Doğru akım makinasının optimal kontrolu

Abstract

ÖZET Bu çalışmada bir doğru akım makinasının,hızı mikroişlemci bazlı bir sistem tarafından gerçek zamanda optimal olarak kontrol edilmiş ve sonuçlar alınmıştır. Kontrolü yapılan doğru akım makinası serbest olarak uyarılmış ve kontrol, rotor ge rilimi üç faz tam dalga kontrollü bir doğrultucu ile değiştirilerek gerçekleştirilmiş tir. Optimal kontrol kuralı belirlenirken, sistemin geçici rejimde (rotor akım ve gerilimine ilişkin sınırlandırmalarda göz önüne alınarak) aşım sız ve minumum za manda istenilen referans değere yerleştirilmesi amaçlanmıştır. Rotor geriliminin de ğişimine ilişkin kontrol kuralı D.. A. makinasına kontrollü doğrultucunun tetikleme açılarının değişimi şeklinde uyarlanmıştır. Kontrollü doğrultucunun giriş kontrol gerilimi (tetikleme açısı) ve çıkış uç ge rilimi arasındaki bağıntının lineer olmayışı, sinüzoidal olarak değişmesi ve çıkış geri lim ifadesinin tetikleme açısına bağımlılığının yükleme durumlarında farklılık gös termesi, aslında lineer yapıda bir statik sürücü için önerilen bu optimal kontrol işle minde bir takım sınırlandırmaları beraberinde getirmiştir. Bununla birlikte optimal kontrol kuralı, geçici rejim ile birlikte sürekli rejim çalışmada da iyi bir performans sağlamakta ve rotor gerilimini; yüke bağlı bir sınır değerde sabit tutarak, hızı sürekli rejimde de optimal olarak kontrol etmektedir. Bu çalışmada D. A. makinasına ait parametrelerin her türlü çalışma şartla rında sabit kaldığı düşüncesiyle optimal kontrol kuralı elde edilmiştir. Oysa makina nın rotor devresine ait direnç, endüktans gibi parametreler ve motor eylemsizlik momenti, çeşitli çalışma koşullarında farklı değerler alabilmektedir. Bu ise bir adap tasyon işlemini gerekli kılmaktadır. Bu yüzden elde edilen teorik sonuçlar ile pratik sonuçlar arasında açık bir farklılık göze çarpmaktadır. Bununla birlikte; algoritmanın basit ve uygulanabilir bir yapıda olması he sap işlemlerindeki kolaylığı ve dolayısıyla hızı arttırmış ve istenilen performans bir takım ihmallerle elde edilmiştir.

