Sampled-Data Models for Linear and Nonlinear Systems (Communications and Control Engineering)

"Sampled-data Models for Linear and Nonlinear Systems provides a fresh new look at a subject with which many researchers may think themselves familiar. Rather than emphasising the differences between sampled-data and continuous-time systems, the authors proceed from the premise that, with modern sampling rates being as high as they are, it is becoming more appropriate to emphasise connections and similarities. The text is driven by three motives: · the ubiquity of computers in modern control and signal-processing equipment means that sampling of systems that really evolve continuously is unavoidable; · although superficially straightforward, sampling can easily produce erroneous results when not treated properly; and · the need for a thorough understanding of many aspects of sampling among researchers and engineers dealing with applications to which they are central. The authors tackle many misconceptions which, although appearing reasonable at first sight, are in fact either partially or completely erroneous. They also deal with linear and nonlinear, deterministic and stochastic cases. The impact of the ideas presented on several standard problems in signals and systems is illustrated using a number of applications. Academic researchers and graduate students in systems, control and signal processing will find the ideas presented in Sampled-data Models for Linear and Nonlinear Systems to be a useful manual for dealing with sampled-data systems, clearing away mistaken ideas and bringing the subject thoroughly up to date. Researchers in statistics and economics will also derive benefit from the reworking of ideas relating a model derived from data sampling to an original continuous system. The Communications and Control Engineering series reports major technological advances which have potential for great impact in the fields of communication and control. It reflects research in industrial and academic institutions around the world so that the readership can exploit new possibilities as they become available." -- Font no determinada Sampled-Data Models for Linear and Nonlinear Systems 3 Preface 6 Contents 8 Symbols and Acronyms 13 Chapter 1: Introduction 16 Part I: Deterministic Systems 20 Chapter 2: Background on Sampling of Signals 21 2.1 Fourier Analysis 21 2.2 Sampling of Continuous-Time Signals and Continuous Transforms 23 2.3 Signal Reconstruction from Samples 29 2.4 Anti-aliasing Filters 31 2.5 Summary 32 Further Reading 33 Chapter 3: Sampled-Data Models for Linear Deterministic Systems 34 3.1 Continuous-Time Model 34 3.2 The Hold Device 37 3.3 The Sampled-Data Model 37 3.4 Poles and Zeros 44 3.5 Relative Degree 45 3.6 Summary 50 Further Reading 50 Chapter 4: Incremental Sampled-Data Models 52 4.1 Sampled-Data Models for Fast Sampling Rates 52 4.2 The Delta Operator 53 4.3 Relative Degree Revisited 56 4.4 Summary 57 Further Reading 58 Chapter 5: Asymptotic Sampling Zeros 59 5.1 Pure Integrator Model 59 5.2 General Linear Systems 67 5.3 Summary 69 Further Reading 70 Chapter 6: Generalised Hold Devices 71 6.1 Generalised Hold Functions 71 6.2 Asymptotic Sampling Zeros for Generalised Holds 75 6.3 Summary 81 Further Reading 82 Chapter 7: Robustness 84 7.1 Robustness of Asymptotic Sampled-Data Models 84 7.2 Robustness of Generalised Hold Designs 86 7.3 Summary 88 Further Reading 88 Chapter 8: Approximate Models for Linear Deterministic Systems 89 8.1 Approximate Models in the Frequency Domain 89 8.1.1 Error Quantification in the Frequency