The papers in this edited volume aim to provide a better understanding of the dynamics and control of a large class of hybrid dynamical systems that are described by different models in different state space domains. They not only cover important aspects and tools for hybrid systems analysis and control, but also a number of experimental realizations. Special attention is given to synchronization - a universal phenomenon in nonlinear science that gained tremendous significance since its discovery by Huygens in the 17th century. Possible applications of the results introduced in the book include control of mobile robots, control of CD/DVD players, flexible manufacturing lines, and complex networks of interacting agents. The book is based on the material presented at a similarly entitled minisymposium at the 6th European Nonlinear Dynamics Conference held in St Petersburg in 2008. It is unique in that it contains results of several international and interdisciplinary collaborations in the field, and reflects state-of-the-art technological development in the area of hybrid mechanical systems at the forefront of the 21st century.
Author(s): Gennady Leonov, Gennady Leonov, Henk Nijmeijer, Alexander Pogromsky, Alexander Fradkov
Series: World Scientific Series on Nonlinear Science, Series B
Publisher: World Scientific Pub Co (
Year: 2010
Language: English
Pages: 261
Contents......Page 14
Preface......Page 8
Biography: Ilya Izrailevich Blekhman......Page 10
1. Huijgens' Synchronization: A Challenge H. Nijmeijer, A.Y. Pogromsky......Page 20
References......Page 24
Abstract......Page 26
2.1 Introduction......Page 27
2.2 Computation of Lyapunov quantities and small limit cycles......Page 28
2.2.1 Computation of Lyapunov quantities in Euclidean coordinates and in the time domain......Page 30
2.2.1.1 Approximation of solution in Euclidean coordinates......Page 31
2.2.1.2 Computation of Lyapunov quantities in the time domain......Page 35
2.2.3 Lyapunov quantities of Lienard equation......Page 38
2.3 Transformation between quadratic systems and Lienard equation......Page 40
2.4 Method of asymptotic integration for Lienard equation with discontinuous right–hand side and large limit cycles......Page 41
2.5 Four limit cycles for Lienard equation and the corresponding quadratic system......Page 43
References......Page 45
3.1 Introduction......Page 48
3.2 Evolutionary variational inequalities......Page 50
3.3 Basic assumptions......Page 52
3.4 Absolute observation - stability of evolutionary inequalities......Page 53
3.5 Application of observation stability to the beam equation......Page 56
References......Page 60
4.1 Introduction......Page 62
4.2 Reduction to a normal form......Page 63
4.3 Existence of invariant domain......Page 64
4.4 Conditions of hyperbolicity......Page 67
References......Page 71
5.1 Introduction......Page 72
5.2.1 System description......Page 74
5.2.2 Case: electromechanical system......Page 76
5.2.3 Motivating example: nonlinear behavior......Page 77
5.3.1 Convergent systems......Page 79
5.3.3 Performance analysis in frequency domain......Page 81
5.4.1 Describing function method......Page 84
5.4.2 Performance analysis example......Page 87
References......Page 88
6.1 Introduction......Page 90
6.2 Experimental set-up......Page 92
6.3 Steady-state behavior of the uncontrolled system......Page 93
6.4 Control objectives and PD controller design approach......Page 94
6.5 Dynamic model......Page 96
6.6 Effects of separate P-action and separate D-action......Page 98
6.7 PD control......Page 99
6.7.1 Control objective 1......Page 100
6.7.2 Control objective 2......Page 101
6.8 Comparison with passive control via a linear DVA......Page 103
6.9 Conclusions......Page 105
References......Page 106
7.1 Introduction......Page 108
7.2 Description of state estimation over the limited-band communication channel......Page 110
7.3 Coding procedure......Page 113
7.4 Evaluation of state estimation error......Page 114
7.5 Example. State estimation of nonlinear oscillator......Page 116
Acknowledgments......Page 118
References......Page 119
8.1 Introduction......Page 122
8.2.1 Typical task......Page 127
8.2.2 Exogenous signals entering the dynamics......Page 128
8.2.3 Stability analysis criteria......Page 131
8.3 Controller design......Page 133
8.4.1 Design of the desired trajectories......Page 134
8.4.2 Design of q¤d(¢) and qd(¢) on the phases Ik......Page 135
8.5 Design of the desired contact force during constraint phases......Page 137
8.6 Strategy for take-off at the end of constraint phases......Page 138
8.7 Closed-loop stability analysis......Page 139
8.8 Illustrative example......Page 141
8.9 Conclusions......Page 142
References......Page 144
Abstract......Page 148
9.1 Introduction......Page 149
9.2 Preliminaries......Page 152
9.3 Measure differential inclusions......Page 153
9.4 Convergent systems......Page 155
9.5 Convergence properties of Lur'e-type measure di®erential inclusions......Page 156
9.6 Tracking control of Lur'e-type measure differential inclusions......Page 161
9.7 Example of a mechanical system with a unilateral constraint......Page 163
9.8 Conclusions......Page 165
References......Page 168
10.1 Introduction......Page 172
10.2 Experimental set-up......Page 173
10.3.1 Problem statement and analysis......Page 175
10.3.2 Experimental and numerical results......Page 177
10.4.1 Problem statement......Page 180
10.5 Conclusion and future research......Page 181
References......Page 183
11.1 Introduction......Page 186
11.2 Control algorithm: formulation of the problem and approach......Page 187
11.2.1 Special case: Energy control for Hamiltonian systems......Page 189
11.3 Two pendulums under a single force......Page 191
11.3.1 Control problem formulation......Page 192
11.3.3 Control algorithm analysis......Page 193
11.4 Conclusion......Page 197
References......Page 198
12.1 Introduction......Page 200
12.2 Problem statement......Page 201
12.3 Synchronization of oscillators driven by Van der Pol control input......Page 204
12.4 Synchronization of oscillators driven by Van der Pol-Du±ng control input......Page 209
Acknowledgments......Page 212
References......Page 213
13.1 Introduction......Page 214
13.2 Preliminaries......Page 215
13.3 Synchronization of diffusively coupled Hindmarsh-Rose oscillators......Page 217
13.4 Experimental setup......Page 220
13.5 Synchronization experiments......Page 222
13.6 Conclusions......Page 227
Acknowledgments......Page 228
References......Page 229
14.1 Introduction......Page 230
14.2 Design of mechanical part......Page 232
14.3 Electronics of the multipendulum setup......Page 233
14.3.1 System for data exchange with control computer......Page 234
14.3.3 Computer-process interface......Page 235
14.3.4 Electronic modules of the set-up......Page 236
14.4 Conclusions......Page 237
References......Page 239
15.1 Introduction......Page 242
15.2 Mechanical model......Page 246
15.3 Band gap effects in periodic structures......Page 247
15.4 Approximate equations governing the slow and the fast motion......Page 248
15.4.2 Example 2: essentially non-linear damping and restoring forces......Page 252
15.5 Numerical examples .......Page 253
15.5.1 Inclusions with linear plus cubic non-linear behaviour......Page 254
15.5.2 Non-linear inclusions with non-local interaction......Page 255
15.5.3 Linear chains with non-linear damping forces......Page 256
15.6 Conclusions......Page 258
References......Page 259
