© 2017

L2-Gain and Passivity Techniques in Nonlinear Control

  • Enriches understanding of a fundamental and rapidly developing area of nonlinear control theory

  • Explains the small-gain and passivity theorems and their implications for stability and stabilization

  • Expands on earlier editions with new examples of port-Hamiltonian network dynamics and coverage of converse and incremental/shifted passivity


Part of the Communications and Control Engineering book series (CCE)

Table of contents

  1. Front Matter
    Pages i-xviii
  2. Arjan van der Schaft
    Pages 1-12
  3. Arjan van der Schaft
    Pages 13-32
  4. Arjan van der Schaft
    Pages 33-58
  5. Arjan van der Schaft
    Pages 59-99
  6. Arjan van der Schaft
    Pages 101-112
  7. Arjan van der Schaft
    Pages 113-171
  8. Arjan van der Schaft
    Pages 173-197
  9. Arjan van der Schaft
    Pages 199-211
  10. Arjan van der Schaft
    Pages 213-240
  11. Arjan van der Schaft
    Pages 241-267
  12. Arjan van der Schaft
    Pages 269-302
  13. Back Matter
    Pages 303-321

About this book


This standard text gives a unified treatment of passivity and L2-gain theory for nonlinear state space systems, preceded by a compact treatment of classical passivity and small-gain theorems for nonlinear input-output maps. The synthesis between passivity and L2-gain theory is provided by the theory of dissipative systems. Specifically, the small-gain and passivity theorems and their implications for nonlinear stability and stabilization are discussed from this standpoint. The connection between L2-gain and passivity via scattering is detailed.

Feedback equivalence to a passive system and resulting stabilization strategies are discussed. The passivity concepts are enriched by a generalised Hamiltonian formalism, emphasising the close relations with physical modeling and control by interconnection, and leading to novel control methodologies going beyond passivity.

The potential of L2-gain techniques in nonlinear control, including a theory of all-pass factorizations of nonlinear systems, and of parametrization of stabilizing controllers, is demonstrated. The nonlinear H-infinity optimal control problem is also treated and the book concludes with a geometric analysis of the solution sets of Hamilton-Jacobi inequalities and their relation with Riccati inequalities for the linearization.


·       L2-Gain and Passivity Techniques in Nonlinear Control (third edition) is thoroughly updated, revised, reorganized and expanded. Among the changes, readers will find:

·         updated and extended coverage of dissipative systems theory

·         substantial new material regarding converse passivity theorems and incremental/shifted passivity

<·         coverage of recent developments on networks of passive systems with examples

·         a completely overhauled and succinct introduction to modeling and control of port-Hamiltonian systems, followed by an exposition of port-Hamiltonian formulation of physical network dynamics

·         updated treatment of all-pass factorization of nonlinear systems


The book provides graduate students and researchers in systems and control with a compact presentation of a fundamental and rapidly developing area of nonlinear control theory, illustrated by a broad range of relevant examples stemming from different application areas.


Hamilton–Jacobi Inequalities Nonlinear Control Passive Systems Small-gain Theorem Input–Output Stability Port-Hamiltonian Systems Physical Network Dynamics Nonlinear Minimum-Phase Nonlinear H-infinity Control

Authors and affiliations

  1. 1.Johann Bernoulli Institute for Mathematics and Computer ScienceUniversity of GroningenGroningenThe Netherlands

About the authors

A.J. (Arjan) van der Schaft (1955) received the undergraduate and Ph.D. degrees in Mathematics from the University of Groningen, The Netherlands, in 1979 and 1983, respectively. In 1982 he joined the Department of Applied Mathematics, University of Twente, Enschede, where he was appointed as a full professor in Mathematical Systems and Control Theory in 2000. In September 2005 he returned to Groningen as a full professor in Mathematics.

He has served as Associate Editor for Systems & Control Letters, Journal of Nonlinear Science, SIAM Journal on Control and Optimization, IEEE Transactions on Automatic Control and the European Journal of Control. Currently he is on the Editorial Board for the Journal of Geometric Mechanics and Annual Reviews in Control.

Arjan van der Schaft is Fellow of the Institute of Electrical and Electronics Engineers (IEEE) and of the International Federation of Automatic Control (IFAC). He was Invited Lecturer at the International Congress of Mathematicians, Madrid, Spain, 2006. He is the 2013 recipient of the 3-yearly awarded Certificate of Excellent Achievements} of the IFAC Technical Committee on Nonlinear Systems. 

Arjan van der Schaft is (co-)author of the following other books: System Theoretic Descriptions of Physical Systems (1984), Variational and Hamiltonian Control Systems (1987, with P.E. Crouch), Nonlinear Dynamical Control Systems (1990, with H. Nijmeijer), An Introduction to Hybrid Dynamical Systems (2000, with J.M. Schumacher), Modeling and Control of Complex Physical Systems: the Port-Hamiltonian Approach( Geoplex Consortium, 2009), and Port-Hamiltonian Systems Theory: An Introductory Overview (2014, with D. Jeltsema). He (co-)edited five conference proceedings.

He (co-)authored around 125 journal articles and 265 refereed contributions to proceedings or book chapters.

Bibliographic information

  • Book Title L2-Gain and Passivity Techniques in Nonlinear Control
  • Authors Arjan van der Schaft
  • Series Title Communications and Control Engineering
  • Series Abbreviated Title Communicat., Control Engineering
  • DOI
  • Copyright Information Springer International Publishing AG 2017
  • Publisher Name Springer, Cham
  • eBook Packages Engineering Engineering (R0)
  • Hardcover ISBN 978-3-319-49991-8
  • Softcover ISBN 978-3-319-84294-3
  • eBook ISBN 978-3-319-49992-5
  • Series ISSN 0178-5354
  • Series E-ISSN 2197-7119
  • Edition Number 3
  • Number of Pages XVIII, 321
  • Number of Illustrations 37 b/w illustrations, 0 illustrations in colour
  • Additional Information Originally published as Volume 218 in the series: Lecture Notes in Control and Information Sciences
  • Topics Computational Intelligence
    Control, Robotics, Mechatronics
    Systems Theory, Control
  • Buy this book on publisher's site
Industry Sectors
Chemical Manufacturing
IT & Software
Consumer Packaged Goods
Materials & Steel
Finance, Business & Banking
Energy, Utilities & Environment
Oil, Gas & Geosciences


“The book presents a well written and systematically organized text focused on a unified treatment of classical input-output stability theory and recent results in nonlinear robust and passivity-based control. … The book will be useful to advanced graduate students, researchers and applied mathematicians interested in this emerging area.” (Lubomír Bakule, zbMath 1410.93004, 2019)

“This text is sure to continue to be one of the go-to references for nonlinear control researchers and practitioners interested in both the classical and modern interpretations and applications of L2-gain and passivity methods for nonlinear systems.” (James Richard Forbes, IEEE Control Systems Magazine, December, 2017)