Non-linear Control (Tue, virtual class room)




Responsible Instructor: T.J.J. van den Boom (Ton)

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This course is taught by prof.dr. H. Nijmeijer via telecolleges (video conference).

Control of nonlinear systems deals in the broadest sense with the unavoidable question: given a nonlinear control system, how can I control it? Even though many control systems are designed as simple as possible- that is linear- most often there are intrinsic nonlinearities such as mechanical friction, play, or switching characteristics, airdrag, that make the overall system nonlinear. In addition, most robotic and automotive systems possess geometric nonlinearities, that in their simplest form appear in the translation from joint space to (Euclidean) space. Often controlling a system with nonlinearities is done using linear tools, and depending on the specific application, with success. This is particularly true in case the nonlinear system is linearized around a suitable working point, or, else by aiming at certain robustness –properties in the controlled system. In the course Nonlinear Control attention is focused on techniques that avoid linearization and which thus include the full nonlinearity of the system dynamics. In the course attention is paid to Lyapunov stability, passivity, and feedback stabilization, using Lyapunov design and/or backstepping. Then an extensive study on input-output linearization and input-state feedback linearization is given, in combination with a set of well-chosen classroom examples. The underlying coordinate transformations, that lead to simplified normal forms, are extensively used. In analogy to linear control systems, notions of nonlinear observability and controllability are developed. The notion of nonlinear observability is then exploited in the development of nonlinear observers for autonomous nonlinear systems.

Study Goals:

- Understand the relevance of nonlinear control in the context of control systems
- Understand basics of nonlinear systems
- Create stabilizing controllers for nonlinear systems
- Apply input output linearization techniques for siso and mimo systems
- Understand and apply feedback linearization techniques for nonlinear control systems
- Create normal forms as tools for system analysis
- Understand basics of nonlinear controllability and nonlinear observability
- Understand the role of coordinate transformations and feedback transformations
- Apply Lyapunov techniques for stabilization and tracking
- Create observers for autonomous nonlinear systems

Education Method:

7 weeks independent learning under supervision, 2 hours

7 weeks lecture in 2 blocks of 2 hours per week

Studyload:140 h, Lectures: 28 h, Self-study 32 h, Assignments: 32 h, Preparation of exam :48 h

Literature and Study Materials:

Hassan Khalil, Nonlinear Systems, 3rd edition, Pearson New International Edition

Lecture notes and exercise material


Written exam


3mE Department Delft Center for Systems and Control


prof.dr. H. Nijmeijer
© Copyright Delft Center for Systems and Control, Delft University of Technology, 2017.