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Doğru akım makinasının adaptif ve optimal kontrolunun pratik gerçeklenmesi

Practical implementation of optimal model reference adaptive control of direct current machine

  1. Tez No: 14352
  2. Yazar: VEDAT DEVECİ
  3. Danışmanlar: PROF.DR. M. KEMAL SARIOĞLU
  4. Tez Türü: Yüksek Lisans
  5. Konular: Elektrik ve Elektronik Mühendisliği, Electrical and Electronics Engineering
  6. Anahtar Kelimeler: Belirtilmemiş.
  7. Yıl: 1991
  8. Dil: Türkçe
  9. Üniversite: İstanbul Teknik Üniversitesi
  10. Enstitü: Fen Bilimleri Enstitüsü
  11. Ana Bilim Dalı: Belirtilmemiş.
  12. Bilim Dalı: Belirtilmemiş.
  13. Sayfa Sayısı: 62

Özet

ÖZET Bu çalışmada, serbest uyarmalı bir doğru akım moto runun model referans adaptif ve optimal kontrolü, kontrolör olarak bir bilgisayar kullanılarak gerçek leştirilmiştir. Değişik şartlar altında çalıştırılan motorun, kararlılığı garanti altına alan bir adaptif algoritma ile optimal model gibi davranması sağlanmış tır. Birinci bölümde adaptıf kontrol kavramı tanıtılmış, bir liner model izleme sistemin Lyapunov tasarımı yapılarak pratik çalışmada kullanılacak algoritma çıkarılmıştır. ikinci bölümde doğru akım motoru için en uygun çalışma yörüngesi olan optimal model çıkarılmıştır. üçüncü bölümde de kıyıcılar tanıtılmış, optimal modelin ayrık karşılığı bulunmuş ve plantın; yükün, re-ferans geriliminin, motorun armatürüne eklenen direnç değerlerinin değişimleri durumlarında model gibi davranması sağlanmış ve elde edilen gerçek zaman so nuçları grafik olarak verilmiştir.

Özet (Çeviri)

