Minimum fazlı olmayan kontrol sistemlerinde sıfır etkisinin emülatör yardımıyla giderilmesi

Abstract

Bu çalışmada, minimum fazlı olmayan sıfırların sistem üzerindeki etkileri incelenmiş ve sistemde emulator kullanarak, bu sıfırların etkisi giderilmeye çalışılmıştır. Ayrıca, emulator parametrelerini bulmak için, Diyofant denkleminin çözümü üzerinde durulmuş ve bu denklemi çözmek için bir algoritma geliştirilmiştir. Yapılan çalışmada, ilk olarak sağ yarı s-düzleminde sıfıra sahip olan sistemlerin davranışı incelenmiş ve bu sıfırların etkisi önce sisteme bir önfiltre eklenerek daha sonra da sıfırın üzerine bir kutup getirilmek suretiyle giderilmeye çalışılmıştır. Simülasyon sonuçlan, sisteme önfiltre eklendiğinde sıfır etkisinin bir miktar giderildiğini, sıfırın üzerine kutup getirildiğinde ise sıfır etkisinin giderilmediğini ve sistemin belli bir süre sonra kararsızlığa gittiğini göstermektedir. Tezin üçüncü bölümünde ise, emulator kavramı ortaya atılmış ve sistemdeki sıfır etkisinin emulator yardımıyla tamamen giderildiği gözlenmiştir. Tezin son kısmında ise, emulator parametrelerini bulduran algoritma bilgisayar ortamında yazdırılmıştır. Ayrıca ikinci ve üçüncü mertebeden olmak üzere iki sistem ele alınarak bu sistemlere ait emulator parametreleri buldurulmuş ve sisteme emulator uygulanması halindeki simülasyon sonuçları verilmiş ve irdelenmiştir.In this study, the effects of non-minimum phase zeros and the cancellation of the zeros have been examined. The systems which have a pole or a zero in the right half-plane are called " non-minimum phase systems ". As it is known, the root-locus always starts from the poles of system and ends in its zeros. The zero which is in the right half plane will cause the root-locus plot move to the right. Because of this, the system will be unstable in some areas. A pole is placed to cancel the zero which causes the system to be unstable in these areas. However, the simulations have shown that the pole which is placed to cancel the zero has not actually canceled it and system has remained to be unstable. Moreover, using a prefilter which is l/{l-(s/b(D0)J in system, the undershoots which are caused by non-minimum phase zeros have been shown that they have decreased to some certain proportion. However, the filter used cannot cancel the effects of the zero totally. In the third part of my thesis, the concept of emulator has been put forward. The dynamic systems which emulate unrealisable operations such as prediction, taking system output derivatives and the cancellation of non-minimum phase zeros are called " emulators ". The emulator is put into a feedback loop within a control system. On the other hand, the difficulty with emulators is that an accurate system model is required before the emulator can be designed. The emulator which cancel non-minimum phase zero can be obtained as well. Let us suppose a system is given by, "" B(s)TT_ C(s)," s Y(s) = ^U(s)+^V(s) (1) where U(s) is the system input signal, Y(s) is the system output signal, V(s) is white noise. Let's define the emulator output signal as follows, ?(s) =Y(s) (2) where P(s) and Z(s) are emulator design polynomials and the polynomial Z(s) is selected to be the same as the polynomial B(s). Let's write equation (1) in equation (2), P(s)B(s) P(s)C(s),Yc,,~ *(s) = ZÖ5ÂCÖ U(s) ~mm V(s) (3) In equation (3) P(s)C(s)/Z(s)A(s) term is divided into realisable and non-realisable parts as follows, P(s)C(s) _ E(s) F(s) Z(s)A(s) Z-(s) Z+(s)A(s) W where Z+(s) is regarded as the realisable part and Z"(s) the non-realisable part. The emulator output is divided into realisable and non-realisable parts. deg (ajj. Hence the reminder tx and the quotient qj is computed. Next, the polynomial a2 is equal to the reminder rt and the polynomial at is divided to the polynomial a2. It is continued until the last remainder become zero. The last non-zero remainder before the zero remainder is the greatest common divisor of the polynomials a0 and ax. As a result, the polynomials (Z+(s)A(s)) and Z~(s) or the polynomials a(s) and b(s) are relatively prime and the GCD of these is "1". Hence, let's solve the following equation. a(s)e(s)+b(s)f(s)=l (11) To compute the polynomials e(s) and f(s), the following formula is utilized with the starting conditions to be w0 = 0, wl = 1, v0 = 1, V! = -qls Wj = Wj-2 Vj = Wi-i 1 Vi-2 Vi_! qi i i=2,3,. (12) XII Hence the polynomials e(s) and f(s) are obtained as follows, wp e(s)= p rP f(s) = ^ (13) where p is the division step which is obtained the last non-zero remainder. After computing the polynomials e(s) and f(s), let's multiply the equation (11) with the polynomial P(s)C(s). Thus, we have obtained a particular solution for the polynomials E(s) and F(s) in the equation (9) as follows, Es(s) = e(s)P(s)C(s) (14) Fö(s) = f(s)P(s)C(s) However, the most important particular solution is the one which minimizes the degree of one of the polynomials, say E(s). This will be called the "minimum degree solution" with respect to E(s) and can be found as follows, Eö(s) = b(s)u(s)+v(s) (15) with deg v