Dynamical system theory for engineers

COM-502 / 6 credits

Teacher: Thiran Patrick

Language: English

Remark: Cours biennal

Summary

Linear and nonlinear dynamical systems are found in all fields of science and engineering. After a short review of linear system theory, the class will explain and develop the main tools for the qualitative analysis of nonlinear systems, both in discrete-time and continuous-time.

Content

  • Introduction: Dynamics of linear and non linear systems. Definitions; Unicity of a solution; Limit Sets, Attractors.
  • Linear Systems: Solutions; Stability of autonomous systems, Geometrical analysis, connection with frequency domain analysis.
  • Nonlinear Systems: Solutions; Examples. Large-scale notions of stability (Lyapunov functions). Hamiltonian systems, gradient systems. Small-scale notions of stability (Linearization; stability and basin of attraction of an equilibrium point, stability of periodic solutions, Floquet Multipliers). Graphical methods for the analysis of low-dimensional systems. Introduction to structural stability, Bifurcation theory. Introduction to chaotic systems (Lyapunov exponents); time permitting: a review of tools of measure theory to compute Lyapunov exponents.
  • The class is methodology-driven. It may present some limited examples of applications, but it is not application-driven.

Keywords

Dynamical Systems, Attractors, Equilibrium point, Limit Cycles, Stability, Lyapunov Functions, Bifurcations, Lyapunov exponents.

Learning Prerequisites

Required courses

  • Linear algebra (MATH 111 or equivalent).
  • Analysis I, II, III (MATH 101, 106, 203 or equivalent).
  • Circuits & Systems II (EE 205 or equivalent) or a Systems & Signals class (MICRO 310/311 or equivalent).

Recommended courses

  • A first-year Probabilty class, such as MATH-232, MATH-231, MATH-234(b), MATH-234(c), or equivalent.
  • Analysis IV (MATH 207 or equivalent)

Important concepts to start the course

  • Linear Algebra (vector spaces, matrix operations, including inversion and eigendecomposition).
  • Calculus (linear ordinary differential equations; Fourier, Laplace and z-Transforms).
  • Basic notions of topology.
  • Basic notions of probability.

Learning Outcomes

By the end of the course, the student must be able to:

  • Analyze a linear or nonlinear dynamical system.
  • Anticipate the asymptotic behavior of a dynamical system.
  • Assess / Evaluate the stability of a dynamical system.
  • Identify the type of solutions of a dynamical sytem.
  • Analyze a linear or nonlinear dynamical system
  • Anticipate the asymptotic behavior of a dynamical system
  • Assess / Evaluate the stability of a dynamical system
  • Identify the type of solutions of a dynamical system

Teaching methods

  • Lectures (blackboard), 2h per week
  • Exercise session, 1h per week

Expected student activities

Exercises in class and at home (paper and pencil, and Matlab)

Assessment methods

  1. Mid-term 20% (conditionally on the Covid situation allowing for it to be taken at EPFL).
  2. Final exam 80%

Supervision

Office hours Yes
Assistants Yes
Forum Yes

Resources

Bibliography

Course notes; textbooks given as reference on the moodle page of the course.

Notes/Handbook

Course notes, exercises and solutions provided on the moodle page of the course.

Moodle Link

Prerequisite for

Classes using methods from dynamical systems.

