Physiological Control Systems
Analysis, Simulation, and Estimation
2nd Edition
Published on 3. July 2018
Book
Hardback
464 pages
978-1-119-05533-4 (ISBN)
Description
A guide to common control principles and how they are used to characterize a variety of physiological mechanisms
The second edition of Physiological Control Systems offers an updated and comprehensive resource that reviews the fundamental concepts of classical control theory and how engineering methodology can be applied to obtain a quantitative understanding of physiological systems. The revised text also contains more advanced topics that feature applications to physiology of nonlinear dynamics, parameter estimation methods, and adaptive estimation and control. The author-a noted expert in the field-includes a wealth of worked examples that illustrate key concepts and methodology and offers in-depth analyses of selected physiological control models that highlight the topics presented.
The author discusses the most noteworthy developments in system identification, optimal control, and nonlinear dynamical analysis and targets recent bioengineering advances. Designed to be a practical resource, the text includes guided experiments with simulation models (using Simulink/Matlab). Physiological Control Systems focuses on common control principles that can be used to characterize a broad variety of physiological mechanisms. This revised resource:
Offers new sections that explore identification of nonlinear and time-varying systems, and provide the background for understanding the link between continuous-time and discrete-time dynamic models
Presents helpful, hands-on experimentation with computer simulation models
Contains fully updated problems and exercises at the end of each chapter
Written for biomedical engineering students and biomedical scientists, Physiological Control Systems, offers an updated edition of this key resource for understanding classical control theory and its application to physiological systems. It also contains contemporary topics and methodologies that shape bioengineering research today.
The second edition of Physiological Control Systems offers an updated and comprehensive resource that reviews the fundamental concepts of classical control theory and how engineering methodology can be applied to obtain a quantitative understanding of physiological systems. The revised text also contains more advanced topics that feature applications to physiology of nonlinear dynamics, parameter estimation methods, and adaptive estimation and control. The author-a noted expert in the field-includes a wealth of worked examples that illustrate key concepts and methodology and offers in-depth analyses of selected physiological control models that highlight the topics presented.
The author discusses the most noteworthy developments in system identification, optimal control, and nonlinear dynamical analysis and targets recent bioengineering advances. Designed to be a practical resource, the text includes guided experiments with simulation models (using Simulink/Matlab). Physiological Control Systems focuses on common control principles that can be used to characterize a broad variety of physiological mechanisms. This revised resource:
Offers new sections that explore identification of nonlinear and time-varying systems, and provide the background for understanding the link between continuous-time and discrete-time dynamic models
Presents helpful, hands-on experimentation with computer simulation models
Contains fully updated problems and exercises at the end of each chapter
Written for biomedical engineering students and biomedical scientists, Physiological Control Systems, offers an updated edition of this key resource for understanding classical control theory and its application to physiological systems. It also contains contemporary topics and methodologies that shape bioengineering research today.
More details
Series
Edition
2nd edition
Language
English
Place of publication
United States
Publishing group
John Wiley & Sons Inc
Target group
Professional and scholarly
Dimensions
Height: 231 mm
Width: 158 mm
Thickness: 28 mm
Weight
862 gr
ISBN-13
978-1-119-05533-4 (9781119055334)
Copyright in bibliographic data and cover images is held by Nielsen Book Services Limited or by the publishers or by their respective licensors: all rights reserved.
Schweitzer Classification
Other editions
Additional editions
E-Book
04/2018
2nd Edition
Wiley
€123.99
Available for download
E-Book
04/2018
2nd Edition
Wiley
€123.99
Available for download
Previous edition
Book
09/1999
3rd Edition
IEEE Publications,U.S.
€203.50
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Person
MICHAEL C.K. KHOO is a Professor of Biomedical Engineering and Pediatrics in the Biomedical Engineering Department at the University of Southern California. He is a fellow of the IEEE, Biomedical Engineering Society, the American Institute of Medical and Biological Engineering, and the International Academy of Medical and Biological Engineering.
