Kalmanovi filteri

master's thesis

• Author: Mihael Alapić
• Mentor: Igor Velčić (mentor)
• Mentor: Bojan Basrak (mentor)
• Committee member: Igor Velčić (predsjednik povjerenstva)
• Committee member: Bojan Basrak (član povjerenstva)
• Committee member: Mladen Jurak (član povjerenstva)
• Committee member: Miljenko Marušić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2019-03-07, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu opisana je teorija koja stoji iza algoritma Kalmanovog filtera. Kalmanov filter je estimator stanja linearnog dinamičkog sustava popraćen šumovima. Primjenu pronalazi u mnogim područjima kao na primjer u navigaciji i robotici, a korišten je i od strane NASA-e za Apollo projekt. U prvom poglavlju opisan je deterministički sustav promatranog modela, objašnjeni su osnovni pojmovi koji će biti korišteni dalje u radu i definirana su svojstva observabilnosti i kontrolabilnosti sustava. U drugom poglavlju promatran je stohastički linearni sustav i uz pretpostavke modela izvedene su jednadžbe za Kalmanov algoritam u diskretnom i kontinuiranom slučaju. Pronađen je najbolji linearni procjenitelj slučajne varijable stanja te u zadnjem poglavlju su dobiveni rezultati prošireni na nelinearne sustave i samim time je opisan prošireni Kalmanov filter.

Abstract (english)

The subject of this thesis is the theory behind Kalman filter algorithms. Kalman filter is a state estimator of linear dynamic system containing statistical noise. Areas of application are navigation, robotics and even spacecraft technology. NASA has used it for the Apollo project. A description of deterministic system of the observed model is provided in the first chapter. Also, in the first chapter basic terms that will be further used are explained and definitions of system properties - observability and controllability are given. In the second chapter, a stohastic linear system is observed and Kalman algorithm equations for discrete and continuous cases are derived. Also, the best linear state estimator has been found. In the last chapter, the results were expanded into nonlinear systems and so is the description of Extended Kalman filter given.

• Keywords: Kalmanov filter; deterministički sustav promatranog modela; observabilnost sustava; kontrolabilnost sustava; stohastički linearni sustav
• Keywords (english): Kalman filter; deterministic system of the observed model; observability; controllability; stohastic linear system
• Language: croatian
• URN:NBN: urn:nbn:hr:217:848768
• Study programme: Title: Finance and Business Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-07-09 11:49:14