Digital Control Systems: Theoretical Problems and Simulation Tools

456 Pages·2017·44.83 MB·english

by  Anastasia Veloni

Digital Control Systems

Digital Control Systems
Theoretical Problems and
Simulation Tools
Anastasia Veloni
Nikolaos I. Miridakis
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Contents
Preface ......................................................................................................................xi
Authors .................................................................................................................xiii
1. Introduction .....................................................................................................1
1.1 Introduction ...........................................................................................1
1.2 Description of Analog and Digital Control Systems .......................2
1.3 Advantages of Digital Control Systems and Applications ..............5
2. z-Transform ......................................................................................................7
2.1 Introduction ...........................................................................................7
2.2 From Laplace Transform to z-Transform ...........................................7
2.2.1 Comparison of s- and z-Planes in the Region
of Convergence .......................................................................10
2.3 z-Transform Properties .......................................................................11
2.3.1 Time Shift ................................................................................11
2.3.2 Linearity ..................................................................................12
2.3.3 Time Reverse ..........................................................................12
2.3.4 Convolution ............................................................................12
2.3.5 Differentiation in z-Domain .................................................12
2.3.6 Initial and Final Value Theorem ..........................................12
2.4 Inverse z-Transform ............................................................................13
2.4.1 Division Method ....................................................................13
2.4.2 Method of Partial Fraction Expansion ................................13
2.4.3 Method of Complex Integration ..........................................14
2.5 Formula Tables ....................................................................................15
2.6 Solved Exercises ..................................................................................15
3. Transfer Function .........................................................................................53
3.1 Introduction .........................................................................................53
3.2 Open-Loop Sampled-Data Control System .....................................53
3.3 Closed-Loop Sampled-Data Control System ...................................54
3.4 Signal Flow Graphs .............................................................................55
3.5 Mason’s Formula .................................................................................57
3.6 Difference Equations ..........................................................................58
3.7 Formula Tables ....................................................................................60
3.8 Solved Exercises ..................................................................................61
vii
viii Contents
4. Transfer Function Discretization ..............................................................89
4.1 Introduction .........................................................................................89
4.2 Discretization Methods ......................................................................90
4.2.1 Impulse-Invariance Method or z-Transform Method .......91
4.2.2 Step-Invariance Method or z-Transform Method
with Sample and Hold ..........................................................92
4.2.3 Backward Difference Method ..............................................94
4.2.4 Forward Difference Method ................................................96
4.2.5 Bilinear or Tustin Method ....................................................96
4.2.6 Frequency Prewarping Method ...........................................98
4.2.7 Matched Pole–Zero Method .................................................99
4.3 Comparison of Discretization Methods ........................................101
4.4 Formulas .............................................................................................102
4.5 Solved Exercises ................................................................................104
5. State-Space Representation ......................................................................159
5.1 Introduction .......................................................................................159
5.2 Discrete-Time State-Space Equations .............................................160
5.2.1 Eigenvalues and Eigenvectors ............................................161
5.3 Solution of State Equations ..............................................................161
5.4 State-Space Representation ..............................................................162
5.4.1 Direct Form ...........................................................................162
5.4.2 Canonical Form ....................................................................164
5.4.3 Controllable Canonical Form .............................................165
5.4.4 Observable Canonical Form ...............................................167
5.4.5 Jordan Canonical Form .......................................................168
5.5 Controllability and Observability ..................................................169
5.6 State-Space Discretization................................................................171
5.6.1 Discretization of Continuous-Time Systems in the
State Space with Numerical Integration Methods ..........171
5.6.2 Discretization of Continuous Time Systems in
State-Space with Numerical Differentiation Methods ....173
5.6.3 Discretization with the Zero-Order Hold Method .........174
5.6.4 Discretization with the First-Order Hold Method ..........174
5.7 Formula Tables ..................................................................................175
5.8 Solved Exercises ................................................................................175
6. Stability of Digital Control Systems ......................................................217
6.1 Stability ...............................................................................................217
6.2 Unit-Circle Criterion .........................................................................219
6.3 Routh Criterion Using the Bilinear Mobius Transformation ......219
6.4 Jury Criterion (Eliahu I. Jury 1955) .................................................221
6.5 Root Locus Method ...........................................................................221
6.5.1 Rules for Approximate Establishment of Root Locus ....222
6.6 Nyquist Stability Criterion ...............................................................224
Contents ix
6.7 Bode Stability Criterion (H.W. Bode—1930) ..................................226
6.8 Formula Table ....................................................................................227
6.9 Solved Exercises ................................................................................229
7. Time and Harmonic Response Analysis Steady-State Errors ...........275
7.1 Time Response ...................................................................................275
7.1.1 Impulse Time Response of First-Order Systems .............276
7.1.2 Impulse Time Response of Second-Order Systems ........277
7.1.3 Step Response of First-Order System ................................278
7.1.4 General Form for the Step Response of Discrete
Time System ..........................................................................278
7.1.5 Correlation between Analog and Discrete Time
Response ...............................................................................281
7.2 Steady-State Errors ............................................................................283
7.3 Harmonic Response of Discrete Systems ......................................287
7.4 Formula Tables ..................................................................................288
7.5 Solved Exercises ................................................................................290
8. Compensation of Digital Control Systems ...........................................325
8.1 Introduction .......................................................................................325
8.2 Indirect Design Methods .................................................................326
8.2.1 Selection of Sampling Rate .................................................328
8.3 Direct Design Methods ....................................................................330
8.3.1 Design via Analytical Methods .........................................330
8.3.2 Design Using the Root Locus Method
of the Characteristic Equation ............................................331
8.3.3 Design in the Frequency Domain .....................................332
8.4 PID Digital Controller ......................................................................333
8.4.1 Digital PID Controller Tuning ...........................................336
8.5 Deadbeat Digital Controller ............................................................338
8.6 Phase Lead/Lag Digital Compensators .........................................338
8.7 Formula Table ....................................................................................339
8.8 Solved Exercises ................................................................................340
9. Simulation Tools: MATLAB, Simulink, LabVIEW,
Comprehensive Control ............................................................................399
9.1 Introduction .......................................................................................399
9.2 Control Systems Simulation Using MATLAB ...............................399
9.2.1 Analysis and System Modeling .........................................400
9.2.2 Control Systems Design ......................................................400
9.2.3 Simulation of Digital Control Systems
Using MATLAB ....................................................................400
9.3 Simulink .............................................................................................406
9.3.1 Introduction ..........................................................................406
9.3.2 Model Creation .....................................................................407
9.4 LabVIEW.............................................................................................411