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L2-Gain and Passivity Techniques in Nonlinear Control

Arjan van der Schaft

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Naturwissenschaften, Medizin, Informatik, Technik / Allgemeines, Lexika


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.

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