Fundamentals of Signals and Control Systems

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 L 2 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