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2020 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Introduction to Infinite-Dimensional Systems Theory

A State-Space Approach

verfasst von: Prof. Ruth Curtain, Dr. Hans Zwart

Verlag: Springer New York

Buchreihe : Texts in Applied Mathematics

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

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Über dieses Buch

Infinite-dimensional systems is a well established area of research with an ever increasing number of applications. Given this trend, there is a need for an introductory text treating system and control theory for this class of systems in detail. This textbook is suitable for courses focusing on the various aspects of infinite-dimensional state space theory. This book is made accessible for mathematicians and post-graduate engineers with a minimal background in infinite-dimensional system theory. To this end, all the system theoretic concepts introduced throughout the text are illustrated by the same types of examples, namely, diffusion equations, wave and beam equations, delay equations and the new class of platoon-type systems. Other commonly met distributed and delay systems can be found in the exercise sections. Every chapter ends with such a section, containing about 30 exercises testing the theoretical concepts as well. An extensive account of the mathematical background assumed is contained in the appendix.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Introduction
Abstract
To motivate the usefulness of developing a theory for linear infinite-dimensional systems, we present some simple examples of control systems which arise in spatial invariant systems, delay systems, and systems describe by partial differential equations. Furthermore, we discuss some system theory concepts for finite-dimensional systems. These concepts will be extended to infinite dimensions in the later chapters.
Ruth Curtain, Hans Zwart
Chapter 2. Semigroup Theory
Abstract
Strongly continuous semigroups and their infinitesimal generators are treated in detail in this chapter. The chapter contains the classical concepts and results on these topics, such as the Hille-Yosida Theorem and the contraction- and dual semigroups. Although the theory can be extended to Banach spaces, we restrict ourselves to separable Hilbert spaces. The relation between semigroup- and generator-invariant subspaces is studied in detail. The chapter ends with a set of 30 exercises and a notes and references section.
Ruth Curtain, Hans Zwart
Chapter 3. Classes of Semigroups
Abstract
The general concept of strongly continuous semigroup was introduced in the previous chapter. In this chapter we focus on examples. This is done by studying three main classes, namely spatially invariant operators, Riesz-spectral operators, and delay equations. Next to existence of the strongly continuous semigroup, we study for these classes the resolvent set and operator, and characterise the semigroup invariant subspaces. The chapter ends with a set of 25 exercises and a notes and references section.
Ruth Curtain, Hans Zwart
Chapter 4. Stability
Abstract
As the title indicates, we study the stability of strongly continuous semigroups. We distinguish between exponential, strong, and weak stability. We show that the stability cannot be concluded from the spectrum of the infinitesimal generator, i.e., the spectrum determined growth assumption does not need to hold. For our classes of systems; spatially invariant operators, Riesz-spectral operators, and delay equations, we show that this growth assumption does hold. Since Sylvester equations can be solved under proper stability assumptions, this topic is part of the chapter. The chapter ends with a set of 22 exercises and a notes and references section.
Ruth Curtain, Hans Zwart
Chapter 5. The Cauchy Problem
Abstract
This chapter studies the abstract differential equation \(\dot{z}(t) = A z(t) + f(t)\) with the initial condition \(z(0)=z_0\), where we assume that A is the infinitesimal generator of a strongly continuous semigroup. We distinguish between different notions of solutions, such as classical and mild (or weak) solution. Furthermore, we study the asymptotic behaviour of these solutions. When f(t) is given by a (time-dependent) feedback, i.e., \(f(t) = D(t) z(t)\), then we show that the abstract equation is still well-posed, and the corresponding solution is given by an evolution operator. If D is time-independent, this evolution operator equals a strongly continuous semigroup. The chapter ends with a set of 14 exercises and a notes and references section.
Ruth Curtain, Hans Zwart
Chapter 6. State Linear Systems
Abstract
