Analysis and Control of Linear Systems E-Book

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 554

Veröffentlichungsjahr: 2013

Beschreibung

Automation of linear systems is a fundamental and essential theory. This book deals with the theory of continuous-state automated systems.

Das E-Book Analysis and Control of Linear Systems wird angeboten von John Wiley & Sons und wurde mit folgenden Begriffen kategorisiert:
Control Systems Technology, Electrical & Electronics Engineering, Elektrotechnik u. Elektronik, Regelungstechnik

Leseprobe

Table of Contents

Preface

Part 1 System Analysis

Chapter 1 Transfer Functions and Spectral Models

1.1. System representation

1.2. Signal models

1.3. Characteristics of continuous systems

1.4. Modeling of linear time-invariant systems

1.5. Main models

1.6. A few reminders on Fourier and Laplace transforms

1.7. Bibliography

Chapter 2 State Space Representation

2.1. Reminders on the systems

2.2. Resolving the equation of state

2.3. Scalar representation of linear and invariant systems

2.4. Controllability of systems

2.5. Observability of systems

2.6. Bibliography

Chapter 3 Discrete-Time Systems

3.1. Introduction

3.2. Discrete signals: analysis and manipulation

3.3. Discrete systems (DLTI)

3.4. Discretization of continuous-time systems

3.5.Conclusion

3.6.Bibliography

Chapter 4 Structural Properties of Linear Systems

4.1. Introduction: basic tools for a structural analysis of systems

4.2. Beams, canonical forms and invariants

4.3. Invariant structures under transformation groups

4.4. An introduction to a structural approach of the control

4.5.Conclusion

4.6.Bibliography

Chapter 5 Signals: Deterministic and Statistical Models

5.1. Introduction

5.2. Signals and spectral analysis

5.3. Generator processes and ARMA modeling

5.4. Modeling of LTI systems and ARMAX modeling

5.5. From the Markovian system to the ARMAX model

5.6. Bibliography

Chapter 6 Kalman's Formalism for State Stabilization and Estimation

6.1. The academic problem of stabilization through state feedback

6.2. Stabilization by pole placement

6.3. Reconstruction of state and observers

6.4. Stabilization through quadratic optimization

6.5. Resolution of the state reconstruction problem by duality of the quadratic optimization

6.6. Control through state feedback and observers

6.7. A few words on the resolution of Riccati’s equations

6.8. Conclusion

6.9. Bibliography

Chapter 7 Process Modeling

7.1. Introduction

7.2. Modeling

7.3. Graphic identification approached

7.4. Identification through criterion optimization

7.5. Conclusion around an example

7.6. Bibliography

Chapter 8 Simulation and Implementation of Continuous Time Loops

8.1. Introduction

8.2. Standard linear equations

8.3. Specific linear equations

8.4. Stability, stiffness and integration horizon

8.5. Non-linear differential systems

8.6. Discretization of control laws

8.7. Bibliography

Part 2 System Control

Chapter 9 Analysis by Classic Scalar Approach

9.1. Configuration of feedback loops

9.2. Stability

9.3. Precision

9.4. Parametric sensitivity

9.5. Bibliography

Chapter 10 Synthesis of Closed Loop Control Systems

10.1. Role of correctors: precision-stability dilemma

10.2. Serial correction

10.3. Correction by combined actions

10.4. Proportional derivative (PD) correction

10.5. Proportional integral (PI) correction

10.6. Proportional integral proportional (PID) correction

10.7. Parallel correction

10.8. Bibliography

Chapter 11 Robust Single-Variable Control through Pole Placement

11.1. Introduction

11.2. The obvious objectives of the correction

11.3. Resolution

11.4. Implementation

11.5. Methodology

11.6. Conclusion

11.7. Bibliography

Chapter 12 Predictive Control

12.1. General principles of predictive control

12.2. Generalized predictive control (GPC)

12.3. Functional predictive control (FPC)

12.4. Conclusion

12.5. Bibliography

Chapter 13 Methodology of the State Approach Control

13.1. Introduction

13.2. H2 control

13.3. Data of a feedback control problem

13.4. Standard H2 optimization problem

13.5. Conclusion

