Model referans adaptif sistem kullanarak parametre kestirim yöntemi ile konum kontrolu

Abstract

Normal çalışması boyunca parametreleri değişen bir sistemin, kontrol sisteminin tasarımında, adaptif kontrol tekniklerinin kullanılması gerekir. Model referans adaptif sistem bu tekniklerin en yaygın olanıdır. Bu tezde, model referans adaptif sistem yaklaşımıyla serbest uyarlamalı bir doğru akım motorunun konum kontrolü gerçekleştirilmiştir. Referans model olarak düşünülen motorun parametreleri ayarlanabilir model yardımıyla kestirilmiştir. Motorun konum kontrolü için optimal model seçiminde lineer regülatör probleminden yararlanılmıştır. Davranış ölçütünün oluşturulmasında konum hatası ve minimum kontrol gücü dikkate alınmıştır. Daha sonra, parametre kestirim kuralları model referans adaptif sistem yaklaşımıyla çıkartılmıştır. Kurallar, Lyapunuv ikinci kriterini kullanarak çıkartılmıştır. Böylece sistemin kararlılığı güvence altına alınmıştır. Oluşturulan konum kontrolü sisteminin bilgisayarda similasyonu yapılmıştır. Simulasyon sonucunda, ayarlanabilir modelin parametrelerin, referans modelin parametrelerindeki değişimi izlediği görülmüştür.

In this thesis, position control of a D.C motor is achieved with parameter identification using model reference adaptive systems. In a process, because of environmental variations, dynamic parameters of the controlled plant change. Therefore, in order to obtain good results in control, these parameters should be identified. Identification of the dymamic parameters of a process can be formulated as a model reference adaptive problem. The process to identified represents the reference model. The adjustable system is constituted by an adjustable model having the structure of the mathematical model of the process. An adaptation mechanism drives the parameters of the adjustable system. Depending on the configuration of the adjustable model, the signals which are fed in, and how the error between the output of the process and the adjustable model is obtain, there are three basic configuration in identification; output error method, the equation error method, and the input error method. In this thesis first method is used. Basic structure of output error is given in Fig. 1. Consider a model reference adaptive system. The process to be identified; - Ill - xp=Apxp+Bpu, xp(0)=xo (1) reference model noise ?^ adjustable system 1 -x - i output error identification algorithm Fig.l. Output error method. The paralle adjustable model; XM=AMXM+BpU (2) The output error e=xp-xM (3) - IV - Lyapunov function candidate which include output error and parameter error can be defined as V=eTPe + tr {[Ap -AM(e,t)]T Fa-i [Ap -AM(e,t)]} +tr {[Bp -BM(e,t)]TFB-i[Bp-BM(e,t)]} (4) where P, Fa-1 ve Fb-i are positive definite matrices. If Ap is a Hurwita matrix; ApTP+PAP=-Q (5) where Q is an arbitrary positive definite matrix. Therefore P can be computed. Then, the first term will be negative definite for all e ^= 0. If one chooses the adaptation law as AM(e,t)=FA(Pe)xMT (6) BM(e,t)=FB(Pe)uT (7) then the rest of terms will be identically null. This design assures global asymptotic stabilty of the model reference adaptive system. If the structure of the mathematical model has been well chosen and the identifiers performs well, the values of the parameters of the adjustable model will converge to parameters of the controlled plant. Therefore, optimal lineer regulator problem is used in modelling the process. In optimal lineer regulator problem, the aim is to find - V - control vector U(t) which minimizes the quadratic performanB index tf 1 1 J= - XT(tf)HX(tf) + - 2 2 {XT(t)Q(t)X(t)+UT(t)R(t)U(t)}-dt to (8) subject to state model of the plan X(t)=A(t)X(t) +b(t)U(t) (9) The Hamiltonian is defined as 1 1 Ha=-XT(t)Q(t)X(t)+ - UT(t)R(t)U(t)+ AT{A(t)X(t)+b(t)U(t)} 2 2 The optimal control vector D*(t) is obtained as follows; U*(t)=-(R(t))-iBT(t)P(t)X*(t) (10) where P(T) is the solution of Kalman Non-Lineer Riccati type matrix diffrential equation P(t)=-P(t)A(t)-AT(t)P(t)+P(t)b(t)(R(t))-ibT(t) -Q(t') (11) If tf is infinite, system is controllable, the H = 0 and A, b, Q and R matrices are constant then the equation 11 reduces to the algebraic matrix equation; -P(t)A(t)-AT(t)P(t)+P(t)b(t)(R(t))-ibT(t)=Q(t) (12) - VI - ü*(t) B *& X(t) X(t) R-1BTP Fig. 2. Optimal control syetem Va Ra Ki Te + 1-T1' B+sJ W Ki Fig. 3. Block diagram of the D.C motor. Fig. 4. Reduced block diagram of the D.C motor, - VII - Block diagram of the model of the D.C motor and its reduced form are given Fig. 3. and Fig. 4. respectively. It is seen that inductance of the motor is ignored. In Fig. 4. Ki K = (13) Ra-J 1 Ki2 T = - (B+ ) (14) J Ra Ra Tl= - Tl' (15) Ki Since the aim to bring motor shaft perdetermined position with minimum control energy, performans index is choosen as ; 1 J = - 2 J (cc?(t)a+0u(t)2)dt (16) where e(t) = 8r-6(t). By using the results in lineer regulator problem, the optimal control vector is found - VIII - u*=F-X+Y....£ Fıı= f (17) T-B / 2-K*Ka Fi2= (l-/l+ ^ - ) (18) (K-Ka)2 T * |3 f-, Tl Y = (19) Ka The resulting optimal control system is used as the mathematical model. To find adaptation laws, a first order adjustable model is put parallel to mathematical model between Va and Wp. The reference model and adjustable model can be described. Wp= -T-Wp + K-(Va-Tl), reference model (20) Wm= -Tm-Wm + KM-(Va-TlM), adjustable model (21) The output error e = Wm-Wp (22) To converge e -> 0, Tm -> T, Km -> K and T1m -> Tl, following Lyapunov function is choosen - IX - V=cl-e2+c2-(T-TM)2+c3-(KM~K)2+c4-(K-Tl-KMTİM)2 (23) where cl, c2, c3 and c4 are positive constants. Using the second method of Lyapunov stability criteria following adaptation laws are obtained. c3 cl-e 1 Va-TlM TIm = ( + ) Km c4 c3 A computer program which simulates the control system is given Appendix C. The outputs show that when environmental variations occur the parameters of the adjustable system follows the variations in the parameters in the reference model.