Content
Preface xiii
About the Companion Website xvii
1 Introduction 1
1.1 Preliminary Considerations, 1
1.2 Historical Background, 2
1.3 Systems Analysis: Fundamental Concepts, 4
1.4 Physiological Control Systems Analysis: A Simple Example, 6
1.5 Differences Between Engineering and Physiological Control Systems, 8
1.6 The Science (and Art) of Modeling, 11
1.7 "Systems Physiology" Versus "Systems Biology", 12
Problems, 13
Bibliography, 15
2 Mathematical Modeling 17
2.1 Generalized System Properties, 17
2.2 Models with Combinations of System Elements, 21
2.3 Linear Models of Physiological Systems: Two Examples, 24
2.4 Conversions Between Electrical and Mechanical Analogs, 27
2.5 Distributed-Parameter Versus Lumped-Parameter Models, 29
2.6 Linear Systems and the Superposition Principle, 31
2.7 Zero-Input and Zero-State Solutions of ODEs, 33
2.8 Laplace Transforms and Transfer Functions, 34
2.8.1 Solving ODEs with Laplace Transforms, 36
2.9 The Impulse Response and Linear Convolution, 38
2.10 State-Space Analysis, 40
2.11 Computer Analysis and Simulation: MATLAB and SIMULINK, 43
Problems, 49
Bibliography, 53
3 Static Analysis of Physiological Systems 55
3.1 Introduction, 55
3.2 Open-Loop Versus Closed-Loop Systems, 56
3.3 Determination of the Steady-State Operating Point, 59
3.4 Steady-State Analysis Using SIMULINK, 63
3.5 Regulation of Cardiac Output, 66
3.5.1 The Cardiac Output Curve, 67
3.5.2 The Venous Return Curve, 69
3.5.3 Closed-Loop Analysis: Heart and Systemic Circulation Combined, 73
3.6 Regulation of Glucose Insulin, 74
3.7 Chemical Regulation of Ventilation, 78
3.7.1 The Gas Exchanger, 80
3.7.2 The Respiratory Controller, 82
3.7.3 Closed-Loop Analysis: Lungs and Controller Combined, 82
Problems, 86
Bibliography, 91
4 Time-Domain Analysis of Linear Control Systems 93
4.1 Linearized Respiratory Mechanics: Open-Loop Versus Closed-Loop, 93
4.2 Open-Loop Versus Closed-Loop Transient Responses: First-Order Model, 96
4.2.1 Impulse Response, 96
4.2.2 Step Response, 97
4.3 Open-Loop Versus Closed-Loop Transient Responses: Second-Order Model, 98
4.3.1 Impulse Responses, 98
4.3.2 Step Responses, 103
4.4 Descriptors of Impulse and Step Responses, 107
4.4.1 Generalized Second-Order Dynamics, 107
4.4.2 Transient Response Descriptors, 111
4.5 Open-Loop Versus Closed-Loop Dynamics: Other Considerations, 114
4.5.1 Reduction of the Effects of External Disturbances, 114
4.5.2 Reduction of the Effects of Parameter Variations, 115
4.5.3 Integral Control, 116
4.5.4 Derivative Feedback, 118
4.5.5 Minimizing Effect of External Disturbances by Feedforward Gain, 119
4.6 Transient Response Analysis Using MATLAB, 121
4.7 SIMULINK Application 1: Dynamics of Neuromuscular Reflex Motion, 122
4.7.1 A Model of Neuromuscular Reflex Motion, 122
4.7.2 SIMULINK Implementation, 126
4.8 SIMULINK Application 2: Dynamics of Glucose-Insulin Regulation, 127
4.8.1 The Model, 127
4.8.2 Simulations with the Model, 131
Problems, 131
Bibliography, 135
5 Frequency-Domain Analysis of Linear Control Systems 137
5.1 Steady-State Responses to Sinusoidal Inputs, 137
5.1.1 Open-Loop Frequency Response, 137
5.1.2 Closed-Loop Frequency Response, 141
5.1.3 Relationship between Transient and Frequency Responses, 143
5.2 Graphical Representations of Frequency Response, 145
5.2.1 Bode Plot Representation, 145
5.2.2 Nichols Charts, 147
5.2.3 Nyquist Plots, 148
5.3 Frequency-Domain Analysis Using MATLAB and SIMULINK, 152
5.3.1 Using MATLAB, 152
5.3.2 Using SIMULINK, 154
