2025 (Current Year) Faculty Courses School of Engineering Department of Systems and Control Engineering Graduate major in Systems and Control Engineering
Nonlinear Control: Geometric Approach
- Academic unit or major
- Graduate major in Systems and Control Engineering
- Instructor(s)
- Mitsuji Sampei
- Class Format
- Lecture
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - Class
- -
- Course Code
- SCE.C532
- Number of credits
- 100
- Course offered
- 2025
- Offered quarter
- 4Q
- Syllabus updated
- Apr 1, 2025
- Language
- English
Syllabus
Course overview and goals
This course focuses on nonlinear control theory based on differential geometry. The basic concepts of differential geometry (differential manifold, vector field, Lie derivative and Lie bracket) are introduced, and their relation to nonlinear control theory (controllability and observability) is discussed. The exact linearization is also introduced.
Matrix theory is a powerful tool for analysis and controller design of linear systems described in linear state equation. Instead of matrix theory, differential geometry should be used for nonlinear systems. This course shows how differential geometry contributes to nonlinear control theory, and gives necessary knowledge for understanding and developing nonlinear control theory.
Course description and aims
At the end of this course, students will be able to:
1) Have an understanding of basic concepts of differential geometry, and based on this, explain accessibility(controllability) and distinguishability(observability) of nonlinear systems.
2) Have an understanding of the nonlinear control theory based on differential geometry, and based on this, design controllers for nonlinear systems.
Keywords
Differential Geometry, Nonlinear System, Nonlinear State Equation, Linearization, Nonlinear Observer
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
Quiz is given in each class.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Differential Geometry and Nonlinear State Equation | Understand the definition of differential manifold. Derive nonlinear state equations of mechanical systems. |
| Class 2 | Vector Field and Coordinate Transformation | Understand the concept of vector fields. Derive the transformation of vector fields associated with coordinate transformation of the manifold. |
| Class 3 | Lie derivative and distinguishability | Understand the concept of Lie derivative. Check the distinguishability(observability) of nonlinear systems. |
| Class 4 | Lie bracket and Accessibility | Understand the concept of Lie bracket. Check the accessibility(controllability) of nonlinear systems. |
| Class 5 | Approximate linearization and Exact Linearization | Understand the concepts of approximate linearization and exact linearization. Approximately and/or exactly linearize nonlinear systems. |
| Class 6 | Exact Linearization: Example | Design state feedbacks using Exact Linearization. |
| Class 7 | Exact Linearization (Proof) | Understand the proof of exact linearization. |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Textbook(s)
Materials used in class can be found on Science Tokyo LMS.
Reference books, course materials, etc.
Hassan K. Khalil: Nonlinear Control, Prentice Hall (2014)
Alberto Isidori: Nonlinear Control Systems, Springer; 3rd ed.(1995)
Evaluation methods and criteria
Students will be assessed on their understanding of Nonlinear Control Theory based on Differential Geometry and their ability to apply them to solve problems.
Students’ course scores are based on the quiz in each class and the reports.
Related courses
- SCE.C201 ： Fundamentals of Dynamical Systems
- SCE.C301 ： Linear System Theory
Prerequisites
Students require the basic knowledge of linear system theory: state equation, controllability, observability, state feedback and observer.