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2012 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces

verfasst von: Birgit Jacob, Hans J. Zwart

Verlag: Springer Basel

Buchreihe : Operator Theory: Advances and Applications

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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SUCHEN

Über dieses Buch

This book provides a self-contained introduction to the theory of infinite-dimensional systems theory and its applications to port-Hamiltonian systems. The textbook starts with elementary known results, then progresses smoothly to advanced topics in current research.

Many physical systems can be formulated using a Hamiltonian framework, leading to models described by ordinary or partial differential equations. For the purpose of control and for the interconnection of two or more Hamiltonian systems it is essential to take into account this interaction with the environment. This book is the first textbook on infinite-dimensional port-Hamiltonian systems. An abstract functional analytical approach is combined with the physical approach to Hamiltonian systems. This combined approach leads to easily verifiable conditions for well-posedness and stability.

The book is accessible to graduate engineers and mathematicians with a minimal background in functional analysis. Moreover, the theory is illustrated by many worked-out examples.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Introduction
Abstract
In this chapter we provide an introduction to the field of mathematical systems theory. Besides examples we discuss the notion of feedback and we answer the question why feedback is useful. However, before we start with the examples we discuss the following picture, which can be seen as the essence of systems theory.
Birgit Jacob, Hans J. Zwart
Chapter 2. State Space Representation
Abstract
In the previous chapter we introduced models with an input and an output. These models were described by an ordinary or partial differential equation. However, there are other possibilities to model systems with inputs and outputs. In this chapter we introduce the state space representation on a finite-dimensional state space. Later we will encounter these representations on an infinite-dimensional state space. State space representations enable us to study systems with inputs and outputs in a uniform framework. In this chapter, we show that every model described by an ordinary differential equation possesses a state space representation on a finite-dimensional state space, and that it is just a different way of writing down the system. However, this different representation turns out to be very important as we will see in the following chapters. In particular, it enables us to develop general control strategies.
Birgit Jacob, Hans J. Zwart
Chapter 3. Controllability of Finite-Dimensional Systems
Abstract
In this chapter we study the notion of controllability for finite-dimensional systems as introduced in Chapter 2.
Birgit Jacob, Hans J. Zwart
Chapter 4. Stabilizability of Finite-Dimensional Systems
Abstract
This chapter is devoted to the stability and stabilizability of state differential equations. Roughly speaking, a system is stable if all solutions converge to zero, and a system is stabilizable if one can find a suitable control function such that the corresponding solution tends to zero. Thus stabilizability is a weaker notion than controllability.
Birgit Jacob, Hans J. Zwart
Chapter 5. Strongly Continuous Semigroups
Abstract
In Chapter 2 we showed that the examples of Chapter 1, which were described by ordinary differential equations, can be written as a first-order differential equation
$$\dot{x}(t) = Ax(t) + Bu(t), \quad \quad y(t) = C_x(t) = Du(t),$$
(5.1)
where x(t) is a vector in Rn or Cn.
Birgit Jacob, Hans J. Zwart
Chapter 6. Contraction and Unitary Semigroups
Abstract
In the previous chapter we introduced C 0-semigroups and their generators. We showed that every C 0-semigroup possesses an infinitesimal generator. In this section we study the other implication, i.e., when is a closed densely defined operator an infinitesimal generator of a C 0-semigroup?
Birgit Jacob, Hans J. Zwart
Chapter 7. Homogeneous Port-Hamiltonian Systems
Abstract
In the previous two chapters we have formulated partial differential equations as abstract first order differential equations. Furthermore, we described the solutions of these differential equations via a strongly continuous semigroup. These differential equations were only weakly connected to the norm of the underlying state space. However, in this chapter we consider a class of differential equations for which there is a very natural state space norm. This natural choice enables us to show that the corresponding semigroup is a contraction semigroup.
Birgit Jacob, Hans J. Zwart
Chapter 8. Stability
Abstract
This chapter is devoted to stability of abstract differential equations as well as to spectral projections and invariant subspaces. One of the most important aspects of systems theory is stability, which is closely connected to the design of feedback controls. For infinite-dimensional systems there are different notions of stability such as strong stability, polynomial stability, and exponential stability. In this chapter, we restrict ourselves to exponential stability. Strong stability will be defined in an exercise, where it is also shown that strong stability is weaker than exponential stability. The concept of invariant subspaces, which we discuss in the second part of this chapter will play a key role in the study of stabilizability in Chapter 10.
Birgit Jacob, Hans J. Zwart
Chapter 9. Stability of Port-Hamiltonian Systems
Abstract
In this chapter we return to the class of port-Hamiltonian partial differential equations which we introduced in Chapter 7. If a port-Hamiltonian system possesses n (linearly independent) boundary conditions and if the energy is non-increasing, then the associated abstract differential operator generates a contraction semigroup on the energy space.
Birgit Jacob, Hans J. Zwart
Chapter 10. Inhomogeneous Abstract Differential Equations and Stabilization
Abstract
In the previous five chapters we considered the homogeneous (abstract) differential equation
$$\dot x(t) = Ax(t),\,\,\,\,\,\,\,\,\,\,\,\,x(0) = x_0 .$$
(10.1)
However, for control theoretical questions it is essential to add an input to the differential equation, see e.g. Chapters 3 and 4. Section 10.1 is devoted to infinite-dimensional inhomogeneous differential equations and in Section 10.2 we add an output equation. The obtained formulas will be very similar to those found in Chapter 2. However, C0-semigroups are in general not differentiable on an infinite-dimensional state space and thus the proofs are more involved.
Birgit Jacob, Hans J. Zwart
Chapter 11. Boundary Control Systems
Abstract
In this chapter we are in particular interested in systems with a control at the boundary of their spatial domain. We show that these systems have well-defined solutions provided the input is sufficiently smooth.
Birgit Jacob, Hans J. Zwart
Chapter 12. Transfer Functions
Abstract
In this chapter we introduce the concept of transfer functions. In the system and control literature a transfer function is usually defined via the Laplace transform. However, in this chapter we use a different approach.
Birgit Jacob, Hans J. Zwart
Chapter 13. Well-posedness
Abstract
The concept of well-posedness can easily be explained by means of the abstract linear system Σ(A,B,C,D) introduced in Section 10.2, that is, for the set of equations
$$\dot x(t) = Ax(t) + Bx(t),\,\,\,\,\,\,\,\,\,\,\,\,\,x(0) = x_0,$$
(13.1)
$$y(t) = Cx(t) + Dx(t),$$
(13.2)
where x is a X-valued function, u is an U-valued function and y is a Y -valued function. The spaces U, X and Y are assumed to be Hilbert spaces. Further, the operator A is assumed to be the infinitesimal generator of a C0-semigroup, and B,C, and D are bounded linear operators.
Birgit Jacob, Hans J. Zwart
Backmatter
Metadaten
Titel
Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces
verfasst von
Birgit Jacob
Hans J. Zwart
Copyright-Jahr
2012
Verlag
Springer Basel
Electronic ISBN
978-3-0348-0399-1
Print ISBN
978-3-0348-0398-4
DOI
https://doi.org/10.1007/978-3-0348-0399-1

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