5.4 Estimation of Frequency Response from Input-Output Data, 156
5.4.1 Underlying Principles, 156
5.4.2 Physiological Application: Forced Oscillation Technique in Respiratory Mechanics, 157
5.5 Frequency Response of a Model of Circulatory Control, 159
5.5.1 The Model, 159
5.5.2 Simulations with the Model, 160
5.5.3 Frequency Response of the Model, 162
Problems, 164
Bibliography, 165
6 Stability Analysis: Linear Approaches 167
6.1 Stability and Transient Response, 167
6.2 Root Locus Plots, 170
6.3 Routh-Hurwitz Stability Criterion, 174
6.4 Nyquist Criterion for Stability, 176
6.5 Relative Stability, 181
6.6 Stability Analysis of the Pupillary Light Reflex, 184
6.6.1 Routh-Hurwitz Analysis, 186
6.6.2 Nyquist Analysis, 187
6.7 Model of Cheyne-Stokes Breathing, 190
6.7.1 CO2 Exchange in the Lungs, 190
6.7.2 Transport Delays, 192
6.7.3 Controller Responses, 193
6.7.4 Loop Transfer Functions, 193
6.7.5 Nyquist Stability Analysis Using MATLAB, 194
Problems, 196
Bibliography, 198
7 Digital Simulation of Continuous-Time Systems 199
7.1 Preliminary Considerations: Sampling and the Z-Transform, 199
7.2 Methods for Continuous-Time to Discrete-Time Conversion, 202
7.2.1 Impulse Invariance, 202
7.2.2 Forward Difference, 203
7.2.3 Backward Difference, 204
7.2.4 Bilinear Transformation, 205
7.3 Sampling, 207
7.4 Digital Simulation: Stability and Performance Considerations, 211
7.5 Physiological Application: The Integral Pulse Frequency Modulation Model, 216
Problems, 221
Bibliography, 224
8 Model Identification and Parameter Estimation 225
8.1 Basic Problems in Physiological System Analysis, 225
8.2 Nonparametric and Parametric Identification Methods, 228
8.2.1 Numerical Deconvolution, 228
8.2.2 Least-Squares Estimation, 230
8.2.3 Estimation Using Correlation Functions, 233
8.2.4 Estimation in the Frequency Domain, 235
8.2.5 Optimization Techniques, 237
8.3 Problems in Parameter Estimation: Identifiability and Input Design, 243
8.3.1 Structural Identifiability, 243
8.3.2 Sensitivity Analysis, 244
8.3.3 Input Design, 248
8.4 Identification of Closed-Loop Systems: "Opening the Loop", 252
8.4.1 The Starling Heart-Lung Preparation, 253
8.4.2 Kao's Cross-Circulation Experiments, 253
8.4.3 Artificial Brain Perfusion for Partitioning Central and Peripheral Chemoreflexes, 255
8.4.4 The Voltage Clamp, 256
8.4.5 Opening the Pupillary Reflex Loop, 257
8.4.6 Read Rebreathing Technique, 259
8.5 Identification Under Closed-Loop Conditions: Case Studies, 260
8.5.1 Minimal Model of Blood Glucose Regulation, 262
8.5.2 Closed-Loop Identification of the Respiratory Control System, 267
8.5.3 Closed-Loop Identification of Autonomic Control Using Multivariate ARX Models, 273
8.6 Identification of Physiological Systems Using Basis Functions, 276
8.6.1 Reducing Variance in the Parameter Estimates, 276
8.6.2 Use of Basis Functions, 277
8.6.3 Baroreflex and Respiratory Modulation of Heart Rate Variability, 279
Problems, 283
Bibliography, 285
9 Estimation and Control of Time-Varying Systems 289
9.1 Modeling Time-Varying Systems: Key Concepts, 289
9.2 Estimation of Models with Time-Varying Parameters, 293
9.2.1 Optimal Estimation: The Wiener Filter, 293
9.2.2 Adaptive Estimation: The LMS Algorithm, 294
9.2.3 Adaptive Estimation: The RLS Algorithm, 296
9.3 Estimation of Time-Varying Physiological Models, 300
9.3.1 Extending Adaptive Estimation Algorithms to Other Model Structures, 300
9.3.2 Adaptive Estimation of Pulmonary Gas Exchange, 300
9.3.3 Quantifying Transient Changes in Autonomic Cardiovascular Control, 304
