Series Editors’ Foreword	7
	Preface	9
	Website	10
	Expected Audience	11
	About the Content	11
	Pathways Through the Book	12
	Acknowledgements	14
	References[1] Constantinescu, A.: Commande robuste et adaptative d’une suspension active. Thèse de doctorat, Institut National Polytechnique de Grenoble (2001)[2] Alma, M.: Rejet adaptatif de perturbations en contrôle actif de vibrations. Ph.D. thesis, Université de Grenoble (2011)[3] Airimitoaie, T.B.: Robust design and tuning of active vibration control systems. Ph.D. thesis, University of Grenoble, France, and University “Politehnica” of Bucharest, Romania (2012)[4] Castellanos-Silva, A.: Compensation adaptative par feedback pour le contrôle actif de vibrations en présence d’incertitudes sur les paramétres du procédé. Ph.D. thesis, Université de Grenoble (2014)[5] Landau, I.D., Silva, A.C., Airimitoaie, T.B., Buche, G., Noé, M.: Benchmark on adaptive regulation—rejection of unknown/time-varying multiple narrow band disturbances. European Journal of Control 19(4), 237—252 (2013). http://dx.doi.org/10.1016/j.ejcon.2013.05.007#1	14
	Contents	15
	Acronyms	23
	Part I Introduction to Adaptive and Robust Active Vibration Control	25
	1 Introduction to Adaptive and Robust Active Vibration Control	26
	1.1 Active Vibration Control: Why and How	26
	1.2 A Conceptual Feedback Framework	32
	1.3 Active Damping	34
	1.4 The Robust Regulation Paradigm	34
	1.5 The Adaptive Regulation Paradigm	35
	1.6 Concluding Remarks	37
	1.7 Notes and Reference	38
	References	38
	2 The Test Benches	41
	2.1 An Active Hydraulic Suspension System Using Feedback Compensation	41
	2.2 An Active Vibration Control System Using Feedback Compensation Through an Inertial Actuator	44
	2.3 An Active Distributed Flexible Mechanical Structure ƒ	46
	2.4 Concluding Remarks	49
	2.5 Notes and References	50
	References	50
	Part II Techniques for Active Vibration Control	51
	3 Active Vibration Control Systems---Model Representation	52
	3.1 System Description	52
	3.1.1 Continuous-Time Versus Discrete-Time Dynamical Models	52
	3.1.2 Digital Control Systems	53
	3.1.3 Discrete-Time System Models for Control	55
	3.2 Concluding Remarks	58
	3.3 Notes and References	58
	References	58
	4 Parameter Adaptation Algorithms	59
	4.1 Introduction	59
	4.2 Structure of the Adjustable Model	60
	4.2.1 Case (a): Recursive Configuration for System Identification---Equation Error	60
	4.2.2 Case (b): Adaptive Feedforward Compensation---Output Error	62
	4.3 Basic Parameter Adaptation Algorithms	64
	4.3.1 Basic Gradient Algorithm	64
	4.3.2 Improved Gradient Algorithm	67
	4.3.3 Recursive Least Squares Algorithm	72
	4.3.4 Choice of the Adaptation Gain	77
	4.3.5 An Example	81
	4.4 Stability of Parameter Adaptation Algorithms	82
	4.4.1 Equivalent Feedback Representation of the Adaptive Predictors	83
	4.4.2 A General Structure and Stability of PAA	86
	4.4.3 Output Error Algorithms---Stability Analysis	90
	4.5 Parametric Convergence	92
	4.5.1 The Problem	92
	4.6 The LMS Family of Parameter Adaptation Algorithms	96
	4.7 Concluding Remarks	97
	4.8 Notes and References	98
	References	98
	5 Identification of the Active Vibration Control Systems---The Bases	100
	5.1 Introduction	100
	5.2 Input--Output Data Acquisition and Preprocessing	102
	5.2.1 Input--Output Data Acquisition Under an Experimental Protocol	102
	5.2.2 Pseudorandom Binary Sequences (PRBS)	102
	5.2.3 Data Preprocessing	104
	5.3 Model Order Estimation from Data	105
	5.4 Parameter Estimation Algorithms	107
	5.4.1 Recursive Extended Least Squares (RELS)	109
	5.4.2 Output Error with Extended Prediction Model (XOLOE)	111
	5.5 Validation of the Identified Models	113
	5.5.1 Whiteness Test	113
	5.6 Concluding Remarks	115
	5.7 Notes and References	116
	References	116
	6 Identification of the Test Benches in Open-Loop Operation	117
	6.1 Identification of the Active Hydraulic Suspension in Open-Loop Operation	117
	6.1.1 Identification of the Secondary Path	118
	6.1.2 Identification of the Primary Path	123
	6.2 Identification of the AVC System Using Feedback Compensation Through an Inertial Actuator	124
	6.2.1 Identification of the Secondary Path	124
