Haddad. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We extend the notion of dissipative dynamical systems to formalize the concept of the nonlinear analog of strict positive realness and strict bounded realness. In particular, using exponentially weighted system storage functions with appropriate exponentially weighted supply rates, we introduce the concept of exponential dissipativity. The proposed results provide a generalization of the strict positive real lemma and the strict bounded real lemma to nonlinear systems. We also provide a nonlinear analog to the classical passivity and small gain stability theorems for state space nonlinear feedback systems. These results are used to construct globally stabilizing static and dynamic output feedback controllers for nonlinear passive systems that minimize a nonlinear nonquadratic performance criterion.