Advances in analysis and control of time-delayed dynamical systems [collection of papers presented at the International Workshop on Recent Advances in Analysis and Control of Time-delayed Dynamical Systems in Tianjin, China on July 2 to 4, 2012]

Contents 8 Preface 6 Chapter 1 Complete Quadratic Lyapunov-Krasovskii Functional: Limitations, Computational Efficiency, and Convergence Keqin Gu 12 1. Introduction 12 2. Complete Quadratic Lyapunov-Krasovskii Functional 16 3. Discretized Lyapunov Functional Method 17 4. Coupled Differential-difference Equations 22 5. Miscellaneous Issues 24 5.1. Computational Efficiency 24 5.2. Convergence Issue for Multiple Neutral Delays 25 5.3. Lyapunov-Krasovskii Functionals Containing State Derivatives 25 6. SOS Method 25 7. Conclusions and Perspectives 26 References 26 Chapter 2 Recent Approaches for the Numerical Solution of State-dependent Delay Differential Equations with Discontinuities Alfredo Bellen 32 1. Introduction 32 2. Weak Solutions 39 3. Regularization Techniques 41 4. Comparing Regularizations 50 References 50 Chapter 3 Engineering Applications of Time-periodic Time-delayed Systems Gabor Stepan 52 1. Introduction 52 2. Delayed Mathieu Equation 53 3. Semi-discretization Method for Periodic DDEs 55 4. Engineering Applications 58 4.1. Modeling and Stability of Milling Operations 58 4.2. Cutting with Varying Spindle Speed 60 4.3. Act-and-wait Control of Force Controlled Robots 61 5. Conclusions 63 References 65 Chapter 4 Synchronization in Delay-coupled Complex Networks Eckehard Scholl 68 1. Introduction 68 2. Stability of Synchronization for Large Delay 69 3. Cluster Synchronization 74 4. Adaptive Synchronization 76 4.1. Speed-gradient Method 76 4.2. Zero-lag Synchronization 77 4.3. Splay State and Cluster Synchronization 78 4.4. Controlling Several Parameters Simultaneously 80 5. Transitions between Synchronization and Desychronization 80 5.1. Excitability of Type II 81 5.2. Excitability of Type I 83 6. Conclusion and Outlook 85 References 86 Chapter 5 Stochastic Dynamics and Optimal Control of Quasi Integrable Hamiltonian Systems with Time-delayed Feedback Control Weiqiu Zhu, Zhonghua Liu 96 1. Introduction 96 2. Stochastic Averaging Method for Quasi Integrable Hamiltonian Systems with Time-delayed Feedback Control 97 2.1. Gaussian White Noise Excitations 100 2.1.1. Non-resonant Case 100 2.1.2. Resonant Case 101 2.2. Wide-band Random Excitations 103 2.2.1. Non-resonant Case 103 2.2.2. Resonant Case 104 2.3. Narrow-band Bounded Noise Excitation 104 2.3.1. External Resonance Only 105 2.3.2. Both Internal and External Resonances 106 2.4. Combined Excitations of Harmonic Function and One Kind of above Random Processes 107 2.4.1. Internal Resonance Only 107 2.4.2. External Resonance Only 107 2.4.3. Both Internal and External Resonances 108 3. Stochastic Dynamics of Quasi Integrable Hamiltonian Systems with Time-delayed Feedback Control 109 3.1. Response 109 3.2. Stochastic Stability 123 3.3. Stochastic Bifurcation 133 3.4. First Passage Failure 140 3.4.1. Gaussian White Noise Excitation 141 4. Stochastic Optimal Control of Quasi Integrable Hamiltonian Systems with Time-delayed Feedback Control 153 4.1. Response Minimization Control 153 4.2. Stabilization 158 4.3. Minimax Optimal Bounded Control 161 5. Concluding Remark 170 References 171 Chapter 6 Delay Induced Strong and Weak Resonances in Delayed Differential Systems Jian Xu, Wanyong Wang 174 1. Introduction 174 2. Analysis for Double Hopf Bifurcation 177 2.1. The Case μ = μ 178 2.2. The Case μ = 2 μ 181 3. Conditions for Strong Resonances and Weak Resonances 184 3.1. High-order Resonances 184 3.2. Low-order Resonances 185 3.2.1. 