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2014 | Book

Introduction, Motivation, and Background Read chapter Read first chapter

Tautological Control Systems

Author: Andrew D. Lewis

Publisher: Springer International Publishing

Book Series : SpringerBriefs in Electrical and Computer Engineering

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik"

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About this book

This brief presents a description of a new modelling framework for nonlinear/geometric control theory. The framework is intended to be—and shown to be—feedback-invariant. As such, Tautological Control Systems provides a platform for understanding fundamental structural problems in geometric control theory. Part of the novelty of the text stems from the variety of regularity classes, e.g., Lipschitz, finitely differentiable, smooth, real analytic, with which it deals in a comprehensive and unified manner. The treatment of the important real analytic class especially reflects recent work on real analytic topologies by the author. Applied mathematicians interested in nonlinear and geometric control theory will find this brief of interest as a starting point for work in which feedback invariance is important. Graduate students working in control theory may also find Tautological Control Systems to be a stimulating starting point for their research.

Table of Contents

Frontmatter
Chapter 1. Introduction, Motivation, and Background
Abstract
One can study nonlinear control theory from the point of view of applications, or from a more fundamental point of view, where system structure is a key element. From the practical point of view, questions that arise are often of the form, “How can we...”, for example, “How can we steer a system from point \(A\) to point \(B\)?” or, “How can we stabilise this unstable equilibrium point?” or, “How can we manoeuvre this vehicle in the most efficient manner?” From a fundamental point of view, the problems are often of a more existential nature, with, “How can we” replaced with, “Can we”. These existential questions are often very difficult to answer in any sort of generality.
Andrew D. Lewis
Chapter 2. Topologies for Spaces of Vector Fields
Abstract
In this chapter we review the definitions of the topologies we use for spaces of Lipschitz, finitely differentiable, smooth, and real analytic vector fields. We comment that all topologies we define are locally convex topologies, of which the normed topologies are a special case. However, few of the topologies we define, and none of the interesting ones, are normable. We, therefore, begin with a very rapid review of locally convex topologies, and why they are inevitable in work such as we undertake here.
Andrew D. Lewis
Chapter 3. Time-Varying Vector Fields and Control Systems
Abstract
We now turn to utilising the locally convex topologies from the preceding chapter to characterise time-varying vector fields and control systems. We shall see that the use of locally convex topologies allows for a comprehensive and unified treatment of these notions, and allows one to understand in a deep way their structure in a way which has hitherto not been possible. This is especially true in the real analytic case, where we describe new results of Jafarpour and Lewis [6, 7] on the structure of time-varying vector fields and control systems depending on state in a real analytic manner.
Andrew D. Lewis
Chapter 4. Presheaves and Sheaves of Sets of Vector Fields
Abstract
We choose to phrase our notion of control in the language of sheaf theory which we believe has the following benefits: it permits natural formulations of problems lacking a natural formulation in “ordinary” control theory; sheaves are the proper framework for constructing germs of control systems which are often important in the study of local system structure; and sheaf theory provides a natural class of mappings between systems of which we take advantage.
Andrew D. Lewis
Chapter 5. Tautological Control Systems: Definitions and Fundamental Properties
Abstract
In this chapter we introduce the class of control systems, tautological control systems, that we propose as being useful mathematical models for the investigation of geometric system structure. As promised in our introduction in Sect. 1.​2, this class of systems naturally handles a variety of regularity classes; we work with finitely differentiable, Lipschitz, smooth, and real analytic classes simultaneously with comparative ease. Also as indicated in Sect. 1.​2, the framework makes essential use of sheaf theory in its formulation. We shall see in Sect. 5.6 that the natural morphisms for tautological control systems ensure feedback-invariance of the theory.
Andrew D. Lewis
Chapter 6. Étalé Systems
Abstract
The development of tautological control systems in the preceding chapter was focussed in large part on connecting this new class of control systems with more common existing classes of systems. In particular, our notion of a trajectory is a quite natural adaptation to our framework of the usual notion of a trajectory for a control system. However, it turns out that there is a limitation of this sort of definition in terms of being able to use the full power of the tautological control system framework. In this chapter we overcome this limitation, and at the same time more fully integrate the sheaf formalism into the way in which we think about tautological control systems.
Andrew D. Lewis
Chapter 7. Ongoing and Future Work
Abstract
In this chapter we will give a sketchy, but we hope compelling, idea of how the tautological control system framework can be used to say new things about control systems. This will also provide an illustration of how, in practice, one can do control theory within the confines of the tautological control system framework, without reverting to the comforting control parameterisations with which one is familiar. We will emphasise that some of these ideas are in the preliminary stages of investigation, so the final word on what results will look like has yet to be uttered. Nonetheless, we believe that even the clear problem formulations we give make it apparent that there is something “going on” here.
Andrew D. Lewis
Metadata
Title
Tautological Control Systems
Author
Andrew D. Lewis
Copyright Year
2014
Electronic ISBN
978-3-319-08638-5
Print ISBN
978-3-319-08637-8
DOI
https://doi.org/10.1007/978-3-319-08638-5