# Linear Systems Analysis

## Course 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.

## Learning and teaching 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.

## Intended learning outcomes

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.

## Reference nosilca

1. JEMEC, Jurij, PERNUŠ, Franjo, LIKAR, Boštjan, BÜRMEN, Miran. 2D sub-pixel point spread function measurement using a virtual point-like source. International journal of computer vision, 2017, vol. 121, no. 3, str. 391-402.

2. NAGLIČ, Peter, PERNUŠ, Franjo, LIKAR, Boštjan, BÜRMEN, Miran. Adopting higher-order similarity relations for improved estimation of optical properties from subdiffusive reflectance. Optics letters, 2017, vol. 42, no. 7, str. 1357-1360.

3. NAGLIČ, Peter, PERNUŠ, Franjo, LIKAR, Boštjan Likar, BURMEN, Miran. Limitations of the commonly used simplified laterally uniform optical fiber probe-tissue interface in Monte Carlo simulations of diffuse reflectance. Biomedical Optics Express, 2015, vol. 6, no. 10, str. 3973-3988.

4. BREGAR, Maksimilijan, CUGMAS, Blaž, NAGLIČ, Peter, HARTMANN, Daniela, PERNUŠ, Franjo, LIKAR, Boštjan, BURMEN,  Miran. Properties of contact pressure induced by manually operated fiber-optic probes. Journal of Biomedical Optics, 2015, vol. 20, no. 12, str. 127002.

5. USENIK, Peter, BÜRMEN, Miran, FIDLER, Aleš, PERNUŠ, Franjo, LIKAR, Boštjan. Near-infrared hyperspectral imaging of water evaporation dynamics for early detection of incipient caries. Journal of dentistry, 2014, vol. 42, no. 10, str. 1242-1247.

## Study materials

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

## Bodi na tekočem

Univerza v Ljubljani, Fakulteta za elektrotehniko, Tržaška cesta 25, 1000 Ljubljana