LQR kontrola linearnih dinamičkih sistema

master's thesis

• Author: Katarina Kešinović
• Mentor: Zlatko Drmač (mentor)
• Committee member: Zlatko Drmač (predsjednik povjerenstva)
• Committee member: Nela Bosner (član povjerenstva)
• Committee member: Mladen Vuković (član povjerenstva)
• Committee member: Lavoslav Čaklović (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2014-07-11, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Predmet ovog rada je sustavni pregled problema numeričke matematike koji se javlja u inženjerskim primjenama teorije kontrole i upravljanja. Razrađen je problem postizanja optimalne kontrole nad zadanim objektom koji je prikazan kao linearan sistem jednadžbi: \(\dot{x}(t) = A(t)x(t) + B(t)u(t)\), uz poznate vrijednosti početnih podataka \(t_{o}$ i $x(t_{o})\). Primjeri uključuju linearno- kvadratnu regulaciju spomenutog sistema koja je postignuta minimizacijom pripadnog funkcionala : \(I = \int t_{o}^{T}[x^{T}(t)Q(t)x(t) + u^{T}(t)R(t)u(t)]dt + x^{T}(t)Mx(t)\). Korištene su Riccatijeve matrične jednadžbe i u tom kontekstu je zapisan navedeni funkcional. U okviru rezultata Riccatijeve teorije dani su i dovoljni rezultati za optimalnost rješenja problema LQR kontrole. Pokazano je kako se ti koncepti javljaju npr. pri konstrukciji aktivnog ovjesa automobila (kako bi se prigušile oscilacije pri vožnji po neravnoj cesti) ili kod uređaja koji treba balansirati svoju težinu (držeći se u uspravnom položaju na svoja dva kotača uz dopuštene male oscilacije). Moguće šumove na senzorima uklanjalo se Kalmanovim filtrom. Varirao se izbor matrica Q i R i ovisno o tome pratilo ponašanje sistema na iste ili različite ulazne podatke. Dana je i usporedba sa metodom premještanja polova, na čijem se principu bazira metoda LQR kontrole.

Abstract (english)

The subject of this paper is a systematic overview of numerical techniques in engineer applications of the control theory. The problem of achieving the optimal control of a system of linear equations- \(\dot{x}(t) = A(t)x(t) + B(t)u(t)\) is elaborated. Examples include linearquadratic regulation of the before mentioned system, which is achieved by minimization of the related functional: \(I = \int t_{o}^{T}[x^{T}(t)Q(t)x(t) + u^{T}(t)R(t)u(t)]dt + x^{T}(t)Mx(t)\). Riccati matrix equations were used, and in this context, the mentioned functional is noted. Along with the results of Riccati theory, sufficient results of optimality for solving the problem of LQR control are also provided. It is also illustrated how these concepts appear e.g. in constructing active suspension of cars (in order to dampen the oscillations while driving on bumpy roads) or in appliances that need to regulate their own weight (to maintain an upright position on its two wheels with minute oscillations allowed). Possible buzzes on the sensors were removed by Kalman’s filter. The choice of Q and R matrices was varied and depending on it, system behavior on the same or different input data was monitored. The comparison with the Pole-placement method, which the LQR method is based on, is provided as well.

• Keywords: numerička matematika; teorija kontrole i upravljanja; problem postizanja optimalne kontrole; Riccatijeve matrične jednadžbe; LQR kontrola; Kalmanov filter
• Keywords (english): numerical techniques; control theory; problem of achieving optimal control; Riccati theory; LQR control; Kalman's filter
• Language: croatian
• URN:NBN: urn:nbn:hr:217:772901
• Study programme: Title: Applied 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: 2019-02-08 11:20:41