9.4 Adaptive Control of Physiological Systems, 307
9.4.1 General Considerations, 307
9.4.2 Adaptive Buffering of Fluctuations in Arterial PCO2, 308
Problems, 313
Bibliography, 314
10 Nonlinear Analysis of Physiological Control Systems 317
10.1 Nonlinear Versus Linear Closed-Loop Systems, 317
10.2 Phase-Plane Analysis, 320
10.2.1 Local Stability: Singular Points, 322
10.2.2 Method of Isoclines, 325
10.3 Nonlinear Oscillators, 329
10.3.1 Limit Cycles, 329
10.3.2 The van der Pol Oscillator, 329
10.3.3 Modeling Cardiac Dysrhythmias, 336
10.4 The Describing Function Method, 342
10.4.1 Methodology, 342
10.4.2 Application: Periodic Breathing with Apnea, 345
10.5 Models of Neuronal Dynamics, 348
10.5.1 The Hodgkin-Huxley Model, 349
10.5.2 The Bonhoeffer-van der Pol Model, 352
10.6 Nonparametric Identification of Nonlinear Systems, 359
10.6.1 Volterra-Wiener Kernel Approach, 360
10.6.2 Nonlinear Model of Baroreflex and Respiratory Modulated Heart Rate, 364
10.6.3 Interpretations of Kernels, 367
10.6.4 Higher Order Nonlinearities and Block-Structured Models, 369
Problems, 370
Bibliography, 374
11 Complex Dynamics in Physiological Control Systems 377
11.1 Spontaneous Variability, 377
11.2 Nonlinear Control Systems with Delayed Feedback, 380
11.2.1 The Logistic Equation, 380
11.2.2 Regulation of Neutrophil Density, 384
11.2.3 Model of Cardiovascular Variability, 387
11.3 Coupled Nonlinear Oscillators: Model of Circadian Rhythms, 397
11.4 Time-Varying Physiological Closed-Loop Systems: Sleep Apnea Model, 401
11.5 Propagation of System Noise in Feedback Loops, 409
Problems, 415
Bibliography, 416
Appendix A Commonly Used Laplace Transform Pairs 419
Appendix B List of MATLAB and SIMULINK Programs 421
Index 425
About the Companion Website xvii
1 Introduction 1
1.1 Preliminary Considerations, 1
1.2 Historical Background, 2
1.3 Systems Analysis: Fundamental Concepts, 4
1.4 Physiological Control Systems Analysis: A Simple Example, 6
1.5 Differences Between Engineering and Physiological Control Systems, 8
1.6 The Science (and Art) of Modeling, 11
1.7 "Systems Physiology" Versus "Systems Biology", 12
Problems, 13
Bibliography, 15
2 Mathematical Modeling 17
2.1 Generalized System Properties, 17
2.2 Models with Combinations of System Elements, 21
2.3 Linear Models of Physiological Systems: Two Examples, 24
2.4 Conversions Between Electrical and Mechanical Analogs, 27
2.5 Distributed-Parameter Versus Lumped-Parameter Models, 29
2.6 Linear Systems and the Superposition Principle, 31
2.7 Zero-Input and Zero-State Solutions of ODEs, 33
2.8 Laplace Transforms and Transfer Functions, 34
2.8.1 Solving ODEs with Laplace Transforms, 36
2.9 The Impulse Response and Linear Convolution, 38
2.10 State-Space Analysis, 40
2.11 Computer Analysis and Simulation: MATLAB and SIMULINK, 43
Problems, 49
Bibliography, 53
3 Static Analysis of Physiological Systems 55
3.1 Introduction, 55
3.2 Open-Loop Versus Closed-Loop Systems, 56
3.3 Determination of the Steady-State Operating Point, 59
3.4 Steady-State Analysis Using SIMULINK, 63
3.5 Regulation of Cardiac Output, 66
3.5.1 The Cardiac Output Curve, 67
3.5.2 The Venous Return Curve, 69
3.5.3 Closed-Loop Analysis: Heart and Systemic Circulation Combined, 73
3.6 Regulation of Glucose Insulin, 74
3.7 Chemical Regulation of Ventilation, 78
3.7.1 The Gas Exchanger, 80
3.7.2 The Respiratory Controller, 82
3.7.3 Closed-Loop Analysis: Lungs and Controller Combined, 82
Problems, 86
Bibliography, 91
