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Table of Contents

1. Getting Started      1
1.1 Using the Package for the First Time      1
1.2 Structure of the Application      2
1.3 Robustness of Numerical Methods      3
2. Introduction      7
2.1 Quick Reference      7
Solutions of the Lyapunov and Sylvester Matrix Equations      8
Solutions of the Algebraic Riccati Equations      8
Reduction to Controller-Hessenberg and

Observer-Hessenberg Forms      9
Controllability and Observability Tests      9
Pole Assignment      9
Feedback Stabilization      10
Design of the Reduced-Order State Estimator (Observer)      10
Model Reduction      10
Model Identification      10
Miscellaneous Matrix Decompositions and Functions      11
2.2 An Industrial Application: Controlling the Drum Boiler      11
2.2.1 The State-Space Model of a Drum Boiler      11
2.2.2 System Responses, Stability, and Poles      12
2.2.3 Testing the Controllability      14
2.2.4 The Design of the LQR Controller      15
2.2.5 The Controller Design Using Constrained Feedback

Stabilization      19
2.2.6 The Observer Design      20
3. Matrix Equations and Control Applications      26
3.1 Lyapunov Equations      26
3.2 Riccati Equations      29
3.2.1 The Schur Methods for the Riccati Equations      30
3.2.2 The Inverse-Free Generalized Eigenvector and Schur

Methods for the Riccati Equations      32
3.2.3 The Matrix Sign-Function Methods for the Riccati Equations      34
3.2.4 The Newton Methods for the Riccati Equations      36
3.2.5 LQR and LQG Designs Using Riccati Equations      38
4. Block Hessenberg Forms      41
4.1 Controller-Hessenberg Forms      41
4.2 Observer-Hessenberg Forms      45
4.3 Controllability and Observability Tests Using

Block Hessenberg Forms      47
5. Pole Assignment and Stabilization by State Feedback      50
5.1 Pole Assignment Methods      50
5.1.1 The Recursive Algorithms      51
5.1.2 The Explicit and Implicit QR Algorithms      53
5.1.3 The Schur Method      56
5.2 Partial Pole Assignment      58
5.3 Constrained Feedback Stabilization      61
5.4 Lyapunov Feedback Stabilization      66
6. State Estimation      69
6.1 Full-Order State Estimation      69
6.2 Reduced-Order State Estimator      71
6.2.1 Reduced-Order State Estimator via Pole Assignment      74
6.2.2 Reduced-Order State Estimator via Sylvester-Observer

Equation      76
7. Model Reduction      80
7.1 Cholesky Factors of the Controllability and Observability Gramians      80
7.2 Model Reduction Using Schur and Square-Root Methods      84
8. Model Identification      89
8.1 Time-Domain System Identification      89
8.1.1 Identification Using Markov Parameters      89
8.1.2 Identification Using Input-Output Data (Subspace

System Identification Method)      93
8.2 Frequency Domain System Identification      97
9. Generalized Eigenvalue Problem      101
9.1 Generalized Eigenvalue Problem      101
9.2 Generalized Schur Decomposition      104
References      110
Appendix. Collection of Control Systems for Case Studies      115
A.1 Continuous-Time Models      115
A.1.1 The Absorption Column      115
A.1.2 The F-8 Aircraft      116
A.1.3 The L-1011 Aircraft      117
A.1.4 The Tubular Ammonia Reactor      118
A.1.5 The Fluid Catalytic Reactor      118
A.1.6 The Binary Distillation Column      119
A.1.7 The Drum Boiler      120
A.1.8 The Flight Control System      121
A.1.9 The Automobile Gas Turbine      121
A.1.10 The CH-47 Helicopter      122
A.1.11 The Magnetic Tape      123
A.1.12 The Electric Power System      123
A.1.13 The J-100 Jet Engine      124
A.1.14 The "Smart" Highway      125
A.1.15 The Generator Axle in a Power Plant      126
A.2 Discrete-Time Models      127
A.2.1 The Catalytic Cracker      127
A.2.2 The Chemical Plant      127
A.2.3 The Paper Machine      128
A.2.4 The Steam Power System      129
Index      130

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