scroll identifier for mobile
main-content

## Über dieses Buch

With respect to the first edition as Volume 218 in the Lecture Notes in Con­ trol and Information Sciences series the basic idea of the second edition has remained the same: to provide a compact presentation of some basic ideas in the classical theory of input-output and closed-loop stability, together with a choice of contributions to the recent theory of nonlinear robust and 1foo control and passivity-based control. Nevertheless, some parts of the book have been thoroughly revised and/or expanded, in order to have a more balanced presen­ tation of the theory and to include some of the new developments which have been taken place since the appearance of the first edition. I soon realized, how­ ever, that it is not possible to give a broad exposition of the existing literature in this area without affecting the spirit of the book, which is precisely aimed at a compact presentation. So as a result the second edition still reflects very much my personal taste and research interests. I trust that others will write books emphasizing different aspects. Major changes with respect to the first edition are the following: • A new section has been added in Chapter 2 relating L2-gain and passivity via scattering, emphasizing a coordinate-free, geometric, treatment. • The section on stability in Chapter 3 has been thoroughly expanded, also incorporating some recent results presented in [182J.

## Inhaltsverzeichnis

### Chapter 1. Input-Output Stability

Abstract
In this chapter we briefly describe the basic notions of input-output stability; both for input-output systems as well as for input-output systems in standard feedback closed-loop configuration.
Arjan van der Schaft

### Chapter 2. Small-gain and Passivity of Input-Output Maps

Abstract
In this chapter we give the basic versions of the classical small-gain and passivity theorem in the study of closed-loop stability (Sections 2.1 and 2.2). Section 2.3 deals with the relation between passivity and L 2-gain via the scattering representation.
Arjan van der Schaft

### Chapter 3. Dissipative Systems Theory

Abstract
In this chapter we present a state space interpretation of the small gain and passivity approach of the previous chapter. Moreover, we will come to some kind of synthesis between the notions of stability of input-output maps treated in Chapter 1 on the one hand, and classical Lyapunov stability of state space systems on the other hand.
Arjan van der Schaft

### Chapter 4. Hamiltonian Systems as Passive Systems

Abstract
In this chapter we deal with Euler-Lagrange and Hamiltonian systems as an important class of passive state space systems. First we consider the passivity of systems described by Euler-Lagrange equations, with an application to a tracking problem. We define the class of port-controlled Hamiltonian systems, including the examples of LC-circuits and mechanical systems with kinematic constraints. This framework is further extended to include dissipation. Stabilization procedures for port-controlled Hamiltonian systems, which exploit the Hamiltonian structure and the passivity property, are discussed. Finally, the notion of power-conserving interconnection is formalized, leading to the notion of implicit port-controlled Hamiltonian systems.
Arjan van der Schaft

### Chapter 5. Passivity by Feedback

Abstract
In this chapter we give necessary and sufficient conditions under which a nonlinear system is feedback equivalent to a passive system. The main idea is to transform (if possible) the nonlinear system into the feedback interconnection of two passive systems. This idea is further explored in Section 5.2 for the stabilization of cascaded systems.
Arjan van der Schaft

### Chapter 6. Factorizations of Nonlinear Systems

Abstract
In this chapter we apply the dissipativity concepts from Chapter 3, in particular the L 2-gain techniques, to obtain some useful types of representations of nonlinear systems, different from the input-state-output representation. In Section 6.1 we will derive stable kernel and stable image representations of nonlinear systems, and we will use them in order to formulate nonlinear perturbation models (with L 2-gain bounded uncertainties). In Section 6.2 we will employ stable kernel representations in order to derive a parametrization of stabilizing controllers, analogous to the Youla-Kucera parametrization in the linear case. Finally, in Section 6.3 we consider the factorization of nonlinear systems into a series interconnection of a, for instance, minimum phase system and an inner system which preserves the L 2-norm.
Arjan van der Schaft

### Chapter 7. Nonlinear H ∞ Control

Abstract
Consider the following standard control configuration. Let ∑ be a nonlinear system
$$\begin{gathered} \dot x = f\left( {x,u,d} \right) \hfill \\ \Sigma :y = g\left( {x,u,d} \right) \hfill \\ z = h\left( {x,u,d} \right) \hfill \\ \end{gathered}$$
(1)
with two sets of inputs u and d, two sets of outputs y and z, and state x.
Arjan van der Schaft

### Chapter 8. Hamilton-Jacobi Inequalities

Abstract
In the previous chapters we have encountered at various places Hamilton-Jacobi equations, or, more generally, Hamilton-Jacobi inequalities. In this chapter we take a closer look at conditions for solvability of Hamilton-Jacobi inequalities and the structure of their solution set using invariant manifold techniques for the corresponding Hamiltonian vectorfield (Section 8.1), and apply this to the nonlinear optimal control problem in Section 8.2. An important theme will be the relation between Hamilton-Jacobi inequalities and the corresponding Riccati inequalities, in particular for dissipativity (Section 8.3) and nonlinear H control (Section 8.4).
Arjan van der Schaft

### Backmatter

Weitere Informationen