4 Time-Domain Analysis of Linear Control Systems 93
4.1 Linearized Respiratory Mechanics: Open-Loop Versus Closed-Loop, 93
4.2 Open-Loop Versus Closed-Loop Transient Responses: First-Order Model, 96
4.2.1 Impulse Response, 96
4.2.2 Step Response, 97
4.3 Open-Loop Versus Closed-Loop Transient Responses: Second-Order Model, 98
4.3.1 Impulse Responses, 98
4.3.2 Step Responses, 103
4.4 Descriptors of Impulse and Step Responses, 107
4.4.1 Generalized Second-Order Dynamics, 107
4.4.2 Transient Response Descriptors, 111
4.5 Open-Loop Versus Closed-Loop Dynamics: Other Considerations, 114
4.5.1 Reduction of the Effects of External Disturbances, 114
4.5.2 Reduction of the Effects of Parameter Variations, 115
4.5.3 Integral Control, 116
4.5.4 Derivative Feedback, 118
4.5.5 Minimizing Effect of External Disturbances by Feedforward Gain, 119
4.6 Transient Response Analysis Using MATLAB, 121
4.7 SIMULINK Application 1: Dynamics of Neuromuscular Reflex Motion, 122
4.7.1 A Model of Neuromuscular Reflex Motion, 122
4.7.2 SIMULINK Implementation, 126
4.8 SIMULINK Application 2: Dynamics of Glucose-Insulin Regulation, 127
4.8.1 The Model, 127
4.8.2 Simulations with the Model, 131
Problems, 131
Bibliography, 135
5 Frequency-Domain Analysis of Linear Control Systems 137
5.1 Steady-State Responses to Sinusoidal Inputs, 137
5.1.1 Open-Loop Frequency Response, 137
5.1.2 Closed-Loop Frequency Response, 141
5.1.3 Relationship between Transient and Frequency Responses, 143
5.2 Graphical Representations of Frequency Response, 145
5.2.1 Bode Plot Representation, 145
5.2.2 Nichols Charts, 147
5.2.3 Nyquist Plots, 148
5.3 Frequency-Domain Analysis Using MATLAB and SIMULINK, 152
5.3.1 Using MATLAB, 152
5.3.2 Using SIMULINK, 154
5.4 Estimation of Frequency Response from Input-Output Data, 156
5.4.1 Underlying Principles, 156
5.4.2 Physiological Application: Forced Oscillation Technique in Respiratory Mechanics, 157
5.5 Frequency Response of a Model of Circulatory Control, 159
5.5.1 The Model, 159
5.5.2 Simulations with the Model, 160
5.5.3 Frequency Response of the Model, 162
Problems, 164
Bibliography, 165
6 Stability Analysis: Linear Approaches 167
6.1 Stability and Transient Response, 167
6.2 Root Locus Plots, 170
6.3 Routh-Hurwitz Stability Criterion, 174
6.4 Nyquist Criterion for Stability, 176
6.5 Relative Stability, 181
6.6 Stability Analysis of the Pupillary Light Reflex, 184
6.6.1 Routh-Hurwitz Analysis, 186
6.6.2 Nyquist Analysis, 187
6.7 Model of Cheyne-Stokes Breathing, 190
6.7.1 CO2 Exchange in the Lungs, 190
6.7.2 Transport Delays, 192
6.7.3 Controller Responses, 193
6.7.4 Loop Transfer Functions, 193
6.7.5 Nyquist Stability Analysis Using MATLAB, 194
Problems, 196
Bibliography, 198
7 Digital Simulation of Continuous-Time Systems 199
7.1 Preliminary Considerations: Sampling and the Z-Transform, 199
7.2 Methods for Continuous-Time to Discrete-Time Conversion, 202
7.2.1 Impulse Invariance, 202
7.2.2 Forward Difference, 203
7.2.3 Backward Difference, 204
7.2.4 Bilinear Transformation, 205
7.3 Sampling, 207
7.4 Digital Simulation: Stability and Performance Considerations, 211
7.5 Physiological Application: The Integral Pulse Frequency Modulation Model, 216
Problems, 221
Bibliography, 224
8 Model Identification and Parameter Estimation 225
8.1 Basic Problems in Physiological System Analysis, 225
8.2 Nonparametric and Parametric Identification Methods, 228
8.2.1 Numerical Deconvolution, 228
8.2.2 Least-Squares Estimation, 230
8.2.3 Estimation Using Correlation Functions, 233
