Tillämpad matematik - Linjära system

**Valid from:** Spring 2023**Decided by:** Maria Sandsten**Date of establishment:** 2022-11-23

**Division:** Mathematics**Course type:** Course given jointly for second and third cycle**The course is also given at second-cycle level with course code:** FMAF10**Teaching language:** Swedish

The aim of the course is to treat some mathematical concepts and methods, above the level of calculus in several variables, that are important for further studies within e.g. image analyis, signal processing, control theory, electrical engineering and for further professional activities.

*Knowledge and Understanding*

For a passing grade the doctoral student must

- be able to describe different properties of linear systems, and to explain how they can be modelled in the time domain and in the frequency domain.
- be able to define the Laplace transform and account for its significance in connection with input/output relations and the solution of differential equations, and be able to use simple transform tables to determine transforms/inverse transforms.
- be able to use matrix theory to analyse quadratic forms and to solve systems of linear differential equations.

*Competences and Skills*

For a passing grade the doctoral student must

- be able to demonstrate the ability to identify problems that can be modelled with linear systems, and be able to analyse the corresponding models.
- be able to demonstrate the abilty to use the introduced concepts in connection with problem solving.
- with proper terminology, suitable notation, and with clear logic be able to explain the solution to a problem in a well structured manner.

Linear systems: Mathematical models of linear, time invariant systems. Transfer function. Step response and impulse response. The frequency function. The Laplace transform: Step and impulse functions. Computational rules for the two-sided Laplace transform. Inverse transforms, in particular of rational functions. Use of transform tables. Convolution. Matrix algebra: Eigenvalues and eigenvectors. Diagonalization, in particular of symmetric matrices. Quadratic forms, diagonalization and classification. Systems of differential equations: solution by diagonalization, solution using exponential matrix.

- Spanne, S. & Sparr, A.: Föreläsningar i Tillämpad matematik, Lineära system. KFS-Sigma, 1996.
- Spanne, S. & Sparr, A.: Övningar i Tillämpad matematik 2, Lineära system.. KFS-Sigma, 1996.

**Types of instruction:** Lectures, laboratory exercises, exercises

**Examination format:** Written exam**Grading scale:** Failed, pass**Examiner:**

**Assumed prior knowledge:** linear algebra, calculus

**Course coordinators:** **Web page:** https://canvas.education.lu.se/courses/20587