Условна оптимизација дискретних система аутоматског управљања применом потпуне преносне функције
Conditional optimization of discrete-time control systems using the full transfer function
Докторанд
Zarić, VladimirМентор
Jovanović, Radiša Ž.Чланови комисије
Lazić, DraganRistanović, Milan
Ribar, Srđan
Pršić, Dragan
Метаподаци
Приказ свих података о дисертацијиСажетак
U okviru doktorske disertacije istrauje se sinteza upravljaqkog sistema
proporcionalno-diferencno-sumarnog (PDS) tipa na osnovu karakteristiqnog
polinoma potpune prenosne funkcije. Potpuna prenosna funkcija omoguava da
se dobiju taqni rezultati u pogledu skraivanja jednakih nula i polova, do-
bijanja karakteristiqnog polinoma i odreivanja potpunog odziva sistema. Na
poqetku se odreuje oblast relativne stabilnosti u prostoru podexljivih para-
metara upravljaqkog sistema tako da stepen priguxenja ζ ima unapred zahtevanu
vrednost. To je uraeno u sluqaju dva podexljiva parametra na primerima PD
i PS upravljaqkih sistema. Osim toga, pronaena je oblast relativne stabilno-
sti u sluqaju tri podexljiva parametra PDS upravljaqkog sistema. U narednom
koraku se sprovodi metoda uslovne optimizacije pri qemu se koristi izraz za
grexku izlazne veliqine koji uzima u obzir istovremeno dejstvo nenultih poqe-
tnih uslova i spoljaxnjeg ulaza, a xto je omogueno primenom potpune prenosne
funkcije. Na taj naqin se prona...laze optimalni parametri linearnih upravlja-
qkih sistema PDS tipa za koje indeks performanse, u obliku sume kvadrata gre-
xaka, ima minimalnu vrednost.
U drugom delu disertacije prethodno razvijena metoda uslovne optimizacije
linearnih diskretnih sistema se proxiruje i primenjuje na klasu nelinearnih
diskretnih sistema, u obliku Takagi-Sugeno (TS) fazi sistema. U tu svrhu
je iskorixena osobina TS fazi sistema da se dinamika nelinearnog sistema
moe izraziti pomou nekoliko linearnih (linearizovanih) sistema. Neline-
arni Takagi-Sugeno fazi model, koji verodostojno opisuje ponaxanje sistema na
celom prostoru izlaza, se dobija interpolacijom nekoliko linearnih matemati-
qkih modela. Izvrxena je sinteza upravljaqkog sistema tipa paralelno raspode-
ljenog upravljanja (PDC) koji koristi iste funkcije pripadnosti kao i nelinearni
Takagi-Sugeno model objekta. Ovakav upravljaqki sistem interpolira nekoliko
lokalnih linearnih upravljaqkih sistema. Zahvaljujui teoriji potpune preno-
sne funkcije, razmatra se najopxtiji i najrealistiqniji sluqaj uslovne opti-
mizacije lokalnih linearnih upravljaqkih sistema, pri qemu je grexka rezultat
istovremenog delovanja nenultih poqetnih uslova i spoljaxnjeg ulaza. Odreeni
su optimalni parametri za tri linearna proporcionalno-sumarna (PS) kontro-
lera pri nultim i nenultim poqetnim uslovima, uvaavajui zahtev da svi po-
jedinaqni zatvoreni sistemi imaju zahtevani stepen priguxenja ζ. Uraena je
sinteza PDC kontrolera koji koristi iste funkcije pripadnosti kao i fazi TS
model objekta, u dva sluqaja. U prvom sluqaju, PDC kontroler je sastavljen od
tri lokalna linearna PS kontrolera qiji su parametri odreeni pri nultim
poqetnim uslovima. U drugom sluqaju, PDC kontroler qine linearni kontroleri
qiji su parametri odreeni pri nenultim poqetnim uslovima.
The doctoral dissertation investigates the synthesis of a proportional-difference-sum (PDS)
type control system based on the characteristic polynomial of the full transfer function. The full
transfer function makes it possible to obtain accurate results in terms of cancellation of equal
zeros and poles, obtaining the characteristic polynomial and determining the full response of
the system. At the beginning, the relative stability area in the space of adjustable parameters
of the control system is determined so that damping coefficient ζ has the value required in
advance. This was done in the case of two adjustable parameters using the examples of PD
and PS control systems. In addition, an area of relative stability was found in the case of
three adjustable parameters of the PDS control system. In the next step, the method of
conditional optimization is implemented using an expression for the output error that includes
the simultaneous influence of nonzero initial conditions and the ex...ternal output, which is made
possible by applying a full transfer function. Thus, the optimal parameters of the PDS type
linear control systems are found for which the performance index, in the form of the sum of
squared errors, has a minimum value.
In the second part of the dissertation, the previously developed method of conditional
optimization of linear discrete systems is extended and applied to the class of nonlinear discrete
systems, in the form of Takagi-Sugeno (TS) fuzzy systems. For this purpose, the property of
the TS fuzzy system was used that the dynamics of a nonlinear system can be expressed using
several linear (linearized) systems. The nonlinear Takagi-Sugeno fuzzy model, which credibly
describes the behavior of the system over the entire output space, is obtained by interpolating
several linear mathematical models. A synthesis of a PDC (parallel distributed compensation)
control system using the same membership functions as the nonlinear Takagi-Sugeno fuzzy
model of the plant was performed. Such control system interpolates several local linear control
systems. Thanks to the theory of the complete transfer function, the most general and realistic
case of conditional optimization of the local linear control systems is considered, where the
output error is the result of the simultaneous action of nonzero initial conditions and external
input. The optimal parameters for three local linear proportional-sum (PS) controllers at zero
and nonzero initial conditions were determined, considering the requirement that all individual
closed systems have the required damping coefficient ζ. Synthesis of a PDC controller that uses
the same membership functions as the fuzzy TS plant model is made in two cases. In the first
case, the PDC controller is composed of three local linear PS controllers whose parameters are
determined at zero initial conditions. In the second case, the PDC controller consists of linear
controllers whose parameters are determined at nonzero initial conditions.