Условна оптимизација дискретних система аутоматског управљања применом потпуне преносне функције

Abstract

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 pronalaze 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 external 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.