	6.2.2 Identification of the Primary Path	130
	6.3 Identification of the Active Distributed Flexible Mechanical Structure Using Feedforward--Feedback Compensation	131
	6.4 Concluding Remarks	137
	6.5 Notes and References	137
	References	137
	7 Digital Control Strategies for Active Vibration Control---The Bases	139
	7.1 The Digital Controller	139
	7.2 Pole Placement	141
	7.2.1 Choice of HR and HS---Examples	142
	7.2.2 Internal Model Principle (IMP)	144
	7.2.3 Youla--Ku?era Parametrization	145
	7.2.4 Robustness Margins	147
	7.2.5 Model Uncertainties and Robust Stability	150
	7.2.6 Templates for the Sensitivity Functions	152
	7.2.7 Properties of the Sensitivity Functions	152
	7.2.8 Input Sensitivity Function	155
	7.2.9 Shaping the Sensitivity Functions for Active Vibration Control	157
	7.3 Real-Time Example: Narrow-Band Disturbance Attenuation on the Active Vibration Control System Using an Inertial Actuator	161
	7.4 Pole Placement with Sensitivity Function Shaping by Convex Optimisation	164
	7.5 Concluding Remarks	167
	7.6 Notes and References	167
	References	168
	8 Identification in Closed-Loop Operation	170
	8.1 Introduction	170
	8.2 Closed-Loop Output Error Identification Methods	171
	8.2.1 The Closed-Loop Output Error Algorithm	175
	8.2.2 Filtered and Adaptive Filtered Closed-Loop Output Error Algorithms (F-CLOE, AF-CLOE)	176
	8.2.3 Extended Closed-Loop Output Error Algorithm (X-CLOE)	177
	8.2.4 Taking into Account Known Fixed Parts in the Model	178
	8.2.5 Properties of the Estimated Model	179
	8.2.6 Validation of Models Identified in Closed-Loop Operation	180
	8.3 A Real-Time Example: Identification in Closed-Loop and Controller Redesign for the Active Control System Using an Inertial Actuator	182
	8.4 Concluding Remarks	186
	8.5 Notes and References	186
	References	187
	9 Reduction of the Controller Complexity	188
	9.1 Introduction	188
	9.2 Criteria for Direct Controller Reduction	190
	9.3 Estimation of Reduced Order Controllers by Identification in Closed-Loop	192
	9.3.1 Closed-Loop Input Matching (CLIM)	192
	9.3.2 Closed-Loop Output Matching (CLOM)	195
	9.3.3 Taking into Account the Fixed Parts of the Nominal Controller	195
	9.4 Real-Time Example: Reduction of Controller Complexity	197
	9.5 Concluding Remarks	200
	9.6 Notes and References	201
	References	201
	Part III Active Damping	202
	10 Active Damping	203
	10.1 Introduction	203
	10.2 Performance Specifications	204
	10.3 Controller Design by Shaping the Sensitivity Functions Using ƒ	208
	10.4 Identification in Closed-Loop of the Active Suspension ƒ	211
	10.5 Redesign of the Controller Based on the Model Identified in Closed Loop	212
	10.6 Controller Complexity Reduction	214
	10.6.1 CLOM Algorithm with Simulated Data	216
	10.6.2 Real-Time Performance Tests for Nominal and Reduced Order Controllers	218
	10.7 Design of the Controller by Shaping the Sensitivity Function with Band-Stop Filters	219
	10.8 Concluding Remarks	224
	10.9 Notes and References	225
	References	226
	Part IV Feedback Attenuation of Narrow-Band Disturbances	227
	11 Robust Controller Design for Feedback Attenuation of Narrow-Band Disturbances	228
	11.1 Introduction	228
	11.2 System Description	229
	11.3 Robust Control Design	231
	11.4 Experimental Results	234
	11.4.1 Two Time-Varying Tonal Disturbances	235
	11.4.2 Attenuation of Vibrational Interference	237
	11.5 Concluding Remarks	238
	11.6 Notes and References	238
	References	239
	12 Direct Adaptive Feedback Attenuation of Narrow-Band Disturbances	240
	12.1 Introduction	240
	12.2 Direct Adaptive Feedback Attenuation of Unknown and Time-Varying ƒ	241
	12.2.1 Introduction	241
	12.2.2 Direct Adaptive Regulation Using Youla--Ku?era Parametrization	245
	12.2.3 Robustness Considerations	247
	12.3 Performance Evaluation Indicators for Narrow-Band Disturbance Attenuation	248
	12.4 Experimental Results: Adaptive Versus Robust	251
	12.4.1 Central Controller for Youla--Ku?era Parametrization	251
	12.4.2 Two Single-Mode Vibration Control	251
	12.4.3 Vibrational Interference	254
	12.5 Adaptive Attenuation of an Unknown Narrow-Band Disturbance on the Active Hydraulic Suspension	256
