Legacy Documentation

Polynomial Control Systems (2014)

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Table of Contents

1. Getting Started     1
1.1 Using the Package for the First Time     2
1.2 Structure of the Application     2
1.3 Integration into Control System Professional Suite     3
2. Introduction: Controlling a Distillation Column—A Case Study of an Industrial Application     6
2.1 The Transfer-Function Model     6
2.2 System Poles, Stability, and Responses     7
2.3 Analysis Using Alternative Models     9
2.4 Pole Assignment with State Feedback     13
2.5 Frequency-Domain Design     15
3. Linear System Models     21
3.1 The State-Space Description     21
3.2 The Transfer-Function Description     24
3.3 The System Matrix Description     24
3.3.1 Creating a System Matrix in Polynomial Form     26
3.3.2 Creating a System Matrix from Other Control Objects     30
3.3.3 The Left and Right Matrix-Fraction Forms of the System Matrix     32
3.3.4 Minimizing the Dimensions of a System Matrix Object     35
3.4 Matrix-Fraction Descriptions     37
3.4.1 Left Matrix-Fraction Models     38
3.4.2 Right Matrix-Fraction Models     42
3.5 Applying Control System Professional Functions to Polynomial Control Objects     47
3.5.1 Continuous-Time versus Discrete-Time Systems     47
3.5.2 Simulating System Behavior     50
3.5.3 Manipulating a System's Contents     54
3.5.4 Part Count and Consistency Check     55
4. Linear System Analysis     57
4.1 Standard Forms     57
4.1.1 The Smith Form     60
4.1.2 The Invariant Zeros of a System     62
4.1.3 The McMillan Form and McMillan Degree     65
4.1.4 The Transmission Zeros of a System     67
4.2 Functional Controllability     70
4.2.1 Functional Controllability Tests     71
4.3 Coprimeness and Matrix Greatest Common Divisors     75
4.3.1 Left and Right Coprime Tests     77
4.3.2 Left and Right Matrix Greatest Common Divisors     79
4.4 Decoupling Zeros     84
4.4.1 Detecting Input and Output Decoupling Zeros     87
4.4.2 Removing Input and Output Decoupling Zeros     91
4.4.3 Creating Least-Order Systems     95
4.5 Similarity Transformations     99
5. Frequency-Domain Design Methods     103
5.1 The Relative Gain Array and Number     104
5.2 The Nyquist Array and Diagonal Dominance     109
5.2.1 The Direct Nyquist Array     113
5.2.2 The Inverse Nyquist Array     119
5.2.3 Diagonal Dominance Measures: Row Dominance Ratios and Column Dominance Ratios     128
5.3 Ways of Achieving Diagonal Dominance     133
5.3.1 The Perron-Frobenius Eigenvalue and Constant Scaling Compensators     133
5.3.2 The Perron-Frobenius Eigenvector and Dynamic Scaling Compensators     139
5.3.3 Normalizing an Element of the Perron-Frobenius Eigenvector     147
5.3.4 Using Alternative Fitting Methods     154
5.3.5 Pseudo-Diagonalization     157
5.4 The Characteristic Locus Design Method     170
5.4.1 Characteristic Value Plots     173
5.4.2 The High-Frequency Aligning Compensator     177
5.4.3 Reactor System Design Example     179
6. Pole Assignment and State Reconstruction     188
6.1 The Mapping Algorithm     189
6.2 The Spectral Algorithm     194
6.3 The Full-Rank Algorithm     197
6.4 The Luenberger Controllable and Observable Canonical Forms     201
6.4.1 The Luenberger Controllable Canonical Form     201
6.4.2 The Luenberger Observable Canonical Form     205
6.5 State Reconstruction     208
6.6 Pole Assignment Using Output Feedback     216
6.6.1 The Dyadic Output Feedback Algorithm     217
6.6.2 The Full-Rank Output Feedback Algorithm     226
References     235
Index     237