1 : 3 Resonance 185 3.2.2. 1 : 2 Resonance 187 3.2.3. 1 : 1 Resonance 189 4. Weak and Strong Resonances in Delayed Feedback Systems 190 4.1. 1 : 2 Resonance 192 4.2. 1 : 3 Resonance 194 4.3. 1 : 5 Resonance 195 5. Weak and Strong Resonances in Van der Pol Systems with Delay Coupling 195 5.1. 1 : 2 Resonance 196 5.2. 1 : 3 Resonance 198 5.3. 1 : 5 Resonance 199 5.4. 1 :√2 Resonance 199 6. Conclusions 200 References 200 Chapter 7 Stability and Hopf Bifurcation of Time-delayed Systems with Complex Coefficients Zaihua Wang, Junyu Li 204 1. Introduction 204 2. The Crossing Direction for Stability Analysis 206 2.1. The Case with a Single Delay 207 2.2. The Degenerate Case with Real or Complex Coefficients 212 2.3. The Case with Commensurate Delays 214 3. Numerical and Graphical Stability Test 216 3.1. Calculation of the Rightmost Characteristic Root (s) 217 3.2. Calculation of the Number of Stability Switches Graphically 218 4. Pseudo-oscillator Analysis for Hopf Bifurcation 220 4.1. Scalar Time-delayed Systems with Real Coefficients 220 4.2. Scalar Time-delayed Systems with Complex Coefficients 224 5. Conclusions 227 References 227 Chapter 8 Estimation and Control in Time-delayed Dynamical Systems Using the Chebyshev Spectral Continuous Time Approximation and Reduced Liapunov-Floquet Transformation Eric A. Butcher, Oleg Bobrenkov, Morad Nazari, Shahab Torkamani 230 1. Introduction 230 2. Chebyshev Spectral Continuous Time Approximation 233 2.1. Formulation 233 2.2. Examples 237 2.2.1. First Order Scalar Linear DDE 237 2.2.2. Delayed Mathieu Equation with Discontinuous Distributed Delays 239 3. Reduced Liapunov-Floquet Transformation 241 3.1. Formulation 241 3.2. Example: Delayed Mathieu Equation 243 4. Feedback Control of Periodic Delayed Systems 246 4.1. Formulation 247 4.2. Delayed State Feedback Control of the Delayed Mathieu Equation 251 5. Stochastic State, Parameter, and Delay Estimation 253 5.1. Formulation 253 5.2. Parametrically Forced Second Order Nonlinear DDE 258 6. Application to Observer-based Delayed Feedback Control of Spacecraft Attitude 261 6.1. Inverse Dynamics Approach for Feedback Control Law 262 6.2. Observer-based Controller Design 263 6.2.1. Delayed Feedback Control from Estimated States 263 6.2.2. Delayed Feedback Control from Estimated Delay and State 264 6.3. Simulation Results 265 7. Conclusions 267 References 270 Chapter 9 Noise-induced Dynamics of Time-delayed Stochastic Systems Yanfei Jin, Haiyan Hu 276 1. Introduction 276 2. Fundamentals for Time-delayed Stochastic Systems 277 2.1. The Method of Multiple Scales 278 2.2. Stochastic Averaging Method 279 2.3. Delayed Fokker-Planck Equations 280 2.4. Two-state Model 281 3. Dynamical Behaviors of the Stochastic Systems with Timedelayed Feedback Control 283 3.1. Principal Resonance of a Duffing Oscillator with Delayed Feedback Control under Narrow-band Random Excitation 284 3.1.1. Narrow-band Random External Excitation 285 3.1.2. Narrow-band Random Parametric Excitation 293 3.2. Moment Stability of Stochastic Systems with Delayed Feedback Control 301 3.2.1. External Gaussian White Noise 302 3.2.2. Parametric Gaussian White Noise 304 4. Noise-induced Resonances in Delayed Bistable Systems 305 4.1. Coherence Resonance 306 4.2. Stochastic Resonance 310 5. Concluding Remarks 311 References 312 Chapter 10 Some Studies on