SUMMARY PRACTICAL IMPLEMENTATION OF OPTIMAL MODEL REFERENCE ADAPTIVE CONTROL OF DIRECT CURRENT MACHINE In this work, the optimal model reference adaptive control o-f the separately excited DC machine operat ing wide range change o-f system parameters and load variations is considered. Adaptive control is an im portant area of modern control dealing as it does with the control of si stems in the presence of uncertainti es, structural perturbations and enviromental varia tions. Adaptive control techniques have also benefited from the steady and even spectacular reduction in the cost / performance ratio of microelectronic devices in recent years. This has resulted in a wide variety of industrial applications in situations which were not considered easily implementable early. Furthermore, many theorectical problems that had baffled reseat - cher s have also been solved during the past few years. Several adaptive control strategies have been succes- fully applied in diverse practical problem. In recent years model reference adaptive control has become a very efficient and systematic method for controlling plants with unknown (or par tially known ) parameters. In this procedure the ob jective of contol are specified in terms a reference model and the design problem involves the determination, from all available parameters or the control input such that the error between the VIplant and model outputs aproaches zero asymptotically. Chapter one covers motivation, de-finitions and classif ations of adaptive control and Lyapunov design of model reference adaptive control or. In chapter two, optimal model reference adaptive control o-f separately excited DC machine is consider - ed. In chapter three, practical implemantation o-f op timal model reference adaptive control of separetely excited DC machine operating wide range change of system parameters and load variations is presented, the matematical model of a separately excided DC mach ine have been established in state-space. In this model, the armature current and the rotational speed of the machine have been considered as the state vari ables while the armature voltage and load torque on the motor shaft have been defined as the control input and as disturbance respectively. In order to achive the optimal performance, a quad ratic index of performance has been defined in terms of state variables and the control input. Using this above mentioned machine model and the defined index of performance, optimal linear regulator problem has been solved for a given set of machine parameters as if the load is an unknown disturbance. Since the load torque actually is an unknown variable, the realization of the the optimal control law necessitates the estima tion of the load torque on the motor shaft. In order to meet this requirement, first, the torque of electic origine is determined in terms of measured state variables; then, taking into consideration of equta- t i on of rotational motoin the instantanous load torque VI ion the motor shaft is estimated. The use of the es timated load torque in the optimal model of the con trol system is developed. This optimal model created arti-fically is consedered as the refer ance model which specifies the desired performance of real plant com prising the separately excited DC machine. Appliying the desing teqnices of the model reference approach of adaptive control theory, the adaptation rules which force the real system to track the optimal model sys tem, are derived. Matematical model of separately excited dc machi ne in the state space can be given as follows, r d dt W -R* -Gr-I FlF La La J Tt_ Ia Va W IF Ra La GF : Armature current : Armature voltage i Angular velocity i constant field current i Armature resistance : Armature inductance speed voltage coefficient Tı_îLoad torque on motor shaft J s Moment of inertia on motor shaft A: Viscous friction cofficient on motor shaft (1) the above given mathematical model can also be written in a more genaral forms as X= A X +B U +D TL (2) where X=CIa W3t D=CO -l/j3T TL are the state and disturbance vectors respectively, U=Va is control vector, A and B the parameter matrices of the machine VI 11de-fined fay A= -R, -GFI F*F the purpose of the optimal control of DC machine is to generate the control input which will give rise to minimun power, minimum electrical and mechanical los ses and which will minimize a desired performance measure under all loading condition and parameter variation. It is well known -from the optimal control theory that the mathematical -formulation o-f this prob- 1 em is: Minimize 1 1 1= XT(tF) H X(tF) + - 2 2 JO EXT Q x + UT R U] dt (4) subject to X= A X +B U +D Tl. (5) Where H and Q positive semi definite symmetrical matr ices and R is positive definite symmetrical matrices and in the problem under consideration there is only equality constraint given by (2). The boundary condi tions Xo,t0»tF &rB specified and X(t|=-) is free. The solution of this problem is obtained by using Hami- ltonyen. HA= CXT Q X +UT R U: + p CA X +B U +D TL3 (6) 3H« X = =AX + BU + DTl (7) IXp = =-Q X - AT p +Z(t) (10) then subsitution into above equation, and algebric manipulation and simplication give K + KA +ATK - KBR-»BTK + Q=0 (11) Z + (AT - KBR~*BT) Z + KDT,_ =0 (12) and boundary condition is H X(tF) = K(tF) X(tF) +Z(tF) (13) the equation (11) is a matrix Riccatti equation which can be solved when K(tp-) is known. The equation (12) is linear time varying differantial equation in which Z is response o-f adjoint system to the distur bance D Tt_. When Z TL is known and Z(tF) is specified Z(t) can be determined. Since D T|_ actually known it can be estimated according to the -Following equation TLE=-J dW/dt -jSW + 6FlFrIA (14) By measuring IA, W and computig W give the estimated value o-f load torque Ti_ and the disturbance D Tı_. The solution o-f equtation (11) and (12) can be achiev ed -for -finite time t|=- and in-finite time tF. For -finite time solution since X(t*=) is not specifi ed, to determine K(t»=) and Z (tF) make it necessary totake K(tF)=H and Z(tFr)=0. For in-finite time solu tion, since the system is completely controllable and A, B, D,H=0,Q,R are constant matrices K must be equal to zero. Moreover, in order to obtain a bounded con trol input Z must be taken zero -For constant touque T,_..from the solution o-f the equation (11), (12) the -following control law -for -finite time is obtained U-=-R-1 BT K(t) X(t) -R-» BT Z(t) (15) For in-finite time solution, since (11) and (12) reduces the algebraic equations, one can write, KA+ATK-KB R-*BT + Q =0 (16) ( AT -K B R-1 BT) Z +K D TL =0 (17) Which give rise constant values -for K, Z- Thus the optimal model can be writen as -follows. X =(A-BR-*BTK)X+Br-B R~* BT Z +D TLE (18) where r is reference input. In the adaptation strategy, the paralel model re-fe- rece adaptive control will be used. The block diagram o-f adaptive control system is shown in -figure 1. XIOPTİMAL RERERANS MODEL JHKui..» I- nfc» PLANT K,- Where AM=A,»-Bf»R-lBF»TK UM=r-R-lBp.xZ 6^=6,. Af>, Bf» actual system parameter matrices with variable elements and D TLE is given by (12). Since the para meters o-f the actual system subjected to internal or external disturbances differ -from its rated parame ters, the armature voltage must be generated by a con- venent adaptation mechanism so that the dc machine with variable load torque and variale parameters, follows the optimal model. For this purpose, let the generalized state error and adaptive control law be Xllde-Fined as e=XM-X,> (21) U,»=-K*»(e,t) XR+Ku UM (22) in order to determine the adaptation strategy the well known Lyapunov or hiperstabi lity approach is used and the -following adaptive law is obtained. KF>(e,t) =- P NBMTPeXp.Tdir -NPBr,TPBXPT +KP(0) (23) Jo t Ku(e,t) = f MBMxPeUMTd,Nr» are at least symmetrical semi definite positive matrices. K»=.(0), Ku(0) are the controller gains obtained -From the rated plant parameters -for perfect model following. Moreover P has to satis-Fy Lyapunov equation written as - Q=A"T P +P AM, Q =QT >0 (25) Q is the symmetrical positive de-Finite matrix. In dijital control of continuous-time plants, we need to convert continues time state space equation into discrete time state space equations. Such con- vertion can be done by introducing fictitious sampler an fictitious holding devices into continuous time system. For infinite time solution of equation; (22), (23), (18) must be discretized and this discretized equa tions can be used in the equation (22) for obtaining discretezed control input uV. Block diagram of com puter control of a plant is given below. The output of the plant is a continues time signal. This signal is converted into dijital form by the sampler and hold circuit and the analog to dijital converter. The dijital computer processes the sequence of numbers by means of an algorithm and produces a new sequence of Killnumber. This number must be converted to a physical control signal. The dijital to analog converter and hold circuit convert the sequence o-f number into a piecewise continues-time signal. input -9 S/H and A/D Dijital Computer D/A 3|and HOLD Filter £ Disturbance or noi se _L Actuator PLANT t9- noise Transduser or Sensor Fig. 2 Block diagram o-f a dijital control system [IV

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