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## Über dieses Buch

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.

## Inhaltsverzeichnis

### Chapter 1. Nonlinear Input–Output Stability

Abstract
In this chapter, we briefly discuss the basic notions of input–output stability for nonlinear systems described by input–output maps. Also the stability of input–output systems in standard feedback closed-loop configuration is treated.
Arjan van der Schaft

### Chapter 2. Small-Gain and Passivity for Input–Output Maps

Abstract
In this chapter we give the basic versions of the classical small-gain (Sect. 2.1) and passivity theorems (Sect. 2.2) in the study of closed-loop stability.
Arjan van der Schaft

### Chapter 3. Dissipative Systems Theory

Abstract
In this chapter the general theory of dissipative systems is treated, laying much of the foundation for subsequent chapters.
Arjan van der Schaft

### Chapter 4. Passive State Space Systems

Abstract
In this chapter we focus on passive systems as an outstanding subclass of dissipative systems, firmly rooted in the mathematical modeling of physical systems.
Arjan van der Schaft

### Chapter 5. Passivity by Feedback

Abstract
In the previous Chaps. 2 and 4 we have seen the importance of the notion of passivity, both for analysis and for control.
Arjan van der Schaft

### Chapter 6. Port-Hamiltonian Systems

Abstract
As described in the previous Chaps. 3 and 4, (cyclo-)passive systems are defined by the existence of a storage function (nonnegative in case of passivity) satisfying the dissipation inequality with respect to the supply rate $$s(u,y)=u^Ty$$. In contrast, port-Hamiltonian systems, the topic of the current chapter are endowed with the property of (cyclo-)passivity as a consequence of their system formulation. In fact, port-Hamiltonian systems arise from first principles physical modeling. They are defined in terms of a Hamiltonian function together with two geometric structures (corresponding, respectively, to power-conserving interconnection and energy dissipation), which are such that the Hamiltonian function automatically satisfies the dissipation inequality.
Arjan van der Schaft

### Chapter 7. Control of Port-Hamiltonian Systems

Abstract
In this chapter, we will exploit the port-Hamiltonian structure for control, going beyond passivity. We will mainly concentrate on the problem of set-point stabilization. Section 7.1 focusses on control by interconnection, by attaching a controller port-Hamiltonian system to the plant port-Hamiltonian system. Section 7.2 takes a different perspective by emphasizing direct shaping of the Hamiltonian and the structure matrices by state feedback. Other control opportunities will be indicated in Sect. 7.3; see also the Notes at the end of this chapter.
Arjan van der Schaft

### Chapter 8. -Gain and the Small-Gain Theorem

Abstract
In this chapter we elaborate on the characterization of finite $$L_2$$-gain for state space systems, continuing on the general theory of dissipative systems.
Arjan van der Schaft

### Chapter 9. Factorizations of Nonlinear Systems

Abstract
In this chapter, we apply the $$L_2$$-gain concepts and techniques from Chaps. 3 and 8 to obtain some useful types of representations of nonlinear systems, different from the standard input-state-output representation.
Arjan van der Schaft

### Chapter 10. Nonlinear Control

Abstract
Consider the following standard control configuration, see Fig. 10.1.
Arjan van der Schaft

### Chapter 11. Hamilton–Jacobi Inequalities

Abstract
In the previous chapters we have encountered at various places Hamilton–Jacobi equations, or, more generally, Hamilton–Jacobi inequalities.
Arjan van der Schaft

### Backmatter

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