Essentials of nonlinear control theory:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
London
Peregrinus
1983
|
| Series: | Institution of Electrical Engineers: IEE topics in control series
2 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 90 S. graph. Darst. |
| ISBN: | 0906048966 |
Staff View
MARC
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| 100 | 1 | |a Leigh, James R. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Essentials of nonlinear control theory |c J. R. Leigh |
| 264 | 1 | |a London |b Peregrinus |c 1983 | |
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Record in the Search Index
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| adam_text | Contents
Preface v
List of symbols vii
1 Initial orientation 1
1.1 Origins of nonlinearity 1
1.2 Definition of nonlinearity 1
1.3 Types of behaviour found in nonlinear systems 2
1.4 Failure of linear techniques 3
1.5 Approaches to nonlinear system analysis 4
2 Dynamic Systems — definitions and notation 5
2.0 Introduction 5
2.1 Dynamic system axioms 5
2.2 Trajectories and critical points 6
2.3 Definitions of stability 6
2.4 Representation of a system by differential equations 7
2.5 First order scalar differential equations 7
2.5.1 Theorem 2.1 8
2.5.2 Definition 8
2.5.3 Theorem 2.2 8
2.6 Systems of first order equations 8
2.7 Vector field aspects 9
2.8 Limit cycles 9
3 The describing function method 10
3.0 Introduction 10
3.1 Outline of the method 10
3.2 Justification for the describing function method 12
3.3 Stability of self excited oscillations 12
3.4 Derivation of the Fourier series for/(a sin a t) 12
3.5 Determination of Fourier coefficients without integration 13
3.5.1 Simple example 14
3.6 Examples 15
3.6.1 Ideal relay 15
3.6.2 Relay with dead zone 16
3.6.3 Cubic non linearity 17
x Contents
3.6.4 Example 4 19
3.6.5 Linear element with hysteresis 21
3.7 Concluding remarks 22
4 The phase plane portrait 23
4.0 Introduction 23
4.1 Definition 23
4.2 Simple examples illustrating behaviour in the phase plane 23
4.2.1 A second order linear process without damping 23
4.2.2 A second order linear process with damping 24
4.3 The direction of rotation of oscillatory trajectories in the phase plane 25
4.3.1 Example 26
4.4 Switching lines in the phase plane 26
4.5 Discussion 28
5 Linearisation 29
5.0 Introduction 29
5.1 The principle of linearisation 29
5.1.1 Nonlinear second order systems with one input and two outputs 30
5.2 Linearisation about an analytically specified trajectory 31
5.3 Other approaches to linearisation 32
6 Determination of the qualitative behaviour of a nonlinear second order system by
linearisation (Lyapunov s first method) 33
6.0 Introduction 33
6.1 Critical points of a system » 33
6.2 Critical point analysis of a linear second order system 34
6.3 Critical points of nonlinear systems 38
6.4 Critical point analysis of a second order nonlinear system 39
6.5 System behaviour near to a critical point 39
6.6 Special cases not covered in the table of Section 6.2 41
6.6.1 Repeated eigenvalues 41
6.6.2 Repeated eigenvalues — special case 42
6.6.3 One eigenvalue is zero 42
6.7 Examples 42
6.7.1 Example 1 42
6.7.2 Example 2 46
6.8 Time varying problems 46
6.8.1 Theorem 6.1 47
6.9 Obtaining information on global stability properties from the
linear approximation 48
6.9.1 Theorem 6.2 48
7 Lyapunov s second or direct method 49
7.0 Introduction 49
7.1 Preliminary definitions 50
7.2 Lyapunov s second (or direct) stability theorem 50
7.3 Cetaev instability theorem 50
7.4 Geometric interpretation 51
7.5 Determination of the extent of a region of asymptotic stability
around the origin 52
Contents xi
7.6 The search for a Lyapunov function 52
7.7 Examples 52
7.7.1 Example 1 52
7.7.2 Example 2 53
7.7.3 Example 3 54
7.7.4 Example 4 56
7.7.5 Example 5 56
7.8 Lyapunov s direct method for time varying problems 58
7.8.1 Stability theorem 58
7.9 Use of the Lyapunov direct method as a design tool 58
7.10 Concluding remarks 59
8 Envelope methods — the Popov and circle criteria for graphical analysis of a single
feedback loop containing a nonlinear element 60
8.0 Introduction 60