In this chapter the system differential equation \(\dot{z}(t) = A z(t) + B u(t)\), \(y(t) = C z(t) + D u(t)\) is introduced. For this state linear system the concepts of controllability and observability are defined, and it shown that there are different generalisations of their finite-dimensional counterparts. Using the characterisation of invariant subspaces of Chaps. 2 and 3, tests for controllability and observability are derived for the different classes of systems. Since the controllability and observability gramian satisfy a Lyapunov equation, a section on Lyapunov equations is also part of this chapter. The chapter ends with a set of 28 exercises and a notes and references section.
Ruth Curtain, Hans Zwart
Chapter 7. Input-Output Maps
Abstract
For the state linear system as introduced in Chap. 6, two input-output maps are introduced. The first is in time domain, and writes the output y as a convolution of the input with the impulse response. The second one is the transfer function, which is in frequency domain. Instead of introducing the transfer function via the Laplace transform, we do it via exponential solutions, i.e., for the input given by \(u(t) =u_0 e^{st}\) we search for an output of a similar form, \(y(t) = y_0 e^{st}\). The mapping \(u_0 \rightarrow y_0\) is our transfer function. The relation between this transfer function and the Laplace transform of the impulse response is studied. Furthermore, it is shown that for systems which are input-output stable the Laplace transform of the output equals the transfer function times the Laplace transform of the input. The chapter ends with a set of 27 exercises and a notes and references section.
Ruth Curtain, Hans Zwart
Chapter 8. Stabilizability and Detectability
Abstract
One of the most important concepts of systems theory is that of stabilizability and its dual concept detectability. We characterise when a system with finitely many inputs is stabilizable. Additionally, we present tests for the stabilizability/detectability of spatially invariant, Riesz-spectral, and delay systems. Similar as for finite-dimensional systems, we show that a system can be stabilized by a dynamic compensator provided it is stabilizable and detectable. As shown in the charactrization of stabilizability, exponential stabilzability may be impossible for certain classes of systems, such as collocated systems. Therefor we also characterize the weaker concept of strong stabilizability. The chapter ends with a set of 21 exercises and a notes and references section.
Ruth Curtain, Hans Zwart
Chapter 9. Linear Quadratic Optimal Control
Abstract
The quadratic optimal problem is studied for our class of state linear systems. This is done on a finite- and on the infinite-time interval. The solution on the infinite horizon case is related to the smallest non-negative solution of the algebraic Riccati equation. Depending on the stability properties of the open loop system, the optimal feedback system will have some stability properties. The largest non-negative solution of the algebraic Riccati equation will always stabilise the system, and we present an algorithm (Newton-Kleinman) for iteratively finding this maximal solution. The chapter ends with a set of 29 exercises and a notes and references section.
Ruth Curtain, Hans Zwart
Chapter 10. Boundary Control Systems
Abstract
In the previous chapters our focus was on systems in which the control action is within the spatial domain. In this chapter we show that systems in which the control action is at the boundary of the spatial domain can be rewritten into one with an in-domain control action. However, this comes with the price that we introduce a derivative in the input. To calculate the transfer function of a boundary control system, it is not needed to rewrite the system first into a state linear systems. The flexible beam with two differents models for the boundary control serves as an example. The chapter ends with a set of 19 exercises and a notes and references section.
Ruth Curtain, Hans Zwart
Chapter 11. Existence and Stability for Semilinear Differential Equations
Abstract
In the previous chapter we treated linear systems. In this chapter we study the semilinear differential equation \(\dot{z}(t) = A z(t) + f(z(t))\), with A the infinitesimal generator of strongly continuous semigroup. We do this for two cases, the first one in which we assume that f is Lipschitz continuous on the state space, and the second one, where we assume that A generates a holomorphic semigroup and f is Lipschitz continuous on the domain of a fractional power of A, i.e., \(D(A^{\alpha })\), \(\alpha \in (0,1)\). Since for non-linear systems stability is studied via Lyapunov function, this is treated in detail. Among others, an infinite-dimensional version of LaSalle’s invariance theorem is proved. The chapter ends with a set of 25 exercises and a notes and references section.
Ruth Curtain, Hans Zwart
Backmatter
Metadaten
Titel
Introduction to Infinite-Dimensional Systems Theory
verfasst von
Prof. Ruth Curtain
Dr. Hans Zwart
Copyright-Jahr
2020
Verlag
Springer New York
Electronic ISBN
978-1-0716-0590-5
Print ISBN
978-1-0716-0588-2
DOI
https://doi.org/10.1007/978-1-0716-0590-5

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