In the programs

  • Semester: Spring
  • Exam form: Written (summer session)
  • Subject examined: Dynamical system theory for engineers
  • Courses: 3 Hour(s) per week x 14 weeks
  • Exercises: 1 Hour(s) per week x 14 weeks
  • Type: optional
  • Semester: Spring
  • Exam form: Written (summer session)
  • Subject examined: Dynamical system theory for engineers
  • Courses: 3 Hour(s) per week x 14 weeks
  • Exercises: 1 Hour(s) per week x 14 weeks
  • Type: optional
  • Semester: Spring
  • Exam form: Written (summer session)
  • Subject examined: Dynamical system theory for engineers
  • Courses: 3 Hour(s) per week x 14 weeks
  • Exercises: 1 Hour(s) per week x 14 weeks
  • Type: optional
  • Semester: Spring
  • Exam form: Written (summer session)
  • Subject examined: Dynamical system theory for engineers
  • Courses: 3 Hour(s) per week x 14 weeks
  • Exercises: 1 Hour(s) per week x 14 weeks
  • Type: optional
  • Semester: Spring
  • Exam form: Written (summer session)
  • Subject examined: Dynamical system theory for engineers
  • Courses: 3 Hour(s) per week x 14 weeks
  • Exercises: 1 Hour(s) per week x 14 weeks
  • Type: optional
  • Semester: Spring
  • Exam form: Written (summer session)
  • Subject examined: Dynamical system theory for engineers
  • Courses: 3 Hour(s) per week x 14 weeks
  • Exercises: 1 Hour(s) per week x 14 weeks
  • Type: optional
  • Semester: Spring
  • Exam form: Written (summer session)
  • Subject examined: Dynamical system theory for engineers
  • Courses: 3 Hour(s) per week x 14 weeks
  • Exercises: 1 Hour(s) per week x 14 weeks
  • Type: optional
  • Semester: Spring
  • Exam form: Written (summer session)
  • Subject examined: Dynamical system theory for engineers
  • Courses: 3 Hour(s) per week x 14 weeks
  • Exercises: 1 Hour(s) per week x 14 weeks
  • Type: optional
  • Semester: Spring
  • Exam form: Written (summer session)
  • Subject examined: Dynamical system theory for engineers
  • Courses: 3 Hour(s) per week x 14 weeks
  • Exercises: 1 Hour(s) per week x 14 weeks
  • Type: optional
  • Semester: Spring
  • Exam form: Written (summer session)
  • Subject examined: Dynamical system theory for engineers
  • Courses: 3 Hour(s) per week x 14 weeks
  • Exercises: 1 Hour(s) per week x 14 weeks
  • Type: optional
  • Semester: Spring
  • Exam form: Written (summer session)
  • Subject examined: Dynamical system theory for engineers
  • Courses: 3 Hour(s) per week x 14 weeks
  • Exercises: 1 Hour(s) per week x 14 weeks
  • Type: optional
  • Semester: Spring
  • Exam form: Written (summer session)
  • Subject examined: Dynamical system theory for engineers
  • Courses: 3 Hour(s) per week x 14 weeks
  • Exercises: 1 Hour(s) per week x 14 weeks
  • Type: optional
  • Semester: Spring
  • Exam form: Written (summer session)
  • Subject examined: Dynamical system theory for engineers
  • Courses: 3 Hour(s) per week x 14 weeks
  • Exercises: 1 Hour(s) per week x 14 weeks
  • Type: optional
  • Semester: Spring
  • Exam form: Written (summer session)
  • Subject examined: Dynamical system theory for engineers
  • Courses: 3 Hour(s) per week x 14 weeks
  • Exercises: 1 Hour(s) per week x 14 weeks
  • Type: optional
  • Semester: Spring
  • Exam form: Written (summer session)
  • Subject examined: Dynamical system theory for engineers
  • Courses: 3 Hour(s) per week x 14 weeks
  • Exercises: 1 Hour(s) per week x 14 weeks
  • Type: optional
  • Semester: Spring
  • Exam form: Written (summer session)
  • Subject examined: Dynamical system theory for engineers
  • Courses: 3 Hour(s) per week x 14 weeks
  • Exercises: 1 Hour(s) per week x 14 weeks
  • Type: optional
  • Exam form: Written (summer session)
  • Subject examined: Dynamical system theory for engineers
  • Courses: 3 Hour(s) per week x 14 weeks
  • Exercises: 1 Hour(s) per week x 14 weeks
  • Type: optional
  • Semester: Spring
  • Exam form: Written (summer session)
  • Subject examined: Dynamical system theory for engineers
  • Courses: 3 Hour(s) per week x 14 weeks
  • Exercises: 1 Hour(s) per week x 14 weeks
  • Type: optional

Reference week

Wednesday, 8h - 11h: Lecture ELA1

Wednesday, 11h - 12h: Exercise, TP ELA1

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