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Asenkron motorlar için algılayıcısız akı gözlemleyicisi ve kontrolü

Sensorless flux observer and controller for asynchronous motors

  1. Tez No: 68872
  2. Yazar: COŞKUN ŞAHİN
  3. Danışmanlar: DOÇ. DR. METİN GÖKAŞAN
  4. Tez Türü: Doktora
  5. Konular: Bilgisayar Mühendisliği Bilimleri-Bilgisayar ve Kontrol, Computer Engineering and Computer Science and Control
  6. Anahtar Kelimeler: Akı gözlemleyicisi, Asenkron motorlar, Motorlar, Flux observer, Induction motors, Motors
  7. Yıl: 1997
  8. Dil: Türkçe
  9. Üniversite: İstanbul Teknik Üniversitesi
  10. Enstitü: Fen Bilimleri Enstitüsü
  11. Ana Bilim Dalı: Bilgisayar Mühendisliği Ana Bilim Dalı
  12. Bilim Dalı: Belirtilmemiş.
  13. Sayfa Sayısı: Belirtilmemiş.

Özet

ÖZET Bu tezde, asenkron motorlar için, çatırtısız kayan kipli kontrol tabanlı dayanıklı akı gözlemleyicisi geliştirilmiştir, özellikle, geniş çalışma aralığı ve ağır çalışma koşullarında hassas kontrol amaçlandığında, akı gözlemleyicisinin dayanıklılığı (robustness) çok önemlidir. Bu tezde sağlanan en önemli katkı, rotor akı gözlemleme modeline eklenen yakınsama terimleridir. Bu terimlerin eklenmesi ile gözlemleme hatalarının sıfıra yakınsayarak orada kalması sağlanmıştır. Sonuçta parametre, hız, yük gibi tüm etkilere karşı dayanıklı bir gözlemleyici elde edilmiştir. Sistem, ortak endüktans, rotor, stator direnç ve öz endüktans değişimleri, değişken yük ve bozucu moment şartlan altında incelenmiştir. Geliştirilen yöntemlerin sonuçlarının başarılı olduğu ve dayanıklı bir gözlemleyici ve kontrol sistemi elde edildiği benzeşim ve uygulama sonuçlan ile gösterilmiştir. Ayrıca hız, konum ve alam kontrol çevrimlerine de oldukça yeni geliştirilen çatırtısız kayan kipli kontrol yaklaşımı uygulanmış ve çok başarılı sonuçlar elde edilmiştir. Sonuçta dayanıklı, kararlı ve dalgalanmasız moment ve hız kontrolü sağlayabilen, yüksek cevap hızı ve hassasiyete sahip bir kontrol sistemi elde edilmiştir. Asenkron motorlar için zor sayılan sıfır hız ve civarında da, yüksek hızlarda da aynı başarımlar elde edilmiştir. xiv

Özet (Çeviri)

