D.A. makinasının parametre kestirimli adaptif optimal kontrolu

Abstract

ÖZET Bir sistemin kontrolü kısaca, girişini değiştirerek çıkışını istenen değere getirme olarak belirtilebilir. Sistemin verilen herhangi bir davranış ölçütünü, optimize edebilen optimal kontrol yöntemleri tercih edilir. Sisteme ait davranış ölçütünün optimize edilebilmesi için sistemin parametrelerinin bilinmesine ihtiyaç vardır. Sistemin parametreleri çeşitli nedenlerle değişebilir ve bu değişmeler çıkışa yansır. Böylece sürekli ve gerçek zamanda davranış ölçütünün optimize edilmesi gerektiği ortaya çıkar. Bunun için, değişen parametrelerin değerlerinin kestirilebilmesi gerekir. Bu çalışmada Model Referans Adaptif Sistem kullanılarak parametre kestirimi yapılmış ve adaptasyon mekanizmasıyla optimal kontrol parametreleri ve dışardan uygulanan giriş büyüklüğü değiştirilerek iyi bir model izleme yaratılmaya çalışılmıştır. Çalışmamızda bir serbest uyarmalı D.A. makinasının yukarıda açıklanan parametre kestirimli adaptif optimal kontrolü yöntemiyle kontrolü incelenmiştir. Kontrol edilecek büyüklük D.C. makinanın hızı olarak alınmıştır. Hızın kontrolunda optimal kontrol modeli seçilmiştir. Hızın verilen bir referans hızı izlemesi istendiğinden lineer izleme problemi model olarak alınmış ve optimize edilecek olan davranış ölçütü kontrol gücünü, omik kayıpları ve motorun hızı ile referans hızı farkını minimuma indirecek şekilde seçilmiştir. Davranış ölçütü ve Hamiltonian denklemleri kullanılarak optimal kontrol parametreleri elde edilmiştir. Daha sonra durum denklemlerinin sonsuzda denge durumuna ulaşacakları kabulüyle dışarıdan uygulanan giriş bulunmuştur. Sistemin nominal değerlerinde çeşitli nedenlerle meydana gelen değişiklikler nedeniyle çıkışta değişiklikler olabilir. Çıkış hatası metodunu kullanarak optimal kontrollü planta ait yeni parametreler ve buradanda plantın parametreleri kestirilebilir. Kestirilen bu yeni parametreler için optimal kontrol parametreleri hesap edilir. Hesap edilen bu optimal kontrol parametreleri ve kestirilen plant parametreleri ile adaptasyon mekanizması plantın optimal kontrol parametrelerini ve dışarıdan uygulanan girişi değiştirir. Böylece plant istenen çıkışlara optimallikten uzaklaşmadan erişir. (vi)