13.6. Appendices

13.7. Bibliography

Chapter 14 Multi-variable Modal Control

14.1. Introduction

14.2. The eigenstructure

14.3. Modal analysis

14.4. Traditional methods for eigenstructure placement

14.5. Eigenstructure placement as observer

14.6. Conclusion

14.7. Bibliography

Chapter 15 Robust H∞/LMI Control

15.1. The H∞ approach

15.2. The μ-analysis

15.3. The μ-synthesis

15.4. Synthesis of a corrector depending on varying parameters

15.5. Conclusion

15.6. Bibliography

Chapter 16 Linear Time-Variant Systems

16.1. Ring of non-commutative polynomials

16.2. Body of rational fractions

16.3. Transfer function

16.4. Algebra of non-stationary linear systems

16.5. Applications

16.6. Conclusion

16.7. Bibliography

List of Authors

Index

First published in France in 2002 by Hermès Science/Lavoisier entitled “Analyse des systèmes linéaires” and “Commande des systèmes linéaires” First published in Great Britain and the United States in 2007 by ISTE Ltd

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

ISTE Ltd

ISTE USA

6 Fitzroy Square

4308 Patrice Road

London W1T 5DX

Newport Beach, CA 92663

UK

USA

www.iste.co.uk

© ISTE Ltd, 2007

© LAVOISIER, 2002

The rights of Philippe de Larminat to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Cataloging-in-Publication Data

[Analyse des systèmes linéaires/Commande des systèmes linéaires. eng] Analysis and control of linear systems analysis and control of linear systems/edited by Philippe de Larminat.

p. cm.

ISBN-13: 978-1-905209-35-4 ISBN-10: 1-905209-35-5 1. Linear control systems. 2. Automatic control. I. Larminat, Philippe de. TJ220.A5313 2006 629.8′32—dc22

2006033665

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN 10: 1-905209-35-5 ISBN 13: 978-1-905209-35-4

Preface

This book is about the theory of continuous-state automated systems whose inputs, outputs and internal variables (temperature, speed, tension, etc.) can vary in a continuous manner. This is contrary to discrete-state systems whose internal variables are often a combination of binary sizes (open/closed, present/absent, etc.).

The word “linear” requires some explanation. The automatic power control of continuous-state systems often happens through actions in relation to the gaps we are trying to control. Thus, it is possible to regulate cruise control by acting on the acceleration control proportionally to the gap observed in relation to a speed instruction. The word “proportional” precisely summons up a linear control law.

Some processes are actually almost never governed by laws of linear physics. The speed of a vehicle, even when constant, is certainly not proportional to the position of the accelerator pedal. However, if we consider closed loop control laws, the return will correct mistakes when they are related either to external disturbances or to gaps between the conception model and the actual product. This means that modeling using a linear model is generally sufficient to obtain efficient control laws. Limits to the automated systems performances generally come from the restricted power of motors, precision of captors and variability of the behavior of the processes, more than from their possible non-linearity.

It is necessary to know the basics of linear automated systems before learning about the theory of non-linear systems. That is why linear systems are a fundamental theory, and the problems linked to closed-loop control are a big part of it.

Input-output and the state representations, although closely linked, are explained in separate chapters (1 and 2). Discrete-time systems are, for more clarity, explained in Chapter 3. Chapter 4 explains the structural properties of linear systems. Chapter 5 looks into deterministic and statistical models of signals. Chapter 6 introduces us to two fundamental theoretical tools: state stabilization and estimation. These two notions are also covered in control-related chapters. Chapter 7 defines the elements of modeling and identification. All modern control theories rely on the availability of mathematical models of processes to control them.