SUMMARY Abstract A robust flux observer and controller for induction motors is proposed based on sliding mode theory. Robustness of the flux observer is a very important issue, especially when it is aimed for use at wide working range and demanding operation conditions. in this study, the velocity, position and current controllers are also based on sliding mode. The vvhole system is analyzed under rotor, stator resistance, mutual and şelf inductance uncertainties, torque variations and external disturbances. The performance of the proposed control scheme is confirmed by simulation and experimental results. l Introduction Fully digital controlled AÇ drives are extensively used in industry. Many fıeld orientation methods are used in these drives. in these high performance induction motor (asynchronous motor) applications, parameter variations and external disturbances can be effective. Especially when it is aimed to track a certain trajectory, robust control techniques have to be used in the observer and in the velocity, position control loops. The good properties of Sliding Mode Control (SMC) in this respect are well known. The chattering problem inherent to SMC is eliminated in recently developed approaches. in this study, the design of the observer is different from methods proposed until now. The flux model is used in polar form and convergence terms are added to the rotor model. Additionally, the modifıed version of the SMC is used for chattering free control. The position, velocity and current controllers are also based on SMC. Robustness of the system is considered under dynamic load and fast reference signal, same time parameter variation and torque disturbances. 1.1 Proposed Control System in proposed control system, a flux observer is used for the calculation of the flux magnitude, angle and stator current components at rotating coordinate frame (Fig.l). The currents are controlled by the sliding mode based control system (SMC PWM). The flux reference value is selected from velocity operation point for maximal torque output. The flux and position are controlled by chattering free sliding mode algorithms in the outer loop. XVrh l*tf N)ijd4-*-T^ ^FhKCbntjL^.w w ^s j 5Js*.*. SMC“^T 9 (İM tXe) Aİqd PWM -: :i ^^ _jjBos.Gont VW ^ ~V-1^ ^ [0J^J> w ql- FUUX 1OBSERVERl-dıat ^Figüre l Proposed control method Superscript 'd' is used for desired value. The other symbols indicate actual values. 1.2 Chattering free sliding mode control The Variable Structure System (VSS) theory has been applied to nonlinear processes. Öne of the main features of this approach is that öne only needs to drive the error to a ”switching surface“, after which the system is in ”sliding mode“ and robust against modeling uncertainties and/or disturbances. Let us consider the plant x = A(x,f)+B(x,j).uM(1) with rank(B)=m, x ? K”, u e K“. in VSS control, the goal is to keep the system motion on the manifold 5 which is defîned as S=£ | sfrt) = 0}(2) The solution to achieve this goal can be calculated from the requirement that sfat)=0 is stable. The control should be chosen such that the candidate Lyapunov fonction satisfies Lyapunov stability criteria. The aim is to force the system states to the sliding surface defîned by 5=G.(xd-x)(3) Firstly, a candidate Lyapunov fünction is selected as v = /.s/2 > O and v = sr.s <0(4) xviit is aimed that the derivative of the Lyapunov fimction is negative definite. This can be assured if we can somehovv make sure that v = -sT.D.s<0(5) where D is positive definite. Therefore (4) and (5) satisfy the Lyapunov conditions. From these equations s= -D.s.(6) Equating (6) to zero results in what is known as ”equivalent control“. in other words, the control that makes the derivative of the sliding function equal to zero is called equivalent control. Derivative of (3) G.xd-G.(A(M) + B.ueJ =0(7) As a result, the equivalent control can be vvritten in the following form uf,=-(G.B)-'.G.(AOM)-i”)(8) From derivative of (3) and using (8) 5 = (G.B).(ue,-u)(9) is obtained. The equivalent control can also be vvritten as given below u^(o = u(o + (G.B)“1.i(10) By using the definition given by (1) and (3) in (6) G.(xd-x) = G.(xd-A(x,0-B(x).u) =-D.s(11) the control is obtained as u = (G.B)”'.(G.(X“ - A(X,O)+AS)(12) Using the equation (8) for the equivalent control can be vvritten as U(o = u^co + (G.B)”l.D.*(13) By looking at (13) an estimation for ueij can be made using the property that u(t) is continuous and can not change too much in a short time as given below xviiueg(t) = U(t--n + (G.B)~\s(14) where T1 is a short delay time. This estimation is also consistent with the logic that ueq isselectedastheaverageofu. Bysubstitutingthe(14)into(13) U(/) = U(»-r) + (G.B)“1.(Z).5 + j)(15) By using Euler interpolation, we get the last form for the discrete controller (G B)~' u« = u(, - T) + [ ' '.((D.T+ l).j(o - sı, - n)(16) 2 Model of Induction Machine and Load The state equation of the induction motor viewed from the stator frame (a, ft frame) driven by voltage source inverter is given by (17-20) RrRr.M. Va = -~^V« -°>-Vfl + ~[^l«(17> RrRr.M Vp =-rVp+<».¥«+-T-ip(18) jLrLir ^^{T^'^^-^^”}(19) <--3:{f(rr'-“-»'0-*-''*”'}(20) where subscript symbols 'ör' and '^“are used for stotionary axis elements the variables of the equations are i : Stator currentu : Stator voltage y/ : Flux componentû) : Angular velocity and parameters of the equations are RE :R, + Rr.M2 /L,2 Rn Rs : Rotor and stator resistanceL, J.s : Rotor and stator inductance, M : Mutual inductance<r : Leakage factor The control variables of the outer loop are the flux and torque of the motor, it is convenient to introduce the model on the rotor flux frame (d,q frame) that can be achieved by the transformation as presented in (21) FcosCö) sin(ö)l m-[-Sm(0) COS(0)\(21) xvüiThe transformation angle 0 between two frames is given by (22) and the flux magnitude is given by (23) â M-Rr ? 0z= CO-i 1 Lr.