Type of course

Admission requirements

A relevant undergraduate Bachelor degree in Engineering program in power electronics and electrical machines. Basic knowledge in power systems is also an advantage.

In addition, the following requirements must be met:

- minimum 25 credits in mathematics (equivalent to Mathematical Methods 1, 2 og 3), 5 credits in statistics and 7,5 ects i physics on a higher level is required.

Application code: 9371

Course content

Dynamic models:

• Mathematical modelling of linear dynamic systems
• Transfer functions and state space representations

Dynamic response for LTI systems:

• Superposition and time invariance
• Poles and zeros, and relation to stability and response
• Routh’s stability criterion with applications for trivial controller design

Feedback control for LTI systems:

• Open and closed loops
• System types and the effect of integral and derivative action
• PID and feedforward control

State space design:

• Canonical forms (CCF, MCF, OCF) and linear transformations
• Controllability and observability
• State feedback control for state space systems
• State estimators for state space systems
• The separation principle

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)

Recommended prerequisites

Objectives of the course

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

Knowledge

The candidate has knowledge of:

• general system configurations and control loops for LTI systems
• basic LTI modelling principles for selected system types, including transfer functions and state space representations
• basic LTI system principles, and the relation between pole placement and stability
• basic system configurations, system types and PID control for LTI systems
• state space models and canonical forms, controllability and observability, state feedback control and estimation, and the separation principle
• 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:

• model physical systems and derive transfer functions and state space models
• determine stability and final value outputs for LTI systems, and perform trivial control design using Routh’s stability criterion
• derive and transform between state space models, canonical forms and transfer functions
• determine controllability and observability for LTI systems
• design state feedback controllers and state estimators for LTI systems
• 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 linear and nonlinear systems (general for all modules)

Language of instruction and examination

Teaching methods

Information to incoming exchange students

This course is open for inbound exchange student who meets the admission requirements. Please see the Admission requirements" section".

Master Level

Do you have questions about this module? Please check the following website to contact the course coordinator for exchange students at the faculty: https://en.uit.no/education/art?p_document_id=510412.