8.1 The Aizerman and Kalman conjectures 60
8.1.1 Aizerman s conjecture 60
8.1.2 Kalman s conjecture 61
8.2 The Popov stability criterion 61
8.3 The circle method 62
9 Limit cycles and relaxation oscillations 66
9.0 Introduction 66
9.1 Limit cycle — definition 66
9.2 Asymptotic stability 68
9.2.1 Theorem 9.1 68
9.3 The Poincare index and its implications in the phase plane 68
9.3.1 Bendixson s first theorem 69
9.3.2 Implications 69
9.3.3 Bendixson s second theorem 69
9.3.4 Theorem 9.2 70
9.4 Relaxation oscillations 70
9.4.1 Example 70
10 Lienard s equation 72
10.0 Introduction 72
10.1 Lienard s equation — definition 72
10.1.1 Restrictions on the functions/and g 72
10.2 Analysis of Lienards equation by Lyapunov s direct method 73
10.3 Examples 73
10.3.1 Example 1 73
10.3.2 Example 2. Van der Pol s equation 73
10.4 Existence of a limit cycle for Lienards equation 74
10.5 A particular example of Lienards equation that has a stable limit cycle 75
11 Gradient systems and system decomposition 77
11.0 Introduction 77
11.1 Gradient systems 77
11.2 Example of a gradient system 78
11.3 Decomposition of a second order system — introduction 79
11.4 The decomposition method for second order nonlinear systems 80
11.4.1 Theorem 11.1 80
11.4.2 Theorem 11.2 81
xii Contents
11.5 Examples 81
11.5.1 Example 1— Linear system with a stable node 81
11.5.2 Example 2 81
12 References 84
13 Bibliography 86
13.1 Further historical and supporting literature 86
13.2 Theoretical foundations 86
13.3 Multivariable and discrete time systems 87
13.4 Other control theory 88
13.5 Selected application papers 88
|
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| discipline | Mathematik Mess-/Steuerungs-/Regelungs-/Automatisierungstechnik / Mechatronik |
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| illustrated | Illustrated |
| indexdate | 2024-07-09T16:09:34Z |
| institution | BVB |
| isbn | 0906048966 |
| language | English |
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| physical | XII, 90 S. graph. Darst. |
| publishDate | 1983 |
| publishDateSearch | 1983 |
| publishDateSort | 1983 |
| publisher | Peregrinus |
| record_format | marc |
| series | Institution of Electrical Engineers: IEE topics in control series |
| series2 | Institution of Electrical Engineers: IEE topics in control series |
| spelling | Leigh, James R. Verfasser aut Essentials of nonlinear control theory J. R. Leigh London Peregrinus 1983 XII, 90 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Institution of Electrical Engineers: IEE topics in control series 2 Commande non linéaire ram Control systems and control theory sigle Nonlinear control theory Nichtlineare Kontrolltheorie (DE-588)4475218-0 gnd rswk-swf Nichtlineare Kontrolltheorie (DE-588)4475218-0 s DE-604 Institution of Electrical Engineers: IEE topics in control series 2 (DE-604)BV004177807 2 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002608947&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Leigh, James R. Essentials of nonlinear control theory Institution of Electrical Engineers: IEE topics in control series Commande non linéaire ram Control systems and control theory sigle Nonlinear control theory Nichtlineare Kontrolltheorie (DE-588)4475218-0 gnd |
| subject_GND | (DE-588)4475218-0 |
| title | Essentials of nonlinear control theory |
| title_auth | Essentials of nonlinear control theory |
| title_exact_search | Essentials of nonlinear control theory |
| title_full | Essentials of nonlinear control theory J. R. Leigh |
| title_fullStr | Essentials of nonlinear control theory J. R. Leigh |
| title_full_unstemmed | Essentials of nonlinear control theory J. R. Leigh |
| title_short | Essentials of nonlinear control theory |
| title_sort | essentials of nonlinear control theory |
| topic | Commande non linéaire ram Control systems and control theory sigle Nonlinear control theory Nichtlineare Kontrolltheorie (DE-588)4475218-0 gnd |
| topic_facet | Commande non linéaire Control systems and control theory Nonlinear control theory Nichtlineare Kontrolltheorie |
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