2019 Nonlinear Control: Geometric Approach

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Academic unit or major
Graduate major in Systems and Control Engineering
Sampei Mitsuji 
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Access Index

Course description and aims

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, input-output linearization and observers with linear error dynamics are 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.

Student learning outcomes

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, using linearization and observers.


Differential Geometry, Nonlinear System, Nonlinear State Equation, Linearization, Nonlinear Observer

Competencies that will be developed

Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

Class flow

Quizzes are given in each class.

Course schedule/Required learning

  Course schedule Required learning
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 (Proof) Understand the proof of exact linearization.
Class 7 Input-Output Linearization Understand the concept of input-output linearization. Obtain input-output linearization of nonlinear systems.
Class 8 Nonlinear Observer with linear error dynamics Understand the concept of observers with linear error dynamics. Design observer for a limited class of nonlinear systems.


Materials used in class can be found on OCW-i.

Reference books, course materials, etc.

Hassan K. Khalil: Nonlinear Control, Prentice Hall (2014)
Alberto Isidori: Nonlinear Control Systems, Springer; 3rd ed.(1995)

Assessment criteria and methods

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.

Related courses

  • SCE.C201 : Fundamentals of Dynamical Systems
  • SCE.C301 : Linear System Theory

Prerequisites (i.e., required knowledge, skills, courses, etc.)

Students require the basic knowledge of linear system theory: state equation, controllability, observability, state feedback and observer.

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