Balansiranje linearnih dinamičkih sistema i redukcija dimenzije

master's thesis

• Author: Đurđica Blažević
• Mentor: Zlatko Drmač (mentor)
• Committee member: Zlatko Drmač (predsjednik povjerenstva)
• Committee member: Nela Bosner (član povjerenstva)
• Committee member: Mladen Vuković (član povjerenstva)
• Committee member: Tina Bosner (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

Velika potreba za redukcijom modela proizlazi iz industrije, na primjer za razne simulacije. Zbog te potrebe do danas su se razvile mnoge metode za redukciju sistema. Osnova svih je sistem n-tog reda svesti na sistem r-tog reda, gdje je r < n, tako da je reducirani sistem blizu početnog sistema, a da se ne izgube bitna svojstva početnog sistema. Balansirano rezanje je redukcijska metoda koja se temelji na zanemarivanju stanja koja su najmanje upravljiva i osmotriva. Problem loše uvjetovanosti transformacijske matrice može se riješiti Schurovom metodom redukcije modela u kojoj pomoću ortogonalnih matrica dobijemo ekvivalentni sistem. Problem nastaje kod eksplicitnog racunanja umnoška gramijana. Prednosti balansiranog rezanja i Schurove metode spojene su u metodu slobodno balansiranje kvadratnim korjenom. Hankelova norma aproksimacije minimizira normu razlike funkcija prijenosa početnog i reduciranog sistema. Ukoliko nam je bitno samo određeno frekventno područje za redukciju modela primjenjujemo metodu koja koristi težinske matrice. Redukcija sistema koji nisu stabilni također je moguća i zahtjeva odvajanje stabilnog od nestabilnog dijela sistema.

Abstract (english)

A great need for model reduction is due to industrial needs, for example, for various simulations. Therefore, until today there are many methods for the model reduction. The basis of all is system of n-th order reduced to a system of r-th order, where r < n, in such a way that the reduced system is close to the initial system, and not to lose the essential properties of the initial system. Balanced truncation is reduction method based on neglecting of states that are least controllable and observabile. The problem of poorly conditioned transformation matrix can be solved using Schur method of model reduction in which we, with the help of orthogonal matrices, obtain equivalent system of. The problem arises in explicit calculation of product Grammians. Advantages of balanced truncation and Schur method are combined in a balancing - free square root method. Hankel norm approximation minimizes the norm of the difference transfer functions of initial and reduced system. If it is essentially just a a certain frequency range we use a method which uses weighting matrices. It is also possible to reduce the dimension of unstable systems by separating stable of unstable parts of the system.

• Keywords: balansiranje linearnih dinamičkih sistema; redukcija dimenzije; redukcija modela; redukcija sistema; balansirano rezanje; Schurova metoda redukcije modela; slobodno balansiranje kvadratnim korjenom; Hankelova norma aproksimacije
• Keywords (english): balancing linear dynamical systems; dimension reduction; model reduction; system reduction; balanced truncation; Schur method of model reduction; balancing-free square root model; Hankel norm approximation
• Language: croatian
• URN:NBN: urn:nbn:hr:217:275106
• 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: Access restricted to students and staff of home institution
• Created on: 2017-06-06 10:25:17