Fundamentals of Signals and Control Systems

Smain FEMMAM received his MSc and PhD in Signal Processing and Computers from Versailles University in France. He is currently Senior Director of Research at the University of Haute-Alsace and is on the editorial board of several journals. His research interests include signal processing for wireless communication-computers, estimation theory, adaptive filtering, array processing, distributed and embedding systems, complex theory and robotics.

• Preface ixChapter 1 Introduction, Generalities, Definitions of Systems 11.1. Introduction 11.2. Signals and communication systems 21.3. Signals and systems representation 51.3.1. Signal 51.3.2 Functional space L 2 61.3.3. Dirac distribution 81.4. Convolution and composition products – notions of filtering 101.4.1. Convolution or composition product 101.4.2. System 111.5. Transmission systems and filters 121.5.1. Convolution and filtering 131.6. Deterministic signals – random signals – analog signals 151.6.1. Definitions 151.6.2. Some deterministic analog signals 161.6.3. Representation and modeling of signals and systems 201.6.4. Phase–plane representation 231.6.5. Dynamic system 261.7. Comprehension and application exercises 28Chapter 2 Transforms: Time – Frequency – Scale 312.1. Fourier series applied to periodic functions 312.1.1. Fourier series 312.1.2. Spectral representation (frequency domain) 332.1.3. Properties of Fourier series 342.1.4. Some examples 352.2. FT applied to non-periodic functions 362.3. Necessary conditions for the Fourier integral 382.3.1. Definition 382.3.2. Necessary condition 382.4. FT properties 392.4.1. Properties 392.4.2. Properties of the FT 392.4.3. Plancherel theorem and convolution product 402.5. Fourier series and FT 412.6. Elementary signals and their transforms 432.7. Laplace transform 462.7.1. Definition 462.7.2. Properties 492.7.3. Examples of the use of the unilateral LT 502.7.4. Transfer function 522.8. FT and LT 532.9. Application exercises 54Chapter 3 Spectral Study of Signals 593.1. Power and signals energy 593.1.1. Power and energy of random signals 593.2. Autocorrelation and intercorrelation 613.2.1. Autocorrelation and cross-correlation in the time domain 613.2.2. A few examples of applications in steady state 643.2.3. Powers in variable state 653.3. Mathematical application of the correlation and autocorrelation functions 663.3.1. Duration of a signal and its spectrum width 683.3.2. Finite or zero average power signals 723.3.3. Application for linear filtering 743.4. A few application exercises 75Chapter 4 Representation of Discrete (Sampled) Systems 814.1. Shannon and sampling, discretization methods, interpolation, sample and hold circuits 814.1.1. Sampling and interpolation 814.2. Z-transform – representation of discrete (sampled) systems 894.2.1. Definition – convergence and residue 894.2.2. Inverse Z-transform 914.2.3. Properties of the Fourier transform 964.2.4. Representation and modeling of signals and discrete systems 994.2.5. Transfer function in Z and representation in the frequency domain 1024.2.6. Z-domain transform, Fourier transform and Laplace transform 1044.3. A few application exercises 105Chapter 5 Representation of Signals and Systems 1235.1. Introduction to modeling 1235.1.1. Signal representation using polynomial equations 1275.1.2. Representation of signals and systems by differential equations 1275.2. Representation using system state equations 1285.2.1. State variables and state representation definition 1285.2.2. State–space representation for discrete linear systems 1345.3. Transfer functions 1355.3.1. Transfer function: external representation 1355.3.2. Transfer function and state–space representation shift 1355.3.3. Properties of transfer functions 1385.3.4. Associations of functional diagrams 1425.4. Change in representation and canonical forms 1425.4.1. Controllable canonical form 1435.4.2. Controllable canonical form 1455.4.3. Observability canonical form 1455.4.4. Observable canonical form 1465.4.5. Diagonal canonical form 1495.4.6. Change in state-space representations and change in basis 1505.4.7. Examples of systems to be modeled: the inverse pendulum 1525.4.8. System phase–plane representation 1555.5. Some application exercises 160Chapter 6 Dynamic Responses and System Performance 1736.1. Introduction to linear time-invariant systems 1736.2. Transition matrix of an LTI system 1736.2.1. Transition matrix 1736.3. Evolution equation of an LTI system 1746.3.1. State evolution equation 1746.3.2. Transition matrix computation 1766.4. Time response to the excitation of continuous linear systems 1776.4.1. System response 1776.4.2. Solution the state equation 1786.4.3. Role of eigenvalues of the evolution matrix A within the system dynamics 1816.5. Sampling and discretization of continuous systems 1826.5.1. Choice of the sampling period (Shannon) and integration methods 1826.5.2. Euler’s method 1826.5.3. Order n Runge–Kutta method 1836.5.4. Method using the state transition matrix with zeroth-order holder 1846.5.5. Evolution equation for a time-invariant discrete system (DTI) 1856.6. Some temporal responses 1866.6.1. Response to an impulse excitation 1876.6.2. Response to step excitation 1876.7. Transfer function frequency responses 1936.7.1. Bode plot 1936.7.2. Nyquist plot 1956.7.3. Black–Nichols plot 1976.8. Parametric identification 1986.8.1. Identification by analogy 1996.8.2. Parameters identification: examples of systems 2016.8.3. Strejc method (minimal dephasing) 2036.9. Dynamics of linear systems 2046.9.1. Link between frequency domain and time domain 2046.10. System performance and accuracy 2056.10.1. Damping factor of a system 2056.10.2. System speed and transient 2056.10.3. System static error, speed, sensitivity to noise and accuracy 2056.10.4. Conclusion 2086.11. Some application exercises 208Chapter 7. System Stability and Robustness Analysis Methods 2277.1. Introduction 2277.2. Definitions related with the stability of a dynamic system 2287.2.1. Equilibrium state of a system 2297.2.2. Stable system: bounded input bounded output 2297.3. Stability criteria 2307.3.1. Routh criterion and stability algebraic criterion 2307.3.2. Jury criterion and discrete system example 2357.4. Some application exercises 2427.4.1. Exercises: circle criterion, causes of instability and practical cases 242Bibliography 257Index 263