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2009 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Analysis and Control of Nonlinear Systems

A Flatness-based Approach

verfasst von: Jean Levine

Verlag: Springer Berlin Heidelberg

Buchreihe : Mathematical Engineering

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

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SUCHEN

Inhaltsverzeichnis

Frontmatter
Chapter 1. Introduction
This book is made of two parts, Theory and Applications. In the first Part, two major problems of automatic control are addressed: trajectory generation, or motion planning, and tracking of these trajectories. In order to make this book as self-contained as possible we have included a survey of Differential Geometry and Dynamical System Theory. The viewpoint adopted for these topics has been tailored to prepare the reader to the language and tools of flatness-based control design, that is why we have preferred to place them ahead in Chapters 2 and 3 rather than to release them in an Appendix.
Jean Lévine

Part I THEORY

Chapter 2. Introduction to Differential Geometry
This Chapter aims at introducing the reader to the basic concepts of differential geometry such as diffeomorphism, tangent and cotangent space, vector field, differential form. Special emphasis is put on the integrability of a family of vector fields, or distribution1, according to its role in nonlinear system theory, For simplicity’s sake, we have defined a manifold as the solution set to a system of implicit equations expressed in a given coordinate system, according to the implicit function theorem. One can then get rid of the coordinate choice thanks to the notion of diffeomorphism or curvilinear coordinates. Particular interest is given to the notion of straightening out coordinates, that allow to express manifolds, vector fields or distributions in a trivial way.
Jean Lévine
Chapter 3. Introduction to Dynamical Systems
This chapter is devoted to the study of the dynamical behaviors of nonlinear uncontrolled systems: stability, instability of flows around an equilibrium or a periodic orbit and comparison to their tangent linear approximation. We consider the set of (uncontrolled) differential equation, or differential system.
Jean Lévine
Chapter 4. Controlled Systems, Controllability
Controllability is one of the so-called structural properties of systems depending on inputs. It has initially been studied in the framework of linear systems to describe the possibility of generating arbitrary motions. For nonlinear systems, several extensions are possible. They are outlined in the second part of this chapter.
Jean Lévine
Chapter 5. Jets of Infinite Order, Lie-Bäcklund’s Equivalence
A cart of mass M rolls along the axis OX of the overhead crane. Its position is denoted by x. It is actuated by a motor that produces a horizontal force of intensity F. Moreover, the cart carries a winch of radius ρ around which is winding a cable hoisting the load attached at its end. The position of the load in the fixed frame XOZ is denoted by (ξ ς) and its mass is m. The torque exerted on the winch by a second motor is denoted by C.
Jean Lévine
Chapter 6. Differentially Flat Systems
Denition 6.1. We say that the system (\(X \times U \times\mathbb{R}_{\infty}^{m},\overline{f}\)) (resp. (\(X \times\mathbb{R}^{n}_{\infty}, \tau_{X}, F\))), with m inputs, is flat, or, shortly, flat, if and only if it is L-B equivalent to the trivial system (\(\mathbb{R}_{\infty}^{m}, \tau_{m}\)) (resp.(\(\mathbb{R}_{\infty}^{m}, \tau_{m},0\))), where \(\tau_{m}\) is the trivial Cartan field of \(\mathbb{R}_{\infty}^{m}\) with coordinates (\(y,$\textit{\.{y}}\textit{\"{y}},$\ldots\)):
$$\tau_{m}=\sum_{j\geq0}\sum^{m}_{i=1}y^{(j+1)}\frac{\partial}{\partial y_{i}^{(j)}}.$$
(6.1)
Jean Lévine
Chapter 7. Flatness and Motion Planning
Let us consider the nonlinear system \(\textit{\.{x}}=f(x,u)\). Given the initial time ti, the initial conditions
$$x(t_{i})=x_{i}, \, u(t_{i})= u_{i},$$
(7.1)
, the final time tf and the final conditions
$$x(t_{f})=x_{f}, \, u(t_{f})= u_{f},$$
(7.2)
Jean Lévine
Chapter 8. Flatness and Tracking
For the solution of the motion planning problem, all we required was the knowledge of a dynamical model and the time, in other words, anticipative data: the reference trajectory was computed from the present time to some future time according to what we know about the system’s evolution. This type of design is called open-loop. If the system dynamics is precisely known and if the disturbances (all signals not taken into account in the model that might affect the system’s evolution) don’t produce significant deviations from the predicted trajectories in the workspace, the open-loop design may sometimes be sufficient.
Jean Lévine

Part II APPLICATIONS

Chapter 9. DC Motor Starting Phase
This chapter is aimed at showing that, even for DC motor control, a quite standard application of control, described by a single input linear system, the so-called flatness-based approach may dramatically improve its performance in a transient phase.
Jean Lévine
Chapter 10. Displacements of a Linear Motor With Oscillating Masses
The aim of this chapter is to show, on a barely undamped oscillating system, roughly approximated by a simple single-input linear model, the importance of feedforward design to simultaneously achieve, with the same smooth controller (no switch), high precision positioning, fast displacements, and robustness versus modelling errors, objectives which are, at first sight, antinomic. We consider a linear motor moving along a rail and, in a first step, related by one exible rod to an auxiliary mass (see Figure 10.1), and in a second step, by two different flexible rods to two different auxiliary masses (see Figure 10.8). The motor is assumed to be controlled by the force it delivers. All along this study, we assume that the motor position and speed are measured in real time with an arbitrary accuracy, but that the auxiliary mass positions are not measured.
Jean Lévine
Chapter 11. Synchronization of a Pair of Independent Windshield Wipers
In this chapter1, we present an industrial case study whose difficulty lies in the synchronization of two independent systems. Note that the decoupling problem has received a large number of contributions (see e.g. Isidori [1995], Nijmeijer and van der Schaft [1990]), as opposed to the synchronization one, its contrary, especially with decentralized information. Here, the independent subsystems correspond to two independent windshield wipers that must be synchronized in particular to avoid collisions. They are driven by two independent actuators, each one fed by its own position measurement only. The lack of global information therefore makes the synchronization hard. We propose a decentralized flatness-based control design, where the reference trajectories are specifically designed to avoid collisions and with a trajectory tracking loop, supervised by a clock control loop that regulates the time rate at which both reference trajectories are tracked.
Jean Lévine
Chapter 12. Control of Magnetic Bearings
In this chapter, we study various aspects of the motion planning and nonlinear feedback design for a rotating shaft actuated by active magnetic bearings. The shaft is here assumed to be rigid and balanced. The magnetic bearings are made up with pairs of electromagnets, one for each direction to be controlled, possibly completed by a passive thrust. Each electromagnet produces an attractive force, whose intensity depends on the coil current, which explains why they are used by pairs in order to produce forces with both signs.
Jean Lévine
Chapter 13. Crane Control
The present chapter constitutes a continuation of the introductory example of section 5.1, chapter 5, and of example 7.2, chapter 7. We show how simple measurement feedbacks may be designed to track different types of rest-to-rest trajectories and compare them.
Jean Lévine
Chapter 14. Automatic Flight Control Systems
The present chapter1 is devoted to the flatness-based control design for one of the major control applications of the 20th century, namely ight control.
Jean Lévine
Backmatter
Metadaten
Titel
Analysis and Control of Nonlinear Systems
verfasst von
Jean Levine
Copyright-Jahr
2009
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-642-00839-9
Print ISBN
978-3-642-00838-2
DOI
https://doi.org/10.1007/978-3-642-00839-9

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