ISBN: 3540437339
TITLE: Dynamics of Controlled Mechanical Systems with Delayed Feedback
AUTHOR: Hu, Wang
TOC:

1 Modeling of Delayed Dynamic Systems 1
1.1 Mathematical Models 1
1.1.1 Dynamic Systems with Delayed Feedback Control 1
1.1.2 Dynamic Systems with Operator's Retardation 5
1.2 Experimental Modeling 9
1.2.1 Identification of Short Time Delays in Linear Systems 10
1.2.2 Identification of Arbitrary Time Delays in Nonlinear Systems 14
1.2.3 Discussions on Identifiability of Time Delays 21
2 Fundamentals of Delay Differential Equations 27
2.1 Initial Value Problems 27
2.1.1 Existence and Uniqueness of Solution 28
2.1.2 Solution of Linear Delay Differential Equations 33
2.2 Stability in the Sense of Lyapunov 37
2.2.1 The Lyapunov Methods 38
2.2.2 Method of Characteristic Function 42
2.2.3 Stability Criteria 47
2.3 Important Features of Delay Differential Equations 54
3 Stability Analysis of Linear Delay Systems 59
3.1 Delay-independent Stability of Single-degree-of-freedom Systems 60
3.1.1 Stability Criteria 61
3.1.2 Stability Criteria in Terms of Feedback Gains 66
3.2 The Generalized Sturm Criterion for Polynomials 70
3.2.1 Classical Sturm Criterion 70
3.2.2 Discrimination Sequence 72
3.2.3 Modified Sign Table 74
3.2.4 Generalized Sturm Criterion 75
3.3 Delay-independent Stability of High Dimensional Systems 76
3.4 Stability of Single-degree-of-freedom Systems with Finite Time Delays 86
3.4.1 Systems with Equal Time Delays 86
3.4.2 Systems with Unequal Time Delays 90
3.5 Stability Switches of High Dimensional Systems 91
3.5.1 Systems with a Single Time Delay 92
3.5.2 Systems with Commensurate Time Delays 98
3.6 Stability Analysis of an Active Chassis 102
3.6.1 A quarter Car Model of Suspension with a Delayed Sky-hook Damper 102
3.6.2 Four-wheel-steering Vehicle with a Time Delay in Drive's Response 109
4 Robust Stability of Linear Delay Systems 115
4.1 Robust Stability of a One-parameter Family of Quasi-polynomials 116
4.1.1 Non-convexity of the Set of Hurwitz Stable Quasi-polynomials 117
4.1.2 Sufficient and Necessary Conditions for Interval Stability 120
4.2 Edge Theorem for a Polytopic Family of Quasi-polynomials 124
4.2.1 Problem Formulation 125
4.2.2 Edge Theorem 127
4.2.3 Sufficient and Necessary Conditions 127
4.3 Dixon's Resultant Elimination 130
4.3.1 Dixon's Resultant Elimination 130
4.3.2 Robust D-stability of One-parameter Family of Polynomials 136
4.4 Robust Stability of Systems with Uncertain Commensurate Delays 140
4.4.1 Problem Formulation 141
4.4.2 Stability of Vertex Quasi-polynomials 143
4.4.3 Stability of Edge Quasi-polynomials 145
4.4.4 A sufficient and Necessary Condition 147
4.4.5 An Illustrative Example 147
5 Effects of a Short Time Delay on System Dynamics 151
5.1 Stability Estimation of High Dimensional Systems 151
5.1.1 Distribution of Eigenvalues Subject to a Short Time Delay 152
5.1.2 Estimation of Eigenvalues 155
5.1.3 Illustrative Examples 158
5.1.4 A Relation of Orthogonality of Mode Shapes 167
5.2 Stability Test Based on the Pad Approximation 168
5.2.1 Test of Stability 168
5.2.2 Test of Interval Stability 176
5.3 Dynamics of Simplified Systems via the Taylor Expansion 179
5.3.1 Linear Systems with Delayed State Feedback 180
5.3.2 Nonlinear Systems with Delayed Velocity Feedback 182
6 Dimensional Reduction of Nonlinear Delay Systems 189
6.1 Decomposition of State Space of Linear Delay Systems 190
6.1.1 Spectrum of a Linear Operator 192
6.1.2 Decomposition of State Space 194
6.2 Dimensional Reduction for Stiff-soft Systems 198
6.2.1 A quarter Car Model as a Singularly Perturbed System 199
6.2.2 Center Manifold Reduction in Critical Cases 200
6.2.3 Reduction for Singularly Perturbed Differential Equations 202
6.3 Stability Analysis of an Active Suspension 205
6.3.1 Center Manifold Reduction 206
6.3.2 Computation of the Approximated Center Manifold 207
6.3.3 Stability Analysis 210
7 Periodic Motions of Nonlinear Delay Systems 213
7.1 The Hopf Bifurcation of Autonomous Systems 213
7.1.1 Theory of the Hopf Bifurcations 214
7.1.2 Decomposition of Bifurcating Solution 217
7.1.3 Bifurcating Solutions in Normal Form 219
7.2 Computation of Bifurcating Periodic Solutions 222
7.2.1 Method of the Fredholm Alternative 222
7.2.2 Stability of Bifurcating Periodic Solutions 227
7.2.3 Perturbation Method 230
7.3 Periodic Motions of a Duffing Oscillator with Delayed Feedback 234
7.3.1 Stability Switches of Equilibrium 235
7.3.2 Periodic Motion Determined by Method of Fredholm Alternative 237
7.3.3 Periodic Motion Determined by Method of Multiple Scales 243
7.4 Periodic Motions of a Forced Duffing Oscillator with Delayed Feedback 248
7.4.1 Primary Resonance 249
7.4.2 1/3 Subharmonic Resonance 254
7.5 Shooting Scheme for Locating Periodic Motions 259
7.5.1 Basic Concepts and Computation Scheme 259
7.5.2 Case Studies 262
8 Delayed Control of Dynamic Systems 267
8.1 Delayed Linear Feedback for Linear Systems 267
8.1.1 Delayed Linear Feedback and Artificial Damping 267
8.1.2 Delayed Resonator: A Tunable Vibration Absorber 269
8.2 Stabilization to Critically Stable Nonlinear Systems 272
8.2.1 Statement of Problem 274
8.2.2 Analysis on Stabilization 275
8.2.3 Case Studies 278
8.2.4 Discussions on Approximate Integrals 280
8.3 Controlling Chaotic Motion 282
8.3.1 Basic Idea 283
8.3.2 Choice of Feedback Gain 284
References 287
Index 293
END