	12.6 Adaptive Attenuation of an Unknown Narrow-Band Disturbance on the Active Vibration Control System Using an Inertial Actuator	259
	12.6.1 Design of the Central Controller	260
	12.6.2 Real-Time Results	262
	12.7 Other Experimental Results	264
	12.8 Concluding Remarks	264
	12.9 Notes and References	265
	References	266
	13 Adaptive Attenuation of Multiple Sparse Unknown and Time-Varying Narrow-Band Disturbances	269
	13.1 Introduction	269
	13.2 The Linear Control Challenge	269
	13.2.1 Attenuation of Multiple Narrow-Band Disturbances Using Band-Stop Filters	271
	13.2.2 IMP with Tuned Notch Filters	275
	13.2.3 IMP Design Using Auxiliary Low Damped Complex Poles	276
	13.3 Interlaced Adaptive Regulation Using Youla--Ku?era IIR Parametrization	277
	13.3.1 Estimation of AQ	279
	13.3.2 Estimation of BQ(q-1)	281
	13.4 Indirect Adaptive Regulation Using Band-Stop Filters	285
	13.4.1 Basic Scheme for Indirect Adaptive Regulation	286
	13.4.2 Reducing the Computational Load of the Design Using the Youla--Ku?era Parametrization	287
	13.4.3 Frequency Estimation Using Adaptive Notch Filters	288
	13.4.4 Stability Analysis of the Indirect Adaptive Scheme	291
	13.5 Experimental Results: Attenuation of Three Tonal Disturbances with Variable Frequencies	291
	13.6 Experimental Results: Comparative Evaluation of Adaptive Regulation Schemes for Attenuation of Multiple Narrow-Band Disturbances	292
	13.6.1 Introduction	292
	13.6.2 Global Evaluation Criteria	297
	13.7 Concluding Remarks	304
	13.8 Notes and References	304
	References	305
	Part V Feedforward-Feedback Attenuation of Broad-Band Disturbances	307
	14 Design of Linear Feedforward Compensation of Broad-band Disturbances from Data	308
	14.1 Introduction	308
	14.2 Indirect Approach for the Design of the Feedforward Compensator from Data	311
	14.3 Direct Approach for the Design of the Feedforward Compensator from Data	311
	14.4 Direct Estimation of the Feedforward Compensator and Real-Time Tests	315
	14.5 Concluding Remark	321
	14.6 Notes and References	321
	References	322
	15 Adaptive Feedforward Compensation of Disturbances	324
	15.1 Introduction	324
	15.2 Basic Equations and Notations	327
	15.3 Development of the Algorithms	329
	15.4 Analysis of the Algorithms	332
	15.4.1 The Perfect Matching Case	332
	15.4.2 The Case of Non-perfect Matching	334
	15.4.3 Relaxing the Positive Real Condition	336
	15.5 Adaptive Attenuation of Broad-band Disturbances---Experimental Results	337
	15.5.1 Broad-band Disturbance Rejection Using Matrix Adaptation Gain	338
	15.5.2 Broad-band Disturbance Rejection Using Scalar Adaptation Gain	342
	15.6 Adaptive Feedforward Compensation with Filtering of the Residual Error	349
	15.7 Adaptive Feedforward + Fixed Feedback Compensation of Broad-band Disturbances	351
	15.7.1 Development of the Algorithms	353
	15.7.2 Analysis of the Algorithms	355
	15.8 Adaptive Feedforward + Fixed Feedback Attenuation of Broad-band Disturbances---Experimental Results	356
	15.9 Concluding Remarks	358
	15.10 Notes and References	358
	References	359
	16 Youla--Ku?era Parametrized Adaptive Feedforward Compensators	363
	16.1 Introduction	363
	16.2 Basic Equations and Notations	364
	16.3 Development of the Algorithms	366
	16.4 Analysis of the Algorithms	369
	16.4.1 The Perfect Matching Case	369
	16.4.2 The Case of Non-perfect Matching	370
	16.4.3 Relaxing the Positive Real Condition	371
	16.4.4 Summary of the Algorithms	371
	16.5 Experimental Results	373
	16.5.1 The Central Controllers and Comparison Objectives	373
	16.5.2 Broad-band Disturbance Rejection Using Matrix Adaptation Gain	373
	16.5.3 Broad-band Disturbance Rejection Using Scalar Adaptation Gain	376
	16.6 Comparison of the Algorithms	378
	16.7 Concluding Remarks	380
	16.8 Notes and References	380
	References	380
	Appendix A Generalized Stability Margin and Normalized Distance Between Two Transfer Functions	382
	Appendix B Implementation of the Adaptation Gain Updating---The U-D Factorization	386
	Appendix C Interlaced Adaptive Regulation: Equations Development and Stability Analysis	388
	Appendix D Error Equations for Adaptive Feedforward Compensation	392
	Appendix E ``Integral + Proportional'' Parameter Adaptation Algorithm	399
	Index	404