Modeling is therefore upstream of the control engineer. However, pedagogically it is located downstream because the basic systems theory is needed before it can be developed. This same theory also constitutes the beginning of Chapter 8, which is about simulation techniques. These techniques form the basis of the control laws created by engineers.

Chapter 9 provides an analysis of the classic invariable techniques while Chapter 10 summarizes them. Based on the transfer function concept, Chapter 11 addresses pole placement control and Chapter 12 internal control. The three following chapters cover modern automation based on state representation. They highlight the necessary methodological aspects. H2 optimization control is explained in Chapter 13, modal control in Chapter 14 and H∞ control in Chapter 15. Chapter 16 covers linear time-variant systems.

Part 1

System Analysis

Chapter 1

Transfer Functions and Spectral Models1

1.1. System representation

A system is an organized set of components, of concepts whose role is to perform one or more tasks. The point of view adopted in the characterization of systems is to deal only with the input-output relations, with their causes and effects, irrespective of the physical nature of the phenomena involved.

Hence, a system realizes an application of the input signal space, modeling magnitudes that affect the behavior of the system, into the space of output signals, modeling relevant magnitudes for this behavior.

Figure 2.1.System symbolics

In what follows, we will consider mono-variable, analog or continuous systems which will have only one input and one output, modeled by continuous signals.

1.2. Signal models

A continuous-time signal (t R) is represented a priori through a function x(t) defined on a bounded interval if its observation is necessarily of finite duration.

When signal mathematical models are built, the intention is to artificially extend this observation to an infinite duration, to introduce discontinuities or to generate Dirac impulses, as a derivative of a step function. The most general model of a continuous-time signal is thus a distribution that generalizes to some extent the concept of a digital function.

1.2.1. Unit-step function or Heaviside step function U(t)

This signal is constant, equal to 1 for the positive evolution variable and equal to 0 for the negative evolution variable.

Figure 1.2.Unit-step function

This signal constitutes a simplified model for the operation of a device with a very low start-up time and very high running time.

1.2.2. Impulse

Physicists began considering shorter and more intense phenomena. For example, an electric loading Mµ can be associated with a mass M evenly distributed according to an axis.

What density should be associated with a punctual mass concentrated in 0? This density can be considered as the bound (simple convergence) of densities Mμn(σ) verifying:

This bound is characterized, by the physicist, by a “function” δ(σ) as follows:

However, this definition does not make any sense; no integral convergence theorem is applicable.

Nevertheless, if we introduce an auxiliary function φ(σ) continuous in 0, we will obtain the mean formula:

Hence, we get a functional definition, indirect of symbol δ: δ associates with any continuous function at the origin its origin value. Thus, it will be written in all cases:

δ is called a Dirac impulse and it represents the most popular distribution. This impulse δ is also written δ(t).

Figure 1.3.Modeling of a short phenomenon

We notice that in the model based on Dirac impulse, the “microscopic” look of the real signal disappears and only the information regarding the area is preserved.

Finally, we can imagine that the impulse models the derivative of a unit-step function. To be sure of this, let us consider the step function as the model of the real signal uo(t) represented in Figure 1.4, of derivative . Based on what has been previously proposed, it is clear that .

Figure 1.4.Derivative of a step function

1.2.3. Sine-wave signal

1.3. Characteristics of continuous systems

The input-output behavior of a system may be characterized by different relations with various degrees of complexity. In this work, we will deal only with linear systems that obey the physical principle of superposition and that we can define as follows: a system is linear if to any combination of input constant coefficients ∑aixi corresponds the same output linear combination, ∑aiyiaiG(xi).

Obviously, in practice, no system is rigorously linear. In order to simplify the models, we often perform linearization around a point called an operating point of the system.

A system has an instantaneous response if, irrespective of input x, output y depends only on the input value at the instant considered. It is called if its response at a given instant depends on input values at other instants.