Domain 90 8.1.2 Approximate Models Based on Euler Integration 92 8.1.3 Approximate Models Using Asymptotic Sampling Zeros 95 8.2 Approximate Models in the Time Domain 100 8.2.1 Approximate Models Based on Up-Sampling 100 8.2.2 Approximate Models Based on Truncated Taylor Series 102 8.2.3 Approximate Models Based on Near Euler Integration 102 8.2.4 Normal Forms for Linear Systems 102 8.2.5 Variable Truncated Taylor Series Model 104 8.3 Summary 108 Further Reading 108 Chapter 9: Approximate Models for Nonlinear Deterministic Systems 110 9.1 Approximate Models Based on Up-Sampling 110 9.2 Normal Forms for Nonlinear Systems 111 9.3 Variable Truncated Taylor Series Model 113 9.4 Approximation Errors for Nonlinear Sampled Models 114 9.4.1 Links to Numerical Analysis 115 9.4.2 Local and Global Truncation Errors 115 9.4.3 Truncation Errors for the TTS Model 117 9.5 Summary 123 Further Reading 123 Chapter 10: Applications of Approximate Sampled-Data Models in Estimation and Control 125 10.1 When Are Deterministic Sampling Zeros Important? 125 10.2 State Feedback Control for Linear Systems 126 10.3 State Estimation for Linear Systems 127 10.4 Output Feedback Control for Linear Systems 128 10.5 Sampled-Data State Feedback Control for Nonlinear Systems 129 10.6 Continuous-Time System Identification 133 10.7 Predictive Control of an Electrical Machine 138 10.8 Summary 142 Further Reading 143 Part II: Stochastic Systems 144 Chapter 11: Background on Sampling of Stochastic Signals 145 11.1 Continuous-Time Stochastic Processes 145 11.2 Power Spectral Density of a Sampled Process 148 11.3 Anti-aliasing Filtering 150 11.4 Summary 152 Further Reading 152 Chapter 12: Sampled-Data Models for Linear Stochastic Systems 154 12.1 Continuous-Time Stochastic Linear Systems 154 12.2 Sampled-Data Model for Systems Having Relative Degree Greater than Zero 156 12.3 Sampled-Data Models for Systems Having Averaging Filter 157 12.4 Summary 160 Further Reading 161 Chapter 13: Incremental Stochastic Sampled-Data Models 162 13.1 Incremental Model 162 13.2 Incremental Model for Instantaneously Sampled Systems 163 13.3 Incremental Model for Sampled Systems with Averaging AAF 164 13.4 Output Power Spectral Density 166 13.5 Simple Connections Between Continuous-Time and Discrete-Time PSDs 168 13.6 Summary 171 Further Reading 172 Chapter 14: Asymptotic Sampling Zeros for Linear Stochastic Systems 173 14.1 Discrete-Time PSDs Corresponding to Simple Continuous-Time Systems 173 14.1.1 First Order System Having Relative Degree 1 173 14.1.2 Second Order System Having Relative Degree 2 174 14.2 Asymptotic Sampling Zeros of the Output Power Spectral Density 176 14.3 Relating Deterministic and Stochastic Sampling Zeros 178 14.4 Asymptotic Sampling Zeros via Time-Domain Arguments 179 14.5 Summary 184 Further Reading 184 Chapter 15: Generalised Sampling Filters 185 15.1 Sampled-Data Models when Generalised Sampling Filters Are Deployed 185 15.2 Generalised Filters to Assign the Asymptotic Sampling Zeros 190 15.2.1 First Order Systems 191 15.2.2 Second Order Systems 193 15.3 Robustness Issues 195 15.4 Summary 196 Further Reading 197 Chapter 16: Approximate Sampled-Data Models for Linear Stochastic Systems 198 16.1 Adding the Sampling Zeros in the Frequency Domain 198 16.2 Approximate Sampled-Data Model Based on Up-Sampling 201 16.3 Approximate Stochastic Sampled-Data Models Based on Successive Integration 203 16.4 Stochastic Sampling Zeros Revisited 204 16.5 Summary 210 Further Reading 210 Chapter 17: Stochastic Nonlinear Systems 211 17.1 Background on Stochastic Differential