8.2.4 Estimation in the Frequency Domain, 235
8.2.5 Optimization Techniques, 237
8.3 Problems in Parameter Estimation: Identifiability and Input Design, 243
8.3.1 Structural Identifiability, 243
8.3.2 Sensitivity Analysis, 244
8.3.3 Input Design, 248
8.4 Identification of Closed-Loop Systems: "Opening the Loop", 252
8.4.1 The Starling Heart-Lung Preparation, 253
8.4.2 Kao's Cross-Circulation Experiments, 253
8.4.3 Artificial Brain Perfusion for Partitioning Central and Peripheral Chemoreflexes, 255
8.4.4 The Voltage Clamp, 256
8.4.5 Opening the Pupillary Reflex Loop, 257
8.4.6 Read Rebreathing Technique, 259
8.5 Identification Under Closed-Loop Conditions: Case Studies, 260
8.5.1 Minimal Model of Blood Glucose Regulation, 262
8.5.2 Closed-Loop Identification of the Respiratory Control System, 267
8.5.3 Closed-Loop Identification of Autonomic Control Using Multivariate ARX Models, 273
8.6 Identification of Physiological Systems Using Basis Functions, 276
8.6.1 Reducing Variance in the Parameter Estimates, 276
8.6.2 Use of Basis Functions, 277
8.6.3 Baroreflex and Respiratory Modulation of Heart Rate Variability, 279
Problems, 283
Bibliography, 285
9 Estimation and Control of Time-Varying Systems 289
9.1 Modeling Time-Varying Systems: Key Concepts, 289
9.2 Estimation of Models with Time-Varying Parameters, 293
9.2.1 Optimal Estimation: The Wiener Filter, 293
9.2.2 Adaptive Estimation: The LMS Algorithm, 294
9.2.3 Adaptive Estimation: The RLS Algorithm, 296
9.3 Estimation of Time-Varying Physiological Models, 300
9.3.1 Extending Adaptive Estimation Algorithms to Other Model Structures, 300
9.3.2 Adaptive Estimation of Pulmonary Gas Exchange, 300
9.3.3 Quantifying Transient Changes in Autonomic Cardiovascular Control, 304
9.4 Adaptive Control of Physiological Systems, 307
9.4.1 General Considerations, 307
9.4.2 Adaptive Buffering of Fluctuations in Arterial PCO2, 308
Problems, 313
Bibliography, 314
10 Nonlinear Analysis of Physiological Control Systems 317
10.1 Nonlinear Versus Linear Closed-Loop Systems, 317
10.2 Phase-Plane Analysis, 320
10.2.1 Local Stability: Singular Points, 322
10.2.2 Method of Isoclines, 325
10.3 Nonlinear Oscillators, 329
10.3.1 Limit Cycles, 329
10.3.2 The van der Pol Oscillator, 329
10.3.3 Modeling Cardiac Dysrhythmias, 336
10.4 The Describing Function Method, 342
10.4.1 Methodology, 342
10.4.2 Application: Periodic Breathing with Apnea, 345
10.5 Models of Neuronal Dynamics, 348
10.5.1 The Hodgkin-Huxley Model, 349
10.5.2 The Bonhoeffer-van der Pol Model, 352
10.6 Nonparametric Identification of Nonlinear Systems, 359
10.6.1 Volterra-Wiener Kernel Approach, 360
10.6.2 Nonlinear Model of Baroreflex and Respiratory Modulated Heart Rate, 364
10.6.3 Interpretations of Kernels, 367
10.6.4 Higher Order Nonlinearities and Block-Structured Models, 369
Problems, 370
Bibliography, 374
11 Complex Dynamics in Physiological Control Systems 377
11.1 Spontaneous Variability, 377
11.2 Nonlinear Control Systems with Delayed Feedback, 380
11.2.1 The Logistic Equation, 380
11.2.2 Regulation of Neutrophil Density, 384
11.2.3 Model of Cardiovascular Variability, 387
11.3 Coupled Nonlinear Oscillators: Model of Circadian Rhythms, 397
11.4 Time-Varying Physiological Closed-Loop Systems: Sleep Apnea Model, 401
11.5 Propagation of System Noise in Feedback Loops, 409
Problems, 415
Bibliography, 416
Appendix A Commonly Used Laplace Transform Pairs 419
Appendix B List of MATLAB and SIMULINK Programs 421
Index 425