Linearization of Nonlinear Control Systems

ISBN-13: 9789811936425 / Angielski / Twarda / 2022

Hong-Gi Lee

Opis

This textbook helps graduate level student to understand easily the linearization of nonlinear control system. Differential geometry is essential to understand the linearization problems of the control nonlinear systems. In this book, the basics of differential geometry needed in linearization are explained on the Euclidean space instead of the manifold for students who are not accustomed to differential geometry. Many Lie algebra formulas, used often in linearization, are also provided with proof. The conditions in the linearization problems are complicated to check because the Lie bracket calculation of vector fields by hand needs much concentration and time. This book provides MATLAB programs for most of the theorems. The book also includes end-of-chapter problems and other pedagogical aids to help understanding and self study. 

Recenzje

"The presentation of the book is explicit. Some bases that are not familiar to engineering students are explained in the book. It also provides a number of exercises, and so is suitable as a textbook for graduate students in the area of systems and control." (Qianqian Xia, Mathematical Reviews, September, 2023)

"The book provides a compendium of linearization techniques for nonlinear control systems together with their thorough mathematical justification and MATLAB implementation. The style of presentation is accessible for both experienced researchers and undergraduate students. ... A concise appendix containing basics of topology, fundamentals on manifolds and vector fields, and MATLAB codes for subfunctions completes the monograph." (Petro Feketa, zbMATH 1505.93001, 2023)

Spis treści

​1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Trends of Nonlinear Control System Theory . . . . . . . . . . . . . . . . . . . . 1

1.2 Approximate Linearization of the Nonlinear Systems. . . . . . . . . . . . . 4

1.3 Exact Linearization of the Nonlinear Systems . . . . . . . . . . . . . . . . . . . 5

2 Basic Mathematics for Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1 Vector Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 State Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Nonsingular State Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Vector Field and Tangent Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.5 Covector Field and One-Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.6 Distribution and Frobenius Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.7 State Equivalence and Feedback Equivalence . . . . . . . . . . . . . . . . . . . 56

2.8 MATLAB Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

2.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3 Linearization by State Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.2 Single Input Nonlinear Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.3 Multi Input Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.4 MATLAB Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4 Feedback Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.2 Single Input Nonlinear Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.3 Multi Input Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.4 Applications of Feedback Linearization . . . . . . . . . . . . . . . . . . . . . . . . 129

4.5 MATLAB Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

4.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

5 Linearization with Output Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

5.2 State Equivalence to a SISO linear system . . . . . . . . . . . . . . . . . . . . . . 156

5.3 State Equivalence to a MIMO linear system. . . . . . . . . . . . . . . . . . . . . 159

5.4 Feedback Linearization with Output of SISO Systems . . . . . . . . . . . . 162

5.5 Input-Output Linearization of MIMO Systems . . . . . . . . . . . . . . . . . . 174

5.5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

5.5.2 Structure Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

5.5.3 Conditions for input-output linearization . . . . . . . . . . . . . . . . . 185

5.6 Feedback Linearization with Multi Output. . . . . . . . . . . . . . . . . . . . . . 191

5.7 MATLAB Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

5.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

6 Dynamic Feedback Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

6.2 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

6.3 Restricted Dynamic Feedback Linearization . . . . . . . . . . . . . . . . . . . . 226

6.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

6.5 MATLAB Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

6.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

7 Linearization of Discrete-time Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

7.2 Single Input Discrete-time Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

7.3 Multi Input Discrete-time Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

7.4 Linearization of Discrete-time Systems with Single Output. . . . . . . . 294

7.5 MATLAB Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

7.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

8 Observer Error Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

8.2 Single Output Observer Error Linearization. . . . . . . . . . . . . . . . . . . . . 322

8.3 Dynamic Observer Error Linearization. . . . . . . . . . . . . . . . . . . . . . . . . 339

8.4 Multi Output Observer Error Linearization . . . . . . . . . . . . . . . . . . . . . 356

8.5 Discrete Time Observer Error Linearization . . . . . . . . . . . . . . . . . . . . 401

8.6 Discrete Time Dynamic Observer Error Linearization . . . . . . . . . . . . 425

8.7 MATLAB Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464

8.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501

9 Input-output Decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509

9.2 Input-output Decoupling of the Nonlinear Systems. . . . . . . . . . . . . . . 510

9.3 Dynamic Input-output Decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515

9.4 MATLAB Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533

9.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538

A Basics of Topology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547

A.1 Topology of Real Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547

A.2 General Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548

B Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553

B.1 Vector Space and Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554

B.2 Vector Fields on Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556

C MATLAB Subfunctions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561

O autorze

Hong-Gi Lee received the B.S. and M.S. degrees in Department of Electronics Engineering from Seoul National University, in 1981 and 1983, respectively. He received the Ph.D. degree in Department of Electrical and Computer Engineering from The University of Texas at Austin, in 1986. He was an assistant professor in Department of Electrical and Computer Engineering at Louisiana State University from 1986 to 1989. He is currently a professor at the Department of Electrical and Electronics Engineering, Chung-Ang University, Seoul, Korea. His research interests include nonlinear control, genetic algorithm, and robotics. 

Blurb

This textbook helps graduate level student to understand easily the linearization of nonlinear control system. Differential geometry is essential to understand the linearization problems of the control nonlinear systems. In this book, the basics of differential geometry, needed in linearization, are explained on the Euclean space instead of the manifold for the students who are not accustomed to differential geometry. Many Lie algebra formulas, used often in linearization, are also provided with proof. The conditions in the linearization problems are complicated to check because the Lie bracket calculation of vector fields by hand needs much concetration and time. This book provides the MATLAB programs for most of the theorems.