SUMMARY ADAPTIVE OPTIMAL CONTROL WITH PARAMETER ESTIMATION OF A D.C. MACHINE The basic aim of control is to get outputs of the system achieve certain specified objectives with the manipulation of the inputs. There is a vast array of design techniques for generating control strategies when the model of the system is known. When the model is unknown, on-line parameter estimation could be combined with on-line control. This leads to adaptive or self-learning controllers. One can distinguish the following types of control problem in increasing order of difficulty: 1. Deterministic control (when there are no disturbances and the system model is known) 2. Stochastic control (when there are disturbances and model is available for the system and disturbances) 3. Adaptive control (when there may be disturbances and the models are not completely specified) The latter type of control problem will be emphasized in this study. Advanced adaptive control systems attempt to obtain optimum performance automatically. The objective of an optimal control system is to optimise a given criterion of performance. It is necessary to specify explicitly a mathematical model or equation which defines the objective function to be optimized. A number of criteria may be optimized. Optimization of the performance index results in the determination of the values of controlled parameters. The parameters of the systems can change due to different factors and the changes result in an appropriate change in the outputs. So the objective function should be optimized in the on-line real-time mode. To do this we need to estimate the new parameters of the system. In this study the model reference adaptive system (M.R.A.S.) is used to estimate the parameters and by adaptation mechanism to modify the optimization parameters and auxiliary input signal to assure good model following. Output error method (parallel M.R.A.S.) in identification of changes that occur on output because of changes in the parameters and integral adaptation mechanism will be used in this study. It has been aimed to use this control system which is called Adaptive optimal control with parameter estimation, on a separately excited D.C. machine in this study. The state equation of the separately excited D.C. motor is (vii)x(t)=Ax(t)+bu(t)+d To control the speed of the D.C. machine is the aim of this study and in the control of the speed an optimum control system is preferred. Because it is handled as a problem of speed to trace a given reference speed, linear tracking problem is used and the performance index that will be optimized is chosen as to minimize the control power, the ohmic losses and the difference between the speed of motor and a given reference speed. Thus, the performance index tf J=% [xf-xrf]Te[xf-xrf]+VaJo {[x(t)-xp]TQ[x(t)-xr]+u(t)TRu(t)}dt where 0 and Q are real symmetric positive semidefinite matrices and R is a real symmetric positive definite matrix, takes the form tf j=y2J [L2(t) + Q2(t)+u2(t)] dt to where 0, Q, R are chosen as `1 0 0=[1] Q= 0 1 R=[1] and Q = G>(t)-G)r In this study t0=0 and also, because it is decided to have an infinite time control solution tf-><» Using the given performance index, Hamiltonian equations and Kalman infinite time control rules, the optimal control parameters, and thus the optimal control input u*(t) and v optimal state variables x*(t) are obtained. u*(t)=-R`V[Px*(t)+S] x*(t) = [A-bR`1bTP]x*(t) + [d-bR`Vs] where P and S satisfies the following equations 0=-PA+PbR`1bTP-ATP-Q (viii)0=-ATS+PbR`1bTS-Pd+Qxr Later, an auxiliary input r(t) is added to the optimal control input to find the input of optimal controlled system. um(t)=u*(t)+r(t) *m(t) = Amxm(t) + br(t) + dm mw mmw m w m where Am=[A-bR`1bTP] dm=[d-bR`1bTS] It is handled as an infinite time control problem and the state variables are bounded in the infinite time. The auxiliary input can be expressed as a function of reference input and the parameters of the system. r(t)=r=G>r+<pm where (pm is a function of reference input cor and system parameters. *m(t) = Amxm(t) + bo> ` + d+ bra mv ' m mw m r m mTm When variation from nominal values of the parameters of the plant occurs, this results in an appropriate change in the output. The state equation of the motor with changed parameters and from here the state equation of the optimal controlled plant is obtained as *s«=/xs(t)+bsrs+ds where As=[VbPR`VPl s p ds=[dp-bPR`VSl Us = Up* + rS and rs=o>r+cps (ix)Then the state equation of the optimal controlled plant is *s(t)=Asxs(t)+bsur+ds +bm `s s» ' s r s sTs Using output error method M.R.A.S. the new parameters of the optimal controlled plant and from here, the parameters of the plant are estimated. For estimated parameters the new optimal control parameters are calculated. With these optimal control parameters and estimated plant parameters, adaptation mechanism changes the optimal control parameters and the auxiliary input to supply the plant have the desired outputs. This procedure goes on-line and real-time. In this study, a computer simulation of adaptive optimal control with parameter estimation of a D.C. machine is made. The results of the simulation showed that the reference speed or the speed of the reference model which are indeed almost the same, has been traced by the speed of the plant, in a damped wave form. It is same for the parameters of the plant. Using adaptation mechanism which was found from the M.R.A.S. the parameters of the plant are estimated in a damped wave form. Integral adaptation mechanism is used in this study. Also the current of the D.C. machine changes with the change of the parameters, but with M.R.A.S. approximates to a value in a damped wave form. In the change of the given reference speed, the optimal control parameters and the auxiliary input of the model machine change. So the speed of the model machine reaches the given reference speed. Using these new model parameters, the plant also reaches the given reference speed. In the computer simulation study, some numerical solution methods are used. These are the Runge - Kutta method for solving the differential equations and the Newton iteration method for solving the non-linear algebraic equations. Solution of non-linear functions, even on powerful computers can take considerable time. A practical way of solution involves carrying out experiments in realistic workshop conditions to obtain the relevant machining data. This data can subsequently be input to optimisation routines to determine optimal machine parameters under different operating conditions. The resulting optimal values can then be stored in the form of databanks. During actual operations, these values stored in databanks are used to compensate for the changes in the values of process variables thereby making it possible to achieve adaptive control of the machining process. Depending upon the complexity of the system, this adaptive control will involve on-line real-time optimisation or storage of optimal machining data in databanks. In the following pages the scheme of the D.C. machine with adaptive control in the way it has been studied here, and the flow diagram of the computer simulation has been given. (x)PROGRAM FLOW DIAGRAM Start I Give the values of the reference speed o>r, step H, running time T and the parameters of the model machine, plant machine, adaptation mechanizm. Calculate the optimal control parameters by Newton iteration method. Calculate the parameters of reference model and plant, and also auxiliary inputs (pm and <ps. Calculate the state variables (xm,xs) of the. optimal controlled, model and plant by Runge- Kutta method, print their graph. Increase the step (T=T+H). I Yes T= given Stop value No 1 Take the difference (e=x-xj. I By adaptation mechanizm estimate the parameters of the plant. Estimate the parameters of the plant machine. Using these parameters calculate the optimal control parameters and auxiliary input tps. Apply the obtained optimal control parameters and auxiliary input <ps for to the plant. (xi)Figure : The block diagram of adaptive control strategy (xii)