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# FSF3832 Mathematical Systems Theory 7.5 credits

### For course offering

Autumn 2023 Start 30 Oct 2023 programme students

### Target group

PhD students only.

### Part of programme

No information inserted

P2 (7.5 hp)

30 Oct 2023
15 Jan 2024

50%

Normal Daytime

English

KTH Campus

### Number of places

Places are not limited

## Application

### For course offering

Autumn 2023 Start 30 Oct 2023 programme students

51348

## Contact

### For course offering

Autumn 2023 Start 30 Oct 2023 programme students

### Contact

Xiaoming Hu (hu@kth.se)

### Examiner

No information inserted

### Course coordinator

No information inserted

### Teachers

No information inserted
Headings with content from the Course syllabus FSF3832 (Spring 2019–) are denoted with an asterisk ( )

## Content and learning outcomes

### Course contents

Linear control systems: reachability, observability, stability, realization theory, minimality, feedback, pole-assignment, observers. Linear-quadratic optimal control, matrix Riccati equa­tion and theory for the algebraic Riccati equation. Kalman filtering.

### Intended learning outcomes

The overall goal of the course is to provide an understanding of the basic ingredients of linear systems theory and how these are used in analysis and design of control, estimation and filtering systems. In the course we take the state-space approach, which is well suited for efficient control and estimation design.

To pass the course, the student should be able to do the following:

• Analyze the state-space model with respect to minimality, observability, reachability, de­tectability and stabilizability.

• Explain the relationship between input-output (external) models and state-space (internal) models for linear systems and derive such models from the basic principles.

• Derive a minimal state-space model using the Kalman decomposition.

• Use algebraic design methods for state feedback design with pole assignment, and construct stable state observers by pole assignment and analyze the properties of the closed loop sys­tem obtained when the observer and the state feedback are combined to an observer based controller. Apply linear quadratic techniques to derive optimal state feedback controllers. Design a Kalman filter for optimal state estimation of linear systems subject to stochastic disturbances.

• Solve the Riccati equations that appear in optimal control and estimation problems.

• Apply the methods given in the course to solve example problems, including to use the Con­trol System Toolbox in Matlab.

• Integrate the tools learnt during the course and apply them to more complex problems.

• Explain how the above results and methods relate and build on each other.

• Explain the mathematical (mainly linear algebra) foundations of the techniques used in linear systems theory and apply those techniques flexibly to variations of the problems studied in the course.

• Solve fairly simple but realistic control design problems using the methods in the course.

## Literature and preparations

### Specific prerequisites

A Master degree including at least 30 university credits (hp) in in Mathematics (Calculus, Linear algebra, Differential equations and transform method), and further at least 6 hp in Mathematical Statistics, 6 hp in Numerical analysis and 6 hp in Optimization.

### Recommended prerequisites

No information inserted

### Equipment

No information inserted

### Literature

Compendiumfrom the department.

## Examination and completion

If the course is discontinued, students may request to be examined during the following two academic years.

P, F

### Examination

• PRO1 - Project work, 1.5 credits, grading scale: P, F
• TEN1 - Written exam, 6.0 credits, grading scale: P, F

Based on recommendation from KTH’s coordinator for disabilities, the examiner will decide how to adapt an examination for students with documented disability.

The examiner may apply another examination format when re-examining individual students.

Project, Written examination, Exercises.

### Other requirements for final grade

Project, Written examination, Optional homeworks give bonus credits on written examination.

### Opportunity to complete the requirements via supplementary examination

No information inserted

### Opportunity to raise an approved grade via renewed examination

No information inserted

### Ethical approach

• All members of a group are responsible for the group's work.
• In any assessment, every student shall honestly disclose any help received and sources used.
• In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.

## Further information

### Course room in Canvas

Registered students find further information about the implementation of the course in the course room in Canvas. A link to the course room can be found under the tab Studies in the Personal menu at the start of the course.

### Main field of study

This course does not belong to any Main field of study.

Third cycle