Control of Continuous Linear Systems

Kaddour Najim is Professor of Automatic Control in the Process Control Laboratory of the Ecole Nationale Supérieure des Ingénieurs en Arts Chimiques et Technologiques (ENSIACET), Toulouse, France. He is also Docent in learning systems at the University of Oulu, Finland. He is the author of numerous books and articles on the subject and has conducted several industrial applications. He is also a member of the Conference Editorial Board of the IEEE Control Systems Society and the IFAC Applications Committee.

• Introduction xiiiChapter 1. Introduction to Signals and Systems 1Yannick BERTHOUMIEU, Eric GRIVEL and Mohamed NAJIM1.1. Introduction 11.2. Signals: categories, representations and characterizations 11.2.1. Definition of continuous-time and discrete-time signals 11.2.2. Deterministic and random signals 61.2.3. Periodic signals 81.2.4. Mean, energy and power 91.2.5. Autocorrelation function 121.3. Systems 151.4. Properties of discrete-time systems 161.4.1. Invariant linear systems 161.4.2. Impulse responses and convolution products 161.4.3. Causality 171.4.4. Interconnections of discrete-time systems 181.5. Bibliography 19Chapter 2. Discrete System Analysis 21Mohamed NAJIM and Eric GRIVEL2.1. Introduction 212.2. The z-transform 212.2.1. Representations and summaries 212.2.2. Properties of the z-transform 282.2.2.1. Linearity 282.2.2.2. Advanced and delayed operators 292.2.2.3. Convolution 302.2.2.4. Changing the z-scale 312.2.2.5. Contrasted signal development 312.2.2.6. Derivation of the z-transform 312.2.2.7. The sum theorem 322.2.2.8. The final-value theorem 322.2.2.9. Complex conjugation 322.2.2.10. Parseval’s theorem 332.2.3. Table of standard transform 332.3. The inverse z-transform 342.3.1. Introduction 342.3.2. Methods of determining inverse z-transforms 352.3.2.1. Cauchy’s theorem: a case of complex variables 352.3.2.2. Development in rational fractions 372.3.2.3. Development by algebraic division of polynomials 382.4. Transfer functions and difference equations 392.4.1. The transfer function of a continuous system 392.4.2. Transfer functions of discrete systems 412.5. Z-transforms of the autocorrelation and intercorrelation functions 442.6. Stability 452.6.1. Bounded input, bounded output (BIBO) stability 462.6.2. Regions of convergence 462.6.2.1. Routh’s criterion 482.6.2.2. Jury’s criterion 49Chapter 3. Frequential Characterization of Signals and Filters 51Eric GRIVEL and Yannick BERTHOUMIEU3.1. Introduction 513.2. The Fourier transform of continuous signals 513.2.1. Summary of the Fourier series decomposition of continuous signals 513.2.1.1. Decomposition of finite energy signals using an orthonormal base 513.2.1.2. Fourier series development of periodic signals 523.2.2. Fourier transforms and continuous signals 573.2.2.1. Representations 573.2.2.2. Properties 583.2.2.3. The duality theorem 593.2.2.4. The quick method of calculating the Fourier transform 593.2.2.5. The Wiener-Khintchine theorem 633.2.2.6. The Fourier transform of a Dirac comb 633.2.2.7. Another method of calculating the Fourier series development of a periodic signal 663.2.2.8. The Fourier series development and the Fourier transform 683.2.2.9. Applying the Fourier transform: Shannon’s sampling theorem 753.3. The discrete Fourier transform (DFT) 783.3.1. Expressing the Fourier transform of a discrete sequence 783.3.2. Relations between the Laplace and Fourier z-transforms 803.3.3. The inverse Fourier transform 813.3.4. The discrete Fourier transform 823.4. The fast Fourier transform (FFT) 863.5. The fast Fourier transform for a time/frequency/energy representation of a non-stationary signal 903.6. Frequential characterization of a continuous-time system 913.6.1. First and second order filters 913.6.1.1. 1st order system 913.6.1.2. 2nd order system 933.7. Frequential characterization of discrete-time system 953.7.1. Amplitude and phase frequential diagrams 953.7.2. Application 96Chapter 4. Continuous-Time and Analog Filters 99Daniel BASTARD and Eric GRIVEL4.1. Introduction 994.2. Different types of filters and filter specifications 994.3. Butterworth filters and the maximally flat approximation 1044.3.1. Maximally flat functions (MFM) 1044.3.2. A specific example of MFM functions: Butterworth polynomial filters 1064.3.2.1. Amplitude-squared expression 1064.3.2.2. Localization of poles 1074.3.2.3. Determining the cut-off frequency at –3 dB and filter orders 1104.3.2.4. Application 1114.3.2.5. Realization of a Butterworth filter 1124.4. Equiripple filters and the Chebyshev approximation 1134.4.1. Characteristics of the Chebyshev approximation 1134.4.2. Type I Chebyshev filters 1144.4.2.1. The Chebyshev polynomial 1144.4.2.2. Type I Chebyshev filters 1154.4.2.3. Pole determination 1164.4.2.4. Determining the cut-off frequency at –3 dB and the filter order 1184.4.2.5. Application 1214.4.2.6. Realization of a Chebyshev filter 1214.4.2.7. Asymptotic behavior 1224.4.3. Type II Chebyshev filter 1234.4.3.1. Determining the filter order and the cut-off frequency 1234.4.3.2. Application 