Nonlinear and Optimal Systems 7.5 Credits

Olinjära och optimala system
Second cycle, R7004E
Course syllabus valid: Autumn 2011 Sp 1 - Present
The version indicates the term and period for which this course syllabus is valid. The most recent version of the course syllabus is shown first.

Syllabus established
The syllabus was established by the Department of Computer Science and Electrical Engineering February 28, 2007 and is valid from Autumn semester 2007.

Last revised
by huvudansvarig utb.ledare vid SRT, Jonny Johansson 04 Feb 2011

Education level
Second cycle
Grade scale
G U 3 4 5
Control Engineering
Subject group (SCB)
Automation Technology

Entry requirements

General entry requirements, second cycle

Specific entry

Courses of at least 90 credits at first cycle including the following knowledge/courses. Intermediate level knowledge in the subject of Automatic control, specifically regarding frequency response, state-space form, and state feedback. Experience with using Matlab for analysis of control systems is also presumed. These prerequisites correspond to the course R7003E.

More information about English language requirements

Course Aim

The student shall be able to:

  • Analyze nonlinear dynamical systems regarding qualitative behavior and stability
  • Analyze nonlinear feedback systems with respect to stability and oscillations
  • Describe control errors and disturbances regarding frequency properties and different size measures
  • Design optimal estimation algorithms and feedback controllers for linear systems
  • Design estimation algorithms and feedback controllers for nonlinear systems
  • Find open loop control strategies for minimizing consumption of resources
  • Implement and commission a control system


Many technical systems, such as industrial processes, robots, vehicles, motors etc. are best described in the form of nonlinear dynamical systems. Methods to analyze these system descriptions are important to be able to e.g. perform measurement and control in these systems. In this course, we will address a number of methods that are available for analyzing nonlinear systems, specifically for the purpose of estimating and controlling quantities. Phase portrait is a tool for graphical illustration of the behavior of a nonlinear system. Elementary methods for drawing phase portraits are introduced. Stability for nonlinear systems is defined and analyzed with Lyapunov functions. For feedback systems, the circle criterion and describing functions are tools for analyzing stability and oscillations. In order to optimize e.g. a control system, one must be able to describe signals quantitatively. Frequency descriptions and size measures, in the form of signal norms, are covered. For linear systems with white, gaussian disturbances, optimal estimation algorithms and controllers can be derived. These are known as Kalman filters and LQG controllers, respectively, and are described thoroughly in the course. For nonlinear systems, the situation is much more complicated but some methods are treated in this course, e.g. the Extended Kalman filter and Model predictive control.

Control strategies without feedback for minimizing consumption of resourses are also treated in the course. A typical example is Goddard's rocket problem wich concerns the question how to vary the engine thrust to reach a given altitude with minimal fuel. Such problem are solved using Pontryagin's maximum principle.

The teaching consists of lectures and problem seminars. Lab work and a project assignment is performed in groups of no more than two students and accounted for with written reports and a demonstration.

Written exam with differentiated grades and approved lab work.

The course will not be given every year.

Transition terms
Sustainable development has been implemented in this course from Autumn semester 2011.

Andreas Johansson

Literature. Valid from Autumn 2011 Sp 1 (May change until 10 weeks before course start)
Glad, T. and L. Ljung: Control Theory. Multivariable and Nonlinear Methods. Taylor & Francis.

Course offered by
Department of Computer Science, Electrical and Space Engineering

0001Written exam4.5G U 3 4 5
0002Laboratory work3.0U G#

Study guidance
Study guidance for the course is to be found in our learning platform Canvas before the course starts. Students applying for single subject courses get more information in the Welcome letter. You will find the learning platform via My LTU.