Delayed System Dynamics and Control Guo-Ping Cai, Long-Xiang Chen, Kun Liu 320 1. Introduction 320 2. Time Delay Identification 321 3. Two Time-delayed Controllers for Linear Structural Systems 322 3.1. The Discrete Time-delayed Controller2 322 3.2. The Continuous Time-delayed Controller3 323 4. Time-delayed Controller for Nonlinear Structural Systems 324 5. Parameter Robustness of Time-delayed Controller 326 6. Robust H∞ Time-delayed Controller Based on The LMI Technique 327 6.1. Maximum Time Delay with a Known Controller 328 6.2. Controller Design with Known Maximum Time Delay 329 6.3. The Largest Time Delay for System Stability with Unknown Controller 329 7. Delayed Positive Feedback Control Technique 329 8. Time Delay Experiments 330 8.1. Continuous and Discrete Time-delayed Controllers 331 8.2. Parameter Robustness of Time-delayed Controller 331 8.3. Robust H∞ Time-delayed Controller 332 8.4. Delayed Positive Feedback Controller 332 9. Concluding Remarks 332 References 333 Chapter 11 Switching Control of Uncertain Dynamical Systems with Time Delay Jian-Qiao Sun, Xiao-Yan Zhang, Zhi-Chang Qin, Shun Zhong 334 1. Introduction 334 2. Supervisory Control for Systems with Uncertain Time Delay 336 2.1. Optimal Feedback Gains via Mapping 337 2.2. High-order Control 337 2.3. Stability Requirements for Switching 339 2.4. Example of LTI System 339 2.4.1. Low-order Feedback Control with Optimal Gains 339 2.4.2. High-order LQR Optimal Control 342 2.5. Identification of Time Delay 344 3. Sliding Mode Control Design for Uncertain Systems 345 3.1. First Order System with Time Delay 346 3.2. First Order System with Delayed Control 348 3.3. Second Order Uncertain System 349 4. Concluding Remarks 350 References 351 "Analysis and control of time-delayed systems have been applied in a wide range of applications, ranging from mechanical, control, economic, to biological systems. Over the years, there has been a steady stream of interest in time-delayed dynamic systems, this book takes a snap shot of recent research from the world leading experts in analysis and control of dynamic systems with time delay to provide a bird's eye view of its development. The topics covered in this book include solution methods, stability analysis and control of periodic dynamic systems with time delay, bifurcations, stochastic dynamics and control, delayed-hamiltonian systems, uncertain dynamic systems with time delay, and experimental investigations of delayed structural control. This book can be a valuable reference to the researchers in the areas of mechanical, civil, structural, aerospace, naval, and electrical engineers, and can serve as a study guide to graduate students persuing research in the area of dynamics and control."--Page 4 of cover Analysis and control of time-delayed systems have been applied in a wide range of applications, ranging from mechanical, control, economic, to biological systems. Over the years, there has been a steady stream of interest in time-delayed dynamic systems, this book takes a snap shot of recent research from the world leading experts in analysis and control of dynamic systems with time delay to provide a bird's eye view of its development. The topics covered in this book include solution methods, stability analysis and control of periodic dynamic systems with time delay, bifurcations, stochastic dynamics and control, delayed Hamiltonian systems, uncertain dynamic systems with time delay, and experimental investigations of delayed structural control.