Nonlinear Systems and Control

The course can not be taken as a single course.

Nonlinear system principles:

• Mathematical models for nonlinear systems.
• Piecewise continuity and Lipschitz properties.
• Nonlinear state transformations and diffeomorphisms.
• Equilibrium points.

Two-dimensional systems:

• Phase portraits.
• Classification of equilibrium points using linearization.

Stability of equilibrium points:

• Definitions of equilibrium point stability (stability, instability, asymptotic and exponential stability) and region of attraction.
• Lyapunov functions.
• Positive/negative (semi)definite and radially unbounded time-invariant functions.
• Lyapunov indirect and direct methods for stability analysis.
• The invariance principle.
• Converse theorems.

Time-varying and perturbed system:

• Comparison functions.
• Positive/negative (semi)definite, radially unbounded and decrescent time-varying functions.
• Definitions of equilibrium point stability (uniform stability).
• Lyapunov methods for analysing time-varying nonlinear systems.
• Stability of perturbed systems.
• Brief overview of boundedness, ultimate boundedness and input-to-state stability.

State feedback stabilization:

• Definitions of stabilization (local, regional, global, semi-global, practical).
• Controller design for nonlinear systems (linearization, feedback linearization, integrator backstepping).
• Passivity and zero-state observability.
• Passivity-based control.

Robust state feedback design:

• Model dependency and robustness.
• Robust controller design (sliding mode control).

After completing the course, the candidates will have the following learning outcome:

Knowledge

The candidate has knowledge of:

• mathematical models for nonlinear systems and basic properties (piecewise continuity and Lipschitz, equilibrium points, diffeomorphisms).
• basic principles and classification of equilibrium points for 2D systems, and the notion of stability of equilibrium points.
• stability properties for time-invariant nonlinear systems and the stability hierarchy.
• stability properties for time-varying nonlinear systems and comparison functions.
• stability and stabilization (local, regional, global, semiglobal and practical) and controller design methodologies.
• robustness in controller design.

Skills

The candidate can:

• determine equilibrium points for nonlinear systems.
• determine equilibrium point stability properties for nonlinear systems using linearization.
• determine equilibrium point stability properties for time-invariant nonlinear systems using Lyapunov methods.
• determine equilibrium point stability properties for time-varying nonlinear systems and perturbed systems using Lyapunov methods.
• design controllers using linearization, feedback linearization, integrator backstepping and passivity.
• design robust controller design using sliding mode.

General competence

The candidate has a general competence on:

• application of control engineering.
• choosing suitable model representations for different systems.
• stability analysis with definitions and theorems.
• choosing suitable methods for stability analysis for different systems.
• choosing suitable controller structures and design methods for different systems.
• Use of simulation tools (Matlab/Simulink) for analysis and simulation of nonlinear systems (general for all modules).

Recommended prerequisites

MAT-3800 Linear Algebra II, STE-2605 Linear Systems and Control Theory

Obligatory prerequisites

ELE-3616 Linear systems and control

If you pass the examination in this course, you will get an reduction in credits (as stated below), if you previously have passed the following courses:

• ELE-3606 Control Engineering 5 ects

Language of instruction and examination

More info about the coursework requirements

More info about the oral exam

Re-sit examination