L2-Gain and Passivity Techniques in Nonlinear Control

  • Arjan van der Schaft

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

  • Arjan van der Schaft
    • 1
  1. 1.Johann Bernoulli Institute for Mathematics and Computer ScienceUniversity of GroningenGroningenThe Netherlands

Bibliographic information

  • DOI
  • Copyright Information Springer International Publishing AG 2017
  • Publisher Name Springer, Cham
  • eBook Packages Engineering Engineering (R0)
  • Print ISBN 978-3-319-49991-8
  • Online ISBN 978-3-319-49992-5
  • Series Print ISSN 0178-5354
  • Series Online ISSN 2197-7119
  • Buy this book on publisher's site