Equations 211 17.2 The Ito Rule 213 17.3 Ito-Taylor Expansions 215 17.4 Numerical Solution of SDEs 219 17.5 Accuracy of Numerical Solutions 219 17.6 Summary 221 Further Reading 221 Chapter 18: Approximate Sampled-Data Models for Nonlinear Stochastic Systems 223 18.1 Approximate Sampled-Data Models Based on Up-Sampling 223 18.2 Approximate Sampled-Data Models Based on Successive Integration 224 18.3 Sampling Zero Dynamics for Stochastic Nonlinear Systems 228 18.4 Summary 233 Further Reading 233 Chapter 19: Applications of Approximate Stochastic Sampled-Data Models 234 19.1 When Are Stochastic Sampling Zeros Important? 234 19.2 Models for System Identification 234 19.3 Effect of Sampling Zeros in Stochastic System Identification 235 19.4 Restricted Bandwidth Estimation 237 19.5 Identification of Continuous-Time State-Space Models from Nonuniformly Fast-Sampled Data 240 19.5.1 Continuous-Time System Description 241 19.5.2 Sampled-Data Model 242 19.5.3 Maximum Likelihood Identification and the EM Algorithm 244 19.5.4 EM Algorithm 244 19.6 Summary 250 Further Reading 250 Part III: Embellishments and Extensions 252 Chapter 20: The Euler-Frobenius Polynomials 253 20.1 Euler-Frobenius Polynomials 253 20.2 Euler-Frobenius Numbers 255 20.3 Euler-Frobenius Fractions 256 20.4 Combinatorial Interpretation of Eulerian Numbers 257 20.5 Euler and Bernoulli Polynomials 258 20.6 Generalised Eulerian Polynomials 260 20.7 Summary 262 Further Reading 262 Chapter 21: Models for Intersample Response 264 21.1 Frequency Domain 264 21.2 Time Domain 267 21.3 Summary 268 Further Reading 268 Chapter 22: Approximate Sampled-Data Models for Fractional Order Systems 270 22.1 Historical Perspective 270 22.2 Fractional Calculus Background 271 22.3 Sampling Zeros for Fractional Order Systems 272 22.4 Approximate Discrete-Time Models 281 22.5 Summary 283 Further Reading 284 Index 286 Front Matter....Pages I-XVII Introduction....Pages 1-4 Front Matter....Pages 5-5 Background on Sampling of Signals....Pages 7-19 Sampled-Data Models for Linear Deterministic Systems....Pages 21-38 Incremental Sampled-Data Models....Pages 39-45 Asymptotic Sampling Zeros....Pages 47-58 Generalised Hold Devices....Pages 59-71 Robustness....Pages 73-77 Approximate Models for Linear Deterministic Systems....Pages 79-99 Approximate Models for Nonlinear Deterministic Systems....Pages 101-115 Applications of Approximate Sampled-Data Models in Estimation and Control....Pages 117-135 Front Matter....Pages 137-137 Background on Sampling of Stochastic Signals....Pages 139-147 Sampled-Data Models for Linear Stochastic Systems....Pages 149-156 Incremental Stochastic Sampled-Data Models....Pages 157-167 Asymptotic Sampling Zeros for Linear Stochastic Systems....Pages 169-180 Generalised Sampling Filters....Pages 181-193 Approximate Sampled-Data Models for Linear Stochastic Systems....Pages 195-207 Stochastic Nonlinear Systems....Pages 209-220 Approximate Sampled-Data Models for Nonlinear Stochastic Systems....Pages 221-231 Applications of Approximate Stochastic Sampled-Data Models....Pages 233-250 Front Matter....Pages 251-251 The Euler–Frobenius Polynomials....Pages 253-263 Front Matter....Pages 251-251 Models for Intersample Response....Pages 265-270 Approximate Sampled-Data Models for Fractional Order Systems....Pages 271-286 Back Matter....Pages 287-289 In this book, rather than emphasize differences between sampled-data and continuous-time systems, the authors proceed from the premise that, with modern sampling rates as high as they are, it is more appropriate to emphasise connections and similarities.