SUMMARY OPTIMAL CONTROL OF A SEPARATELY EXCITED DC MACHINE Recent developments in modern control theory and their implementation by using computers, made these two subjects, computers and modern control, approach nearer and especially use of controller units in electromechanical systems cannot be abondoned and must be taken into accont as a whole. In this study,the speed of a separately excited DC machine, is controlled by a microprocessor based on, an on-line optimal speed controller system and the results have been recorded. The DC machine, driven by a controlled rectifier, separately ex cited and control is done with varying the rotor voltage. While determining the optimal control law, in system's transient duration, tracking and minimum settling time is aimed satisfying the constarints on rotor cur rent and voltage. This optimal control law related with the change in the rotor voltage. The optimal control law, thus obtained is used to determine the firing angle of the controlled rectifier. The controlled rectifier not being linear in others words the output voltage re lated to the varying of the firing anlges changing as a sinusoidal function, and this expression related to the firing angle, differing under different loads, indeed this op timal control law proposed for the DC chopper brings some problems. Besides, the optimal control law with transient state duration and steady state work had a good performance and bounded the rotor voltage at the desired value and thus speed is controlled optimally. In this study, with the thought of the parameters of the DC machine are fixed, the optimal control law is obtained. However, the parameter like resistance and in ductance of the rotor circuit of the machine, moment of inertia, can get different valu es under different working conditions. VIAfter all the basic and simple structure of the algorithm, ease of the equations and thus this increased the speed and the desired performance with same neglecti- ons, is obtained. Separately excited DC machine drive system has been used in the industrial field when the wide range and high accurate speed control is necessary. In early cont rol applications of DC machines.conventional proportionol plus integral plus derivati ve type controllers have been widely used to achieve the desired performance charac teristics. Then, the new methods have been proposed to design robust speed control system for separately excited DC machine drive system which attains the optimum performance on the basis of optimal control theory. In this thesis an on line optimal speed control methodwhich is convenient for application to a separately excited DC machine controlled by a microprocessor unit is proposed at the same time the simulation and application results have been recor ded. Application of modern control theory to electrical drive systems has gained importance with the recent advancement in microprocessor technology though their processing speed is still a problem for on-line control. However, once this difficulty iso> vercome with appropriate control procedure and with available measurements of state variables, optimal control offers obvious adventages compared with analog cont rol. First of all we must determine the performance index to find out the optimal control law. The state equations of the seperately excited DC machine can be given in nor malized form as follows: *aB-JL^_5in + 5a-vq dt Tq q Tq Tq q dJl = A_ i O- m dt Th q Th y The basic aim for the system under consideration is the determination of an optimal on-line control law that will minimize the speed error in the steady state and also the duration of the transient state. How.ever, the control law must also satisfy the following constraints: vn-the rotor current must not exceed a prescribed value `iqlSlm - the rotor voltage must be bounded with its nominal value lvql£l To minimize the transient state duration the rotor current and / or voltage are allowed track their upper limits until steady state is reached. The most important step at this point is the determination of the performance index which is chosen as follows to satisfy the above stated aims: Jrtf K - n - ^Q- 2 dt With &i and s2 as the weighting factors the choice of which will give rise to different transient and steady-state performances. The first term of the performance index aims a negligible steady state error in speed. The second term provides the system with the necessary input voltage for the desired speed in the transient state while keeping the current at its maximum allo wable level until steady state is reached. Then comes the determination of the final ti met^ which is the time when transient state is assumed to end and is obviously de pendent on the load torque, my and the reference speed Nr. The speed control, of a DC machine constitutes a linear and split boundery value problem with ToUt^, iq (t^ fixed at the initial time t0 and the costates Xf fixed at the final time tf, which is unspecified. The analytical solution of the problem is carried out by calculating the hamil- tonien consisting of state and costate equations and thus, the optimal control law is obtained as Vq =n + Kq`%TT (n(tf)-Nr)(l-e(t-M/Tq) with the bound expressed by Vqm = Nr + -L my Kq Which is the function ' v~ must track once it is exceeded. vmThe n(tf) and tf terms in optimal control algorithm appear as a problem for on-line control as it is almost impossible to know their exact value in advance for a gi ven speed reference and load torque. However, as the performance index also aims the minimization of the steady-state error in speed, the n(tf)-Nr term can be assumed ze ro thus giving the optimum voltage as vq = n + Im Kq while expression Vqm remains the same. This assumption which is also verified by the simulation results, will obvio usly offer great practical advantages as the calculation of Vq in the optimum voltage expression will require a very short processing time on microprocessor. As to the load torque niy, it may be calculated making use of the mechanical dynamic equation: my = me-Th^ through which the my values will be obtained digitally by with nk+1 and n^ as the present and previous speed values and AT, the sampling peri od. Thus obtained control law can very simply be applied to the DC machine over a DC chopper consisting of transistors. The optimal voltage function is actually the variati on of the average voltage that sould be applied to the DC machine for an optimum transient and steady state performance. But in this study a full wave three phase controlled rectifier is used to change the average voltage of rotor circuit of DC machine. The average voltage expression of three phase full wave controlled rectifier is Vda = VdoCosa Where a is firing angle of thyristors and V^0 maximum value of the output vol tage V^ But this expression is only true for continuous current. For discontinuous current the output voltage expression related to the firing angle is different. There fore application results have only been obtained while the DC machine is loaded (For continuous current) IXThe flow chart of the control program has been established as follows: nk-l <- nk nk-l«- 0 Read the present` speed' Read the rotor current vqm=Nr + ia--iî-BçzB-ı Kq Kq AT Count value = K nk-l«- nk N K=K-1 Read the speed nThe real time optimal control algorithm of the DC machine is given abo ve. Firstly the previous speed value is made zero. Then the present speed and current values are read. After that the boundary voltage value Vqm is calculated. The count value K is determined. (This value defines the difference between the transient and steady state voltage algorithms.) And then vq (transient state algorithm) is cal culated. After the comparison the desired output is determined. Finally the previous speed value is made equal to the present speed and returns. XI