L2-Gain and Passivity Techniques in Nonlinear Control

The aim of these lecture notes is to provide a synthesis between classical input-output and closed-loop stability theory, in particular the small-gain and passivity theorems, and recent work on nonlinear H( and passivity-based control. The treatment of the theory of dissipative systems is the main aspect of these lecture notes. Fundamentals of passivity techniques are summarised, and it is shown that the passivity properties of different classes of physical systems can be unified within a generalised Hamiltonian framework. Key developments in linear robust control theory are extended to the nonlinear context using L2-gain techniques. An extensive treatment of nonlinear H( control theory is presented, emphasising its main structural features. Since the application of L2-gain techniques relies on solving Hamilton-Jacobi inequalities the structure of their solution sets and conditions for solvability are derived.

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.