\¥r\ ”(22) Rr ^\ = J-r(MJ“~^r\) (23) (22) and (23) construct the polar form of the rotor model. The electrical torque of the motor can be written as (24) r Mx v (24) This torque should be equal to the load torque given as in (25) Te = Tm=jL.q + Bi.q + G{q) + Td (25) From (25), the acceleration can be written as q = -(Te-(Bi.q + G(q) + Td)) (26) All smooth varying or limited amplitude torque components are summed up in the term T/' and the electrical torque '7V defined in (24) is substituted into (27) to obtain ''T&rM-n (27) where subscript symbols 'tf and V are used for direct and quadrature axis elements in rotor frame The variables of the equations are Te : Electrical torque Tj : External torque disturbance q : Rotor position q, O) : Acceleration and parameters of the equations are BL : Viscous friction of load plus motor Tm : Mechanical torque G(q) : Gravity effect q, û) : Angular velocity | ¥fl : Rotor flux magnitude : Load plus motor inertia XIX3 Observer Design Let us define the observer model as A (&?}”,“,>”Rr.M- yfa = -\ - --/j\.yfa + [û)-Tj).y/fi + İa \Lr / Lr A (Rr \“,”X _ Rr.M- W = -hr - ju).y/fi-[û) - r}).y/a+--ip a 1 M. nT I la = ~ - <-- -.lf/a -Rs.la + Ua\ oLs I Lr J f i r m a“ ~ i The polar form of (28) and (29) can be given as (28) (29). (30) (31) ¥r -Xr.\y/r\ + M.Xr.İd + ju. wr\ (32) 6 - co + M.xr. rf-r WA (33) where ”“”indicate estimated values,“”time derivation and ju, rj are observer control variables. 3.1 Stator currents observer First of all, we have to design an observer for stator currents. Its role is to ensure that the error of the stator currents converge to zero ( A/a, bdp -»0 and Aia,Aifi -»0). For this reason, we will define the auxiliary vector f as (34) f = hi xr-n.cd-tj y/a (S-tj) xr-n\\yp_ (34) where xr = RrjLr. Using the definition (34) the model of observer in (28-3 1) can be written as ¥a Pp. fa A + M.Xr. (35) XX(36) where, k\ = Re/ct.Ls, to = M/cr.Ls.Lr, ki = l/cr.Ls. The estimation errors of stator currents are given by (37) (37) can be written in generalized form (1) with the following definitions * = [*a */»] » M*>t) = -kı.x + kı. ua ufi, B=k2, u(x,t) = [fa fp]. In that case, the reference signals are real stator currents, x' = I ia ifi and the observer control errors are observation errors ( x = Ax = xr - x ). After these definitions and using the result of SMC design in (16), we can write the final equation of observer control (38) (0 1,-n k2-T-\V>r\ Cos(0) Sin(0) Sin(0) - Cos{0) (d.T+l).Aia(l) - A/a(,_r) (d.T+ l).Ai/r(0 - Aifi(l_T) _ (38) The stator current errors and derivatives will converge to zero under the selected control input ( Aia, Aip -»0 and Aia,Aifi ->0). 3.2 Flux observer Derivatives of flux components could be determined from (30) and (31) and the flux could be obtained by integration. This is the classical approach of flux estimation often applied in literature. This method is not robust against parameter uncertainties and has incremental summation error problem. Flux observation errors converge to zero when conditions in (39) are satisfied A|^r| + ^.A|vr| = 0 and A0 + Ky.A0 = O where K¥ is the slope of sliding line. We can write estimation errors of auxiliary function fas follows A/a =fa -fa =Xr.Ay/a + â.Ay p + n.y/a + 7j.$p ¥P =ffi ~fp = xr.Ay/p-S.Aif/a + p.yffi - rj.y/a (39) (40) (41) XXICurrent observation errors converge to zero with the control of stator observer (Ais->0, A(cRJdf) ->0) and the components of estimation error f converge to zero (A/a-»0, A/£->0) simultaneously, as seen from (37). Flux observation errors could be written by equating (40) and (41) to zero. Ly/“ = S +A* İXr-^”-â-Vfi)- vi'!A2-(v>V'a +X“V>(,) (42) -* -. ”T“ %XJ JÇ ı (A) h¥fi = x2+co2'^a ^'^K2+V^a -û-Vp) (43) The polar form of (42) and (43) are given below,, -(jU.Xr + î].S).\lj}r\ Ak= 2 A» (44) xr +Û) - ll.S + TJ.Xr Afl= P? ?t (45) x; + to The convergence term can be added to the observer model in (32) and (33) by using the definition of the estimation errors in (46) and convergence conditions in (39) such as 0=0+A0 and |^r|= y/r +A|^r| (46) \\jrr\ = -*r.|j/J + M.Xr.îd + jU.\ipr\ - Kv. aL,|. (47) 6 = â + M.Xr.X-\-Kv.A0 (48) Kİ By substituting (44, 45) into the (47, 48), we get the final form of the flux observer (49) 0 = û} + M.Xr.TTi + Kv.î--2 - £j- (50) Wr\ The stator part of observer equations are given in (30, 31), the equations of flux observer are given in (49, 50) and the observer controller is given by (38). The full order flux observer of the induction motor is constructed by these simultaneous equations. xxu4.1 Flux Control From (23) it follows x = Wr\ ¥ ~\T,A(x,/) = [0 -JfrfVr|, B = [0 M.Xr],U = id, D=d=constant, and for invertable (GB), G = [g l], ^constant selected. By using the (16) '*'”SS/“”») +jh{(g+d+Tgd^l) -fr+4I^U) <51) '«(*) '-(*-!) " M.x where in yr I = \yrr\ -Wr\, x0c)=x{k.T) 4.2 Position Control From (27) x = [q e>]\ A(x,/) = [<o -^M]r, B(x,/) = [o |^|.Zr/(M.yi)]r, u=/* D=d=cont.t and for invertable (GBj, G = [g l], g=const. selected. By using the (16) = ',Vi)+I^-(fe + dM®(«-Vi)) + ^rf-fe*)-9(*-.,^ (52).d.j Jl.M r where mx =x -x, X(k)=X(k.T). Detailed description of the proposed method and results are given in chapter. 4. Laboratory setup, realization results and photographs are included in chapter. 5. 5 Conclusion Simulation and application results indicated that the flux observer designed is robust against parameter and load variations. Position and velocity loop responses obtained are fast and accurate for large velocity working ranges. The observer performance is good enough for real applications, even when large and fast rotor time constant, stator resistance and mutual inductance variations are introduced. Thus, the observer and controller structure proposed in the study can be used İn servo, CNC and other position control applications where supreme tracking performance, robustness and low torque ripple are required. xxm

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