SYSTEMS AND TRANSFORMS | FMA450 |

**Aim**

The aim of the course is to present mathematical concepts and methods from linear algebra and analysis which are important in systems theory, continuous and discrete, and for further studies within e.g. mathematics, economy, physics, mathematical statistics, mechanics, control theory, signal theory and for future professional work. The aim is also to develop the students' ability to solve problems, to assimilate mathematical text and to communicate mathematics.

*Knowledge and understanding*

For a passing grade the student must

be able to describe and use the concepts of linearity, time and space invariance, stability, causality, impulse response and transfer function, in continuous as well as discrete time..

be able to describe the structure of an exponential matrix, and be able to compute exponential matrices in simple cases.

be able to characterize different types of quadratic forms using eigenvalue methods and via a completion of squares.

be able to define the concept of convolution, continuous and discrete, and to use convolutions in the context of systems and in the description of certain types of integral equations.

have some experience and understanding of mathematical and numerical software.

*Skills and abilities*

For a passing grade the student must

be able to show capability to use eigenvalue techniques, elementary distribution theory, function theory, Fourier and Laplace transforms and convolutions in problem solving within the theory of linear systems.

in connection with problem solving be able to show capability to integrate knowledge from the different parts of the course.

with proper terminology, in a well structured way and with clear logic be able to explain the solution to mathematical problems in the course.

**Contents***Linear algebra*: Spectral theory, quadratic forms.

*Systems of linear differential equations:* Equations in state form. Solution via diagonalization. Stability. Stationary solutions and transients. Solution via exponential matrix.

*Input/output relations:* Linearity, time and space invariance, stability, causality. Convolutions. Elementary distribution theory. Transfer and frequency function. Discrete systems.

*Fourier analysis*: The Laplace and Fourier transforms. Inversion formulae, the convolution theorem and Parseval's formula. Transform theory and analytic functions. Applications in differential equations and systems of differential equations.

**Literature**

Spanne, S: Lineära system. KF-Sigma 1997.