# Linear Systems Analysis

## Subject description

Introduction. What are systems? Classification of systems.

Analysis of systems in the time domain. Linear differential equations. Solving linear differential equations. System transfer function. Convolution of linear continuous-time systems. Stability of continuous-time linear systems. Routh-Hurwitz stability criterion.

Analysis of systems in the frequency domain. Characteristics of the frequency response function. Bode diagrams. Polar diagrams.

State space approach. State space models. State variables and state vector. General solution of the state equation in the time domain. State transition matrix. State space model and the transfer function. Stability analysis in state space. State space canonical forms.

Controllability and observability.

Analysis of linear electrical circuits.

Applications of systems theory. Examples from biomedicine, optics, engineering, economy, management, etc.

Analysis of biological and optical systems. Mathematical modelling of biological and optical systems. Linear models. Analysis of biological systems in the time and frequency domains. State space analysis of biological systems. Applications of convolution in optics.

## Objectives and competences

The purpose of this course is to provide the students with the basic knowledge and tools of modern linear systems theory in several domains. The students will gain knowledge on modelling, time and frequency domain analysis, state space approach, stability, controllability, and observability and illustrative aapplications of systems theory in optics and biological systems. The students will also be introduced to the computational tools for linear systems theory available in Matlab and Python.

## Teaching and learning methods

The lectures provide a theoretical background on particular subjects together with presentation of simple illustrative examples. Additional examples are presented and discussed in tutorials. Practical work is being carried out within laboratory exercises, where students prepare reports for each assignment.

## Expected study results

After successful completion of the course, students should be able to:

-classify linear systems and describe their basic properties,

-use systemic approach to analysis of linear electric circuits,

-analyze the properties of linear systems in time domain, frequency domain and state space,

-use linear system analysis methods for modelling and evaluation of biological and optical systems,

-use some of the existing Matlab and Python libraries for numerical analysis of linear systems.

## Basic sources and literature

1. Antsaklis P.J., Michel A.N. A Linear Systems Primer, Birkhäuser Boston,  2007
2. Strum R.D., Kirk D.E. Contemporary Linear Systems Using MATLAB, Pws Bookware Companion Series, 1999
3. Gajič Z. Linear Dynamic Systems and Signals, Prentice hall, 2002
4. Hoppensteadt F.C., Peskin C. Modeling and Simulation in Medicine and the Life Sciences, Springer; 2. izdaja, 2004
5. Študijsko gradivo izvajalcev predmeta, predloge predavanj in laboratorijskih vaj

## Stay up to date

University of Ljubljana, Faculty of Electrical Engineering Tržaška cesta 25, 1000 Ljubljana