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Fundamentals of Signals and Control Systems

ISBN: 978-1-78630-098-0
290 pages
February 2017, Wiley-ISTE
Fundamentals of Signals and Control Systems (1786300982) cover image

Description

The aim of this book is the study of signals and deterministic systems, linear, time-invariant, finite dimensions and causal. A set of useful tools is selected for the automatic and signal processing and methods of representation of dynamic linear systems are exposed, and analysis of their behavior. Finally we discuss the estimation, identification and synthesis of control laws for the purpose of stabilization and regulation.

The study of signal characteristics and properties systems and knowledge of mathematical tools and treatment methods and analysis, are lately more and more importance and continue to evolve. The reason is that the current state of technology, particularly electronics and computing, enables the production of very advanced processing systems, effective and less expensive despite the complexity.

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Table of Contents

Preface ix

Chapter 1. Introduction, Generalities, Definitions of Systems 1

1.1. Introduction 1

1.2. Signals and communication systems 2

1.3. Signals and systems representation 5

1.3.1. Signal  5

1.3.2. Functional space L2  6

1.3.3. Dirac distribution 8

1.4. Convolution and composition products – notions of filtering 10

1.4.1. Convolution or composition product 10

1.4.2. System 11

1.5. Transmission systems and filters 12

1.5.1. Convolution and filtering 13

1.6. Deterministic signals – random signals – analog signals 15

1.6.1. Definitions 15

1.6.2. Some deterministic analog signals  16

1.6.3. Representation and modeling of signals and systems 20

1.6.4. Phase–plane representation 23

1.6.5. Dynamic system  26

1.7. Comprehension and application exercises 28

Chapter 2. Transforms: Time – Frequency – Scale  31

2.1. Fourier series applied to periodic functions 31

2.1.1. Fourier series  31

2.1.2. Spectral representation (frequency domain)  33

2.1.3. Properties of Fourier series  34

2.1.4. Some examples  35

2.2. FT applied to non-periodic functions  36

2.3. Necessary conditions for the Fourier integral  38

2.3.1. Definition  38

2.3.2. Necessary condition  38

2.4. FT properties  39

2.4.1. Properties  39

2.4.2. Properties of the FT  39

2.4.3. Plancherel theorem and convolution product  40

2.5. Fourier series and FT 41

2.6. Elementary signals and their transforms 43

2.7. Laplace transform 46

2.7.1. Definition  46

2.7.2. Properties  49

2.7.3. Examples of the use of the unilateral LT  50

2.7.4. Transfer function 52

2.8. FT and LT 53

2.9. Application exercises 54

Chapter 3. Spectral Study of Signals 59

3.1. Power and signals energy 59

3.1.1. Power and energy of random signals  59

3.2. Autocorrelation and intercorrelation 61

3.2.1. Autocorrelation and cross-correlation in the time domain 61

3.2.2. A few examples of applications in steady state 64

3.2.3. Powers in variable state  65

3.3. Mathematical application of the correlation and autocorrelation functions  66

3.3.1. Duration of a signal and its spectrum width 68

3.3.2. Finite or zero average power signals 72

3.3.3. Application for linear filtering  74

3.4. A few application exercises  75

Chapter 4. Representation of Discrete (Sampled) Systems 81

4.1. Shannon and sampling, discretization methods, interpolation, sample and hold circuits  81

4.1.1. Sampling and interpolation  81

4.2. Z-transform – representation of discrete (sampled) systems  89

4.2.1. Definition – convergence and residue  89

4.2.2. Inverse Z-transform  91

4.2.3. Properties of the Fourier transform 96

4.2.4. Representation and modeling of signals and discrete systems  99

4.2.5. Transfer function in Z and representation in the frequency domain 102

4.2.6. Z-domain transform, Fourier transform and Laplace transform  104

4.3. A few application exercises  105

Chapter 5. Representation of Signals and Systems  123

5.1. Introduction to modeling 123

5.1.1. Signal representation using polynomial equations 127

5.1.2. Representation of signals and systems by differential equations 127

5.2. Representation using system state equations  128

5.2.1. State variables and state representation definition 128

5.2.2. State–space representation for discrete linear systems 134

5.3. Transfer functions 135

5.3.1. Transfer function: external representation 135

5.3.2. Transfer function and state–space representation shift 135

5.3.3. Properties of transfer functions  138

5.3.4. Associations of functional diagrams 142

5.4. Change in representation and canonical forms 142

5.4.1. Controllable canonical form 143

5.4.2. Controllable canonical form 145

5.4.3. Observability canonical form 145

5.4.4. Observable canonical form  146

5.4.5. Diagonal canonical form 149

5.4.6. Change in state-space representations and change in basis  150

5.4.7. Examples of systems to be modeled: the inverse pendulum  152

5.4.8. System phase–plane representation 155

5.5. Some application exercises  160

Chapter 6. Dynamic Responses and System Performance 173

6.1. Introduction to linear time-invariant systems  173

6.2. Transition matrix of an LTI system 173

6.2.1. Transition matrix 173

6.3. Evolution equation of an LTI system  174

6.3.1. State evolution equation  174

6.3.2. Transition matrix computation  176

6.4. Time response to the excitation of continuous linear systems 177

6.4.1. System response  177

6.4.2. Solution the state equation  178

6.4.3. Role of eigenvalues of the evolution matrix A within the system dynamics  181

6.5. Sampling and discretization of continuous systems  182

6.5.1. Choice of the sampling period (Shannon) and integration methods 182

6.5.2. Euler’s method 182

6.5.3. Order n Runge–Kutta method  183

6.5.4. Method using the state transition matrix with zeroth-order holder  184

6.5.5. Evolution equation for a time-invariant discrete system (DTI)  185

6.6. Some temporal responses 186

6.6.1. Response to an impulse excitation  187

6.6.2. Response to step excitation  187

6.7. Transfer function frequency responses 193

6.7.1. Bode plot  193

6.7.2. Nyquist plot  195

6.7.3. Black–Nichols plot  197

6.8. Parametric identification 198

6.8.1. Identification by analogy 199

6.8.2. Parameters identification: examples of systems  201

6.8.3. Strejc method (minimal dephasing) 203

6.9. Dynamics of linear systems  204

6.9.1. Link between frequency domain and time domain 204

6.10. System performance and accuracy 205

6.10.1. Damping factor of a system 205

6.10.2. System speed and transient 205

6.10.3. System static error, speed, sensitivity to noise and accuracy 205

6.10.4. Conclusion  208

6.11. Some application exercises 208

Chapter 7. System Stability and Robustness Analysis Methods 227

7.1. Introduction  227

7.2. Definitions related with the stability of a dynamic system 228

7.2.1. Equilibrium state of a system 229

7.2.2. Stable system: bounded input bounded output 229

7.3. Stability criteria  230

7.3.1. Routh criterion and stability algebraic criterion  230

7.3.2. Jury criterion and discrete system example 235

7.4. Some application exercises  242

7.4.1. Exercises: circle criterion, causes of instability and practical cases 242

Bibliography 257

Index 263

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