1244.5. Elliptic filters: the Cauer approximation 1254.6. Summary of four types of low-pass filter: Butterworth, Chebyshev type I, Chebyshev type II and Cauer 1254.7. Linear phase filters (maximally flat delay or MFD): Bessel and Thomson filters 1264.7.1. Reminders on continuous linear phase filters 1264.7.2. Properties of Bessel-Thomson filters 1284.7.3. Bessel and Bessel-Thomson filters 1304.8. Papoulis filters (optimum (On)) 1324.8.1. General characteristics 1324.8.2. Determining the poles of the transfer function 1354.9. Bibliography 135Chapter 5. Finite Impulse Response Filters 137Yannick BERTHOUMIEU, Eric GRIVEL and Mohamed NAJIM5.1. Introduction to finite impulse response filters 1375.1.1. Difference equations and FIR filters 1375.1.2. Linear phase FIR filters 1425.1.2.1. Representation 1425.1.2.2. Different forms of FIR linear phase filters 1475.1.2.3. Position of zeros in FIR filters 1505.1.3. Summary of the properties of FIR filters 1525.2. Synthesizing FIR filters using frequential specifications 1525.2.1. Windows 1525.2.2. Synthesizing FIR filters using the windowing method 1595.2.2.1. Low-pass filters 1595.2.2.2. High-pass filters 1645.3. Optimal approach of equal ripple in the stop-band and passband 1655.4. Bibliography 172Chapter 6. Infinite Impulse Response Filters 173Eric GRIVEL and Mohamed NAJIM6.1. Introduction to infinite impulse response filters 1736.1.1. Examples of IIR filters 1746.1.2. Zero-loss and all-pass filters 1786.1.3. Minimum-phase filters1806.1.3.1. Problem 1806.1.3.2. Stabilizing inverse filters 1816.2. Synthesizing IIR filters 1836.2.1. Impulse invariance method for analog to digital filter conversion 1836.2.2. The invariance method of the indicial response 1856.2.3. Bilinear transformations 1856.2.4. Frequency transformations for filter synthesis using low-pass filters 1886.3. Bibliography 189Chapter 7. Structures of FIR and IIR Filters 191Mohamed NAJIM and Eric GRIVEL7.1. Introduction 1917.2. Structure of FIR filters 1927.3. Structure of IIR filters 1927.3.1. Direct structures 1927.32. The cascade structure 2097.3.3. Parallel structures 2117.4. Realizing finite precision filters 2117.4.1. Introduction 2117.4.2. Examples of FIR filters 2127.4.3. IIR filters 2137.4.3.1. Introduction 2137.4.3.2. The influence of quantification on filter stability 2217.4.3.3. Introduction to scale factors 2247.4.3.4. Decomposing the transfer function into first- and second-order cells 2267.5. Bibliography 231Chapter 8. Two-Dimensional Linear Filtering 233Philippe BOLON8.1. Introduction 2338.2. Continuous models 2338.2.1. Representation of 2-D signals 2338.2.2. Analog filtering 2358.3. Discrete models 2368.3.1. 2-D sampling 2368.3.2. The aliasing phenomenon and Shannon’s theorem 2408.3.2.1. Reconstruction by linear filtering (Shannon’s theorem) 2408.3.2.2. Aliasing effect 2408.4. Filtering in the spatial domain 2428.4.1. 2-D discrete convolution 2428.4.2. Separable filters 2448.4.3. Separable recursive filtering 2468.4.4. Processing of side effects 2498.4.4.1. Prolonging the image by pixels of null intensity 2508.4.4.2. Prolonging by duplicating the border pixels 2518.4.4.3. Other approaches 2528.5. Filtering in the frequency domain 2538.5.1. 2-D discrete Fourier transform (DFT) 2538.5.2. The circular convolution effect 2558.6. Bibliography 259Chapter 9. Two-Dimensional Finite Impulse Response Filter Design 261Yannick BERTHOUMIEU9.1. Introduction 2619.2. Introduction to 2-D FIR filters 2629.3. Synthesizing with the two-dimensional windowing method 2639.3.1. Principles of method 2639.3.2. Theoretical 2-D frequency shape 2649.3.2.1. Rectangular frequency shape 2649.3.2.2. Circular shape 2669.3.3. Digital 2-D filter design by windowing 2719.3.4. Applying filters based on rectangular and circular shapes 2719.3.5. 2-D Gaussian filters 2749.3.6. 1-D and 2-D representations in a continuous space 2749.3.6.1. 2-D specifications 2769.3.7. Approximation for FIR filters 2779.3.7.1. Truncation of the Gaussian profile 2779.3.7.2. Rectangular windows and convolution 2799.3.8. An example based on exploiting a modulated Gaussian filter 2809.4. Appendix: spatial window functions and their implementation 2869.5. Bibliography 291Chapter 10. Filter Stability 293Michel BARRET10.1. Introduction 29310.2. The Schur-Cohn criterion 29810.3. Appendix: resultant of two polynomials 31410.4. Bibliography 319Chapter 11. The Two-Dimensional Domain 321Michel BARRET11.1. Recursive filters 32111.1.1. Transfer functions 32111.1.2. The 2-D z-transform 32211.1.3. Stability, causality and semi-causality 32411.2. Stability criteria 32811.2.1. Causal filters 32911.2.2. Semi-causal filters 33211.3. Algorithms used in stability tests 33411.3.1. The jury Table 33411.3.2. Algorithms based on calculating the Bezout resultant 33911.3.2.1. First algorithm 34011.3.2.2. Second algorithm 34311.3.3. Algorithms and rounding-off errors 34711.4. Linear predictive coding 35111.5. Appendix A: demonstration of the Schur-Cohn criterion 35511.6. Appendix B: optimum 2-D stability criteria 35811.7. Bibliography 362List of Authors 365Index 367