REVIEW PAPER                              International J. of Recent Trends in Engineering and Technology, Vol. 3, No. 4, ...
REVIEW PAPER                                 International J. of Recent Trends in Engineering and Technology, Vol. 3, No. ...
REVIEW PAPER                                      International J. of Recent Trends in Engineering and Technology, Vol. 3,...
REVIEW PAPER                                          International J. of Recent Trends in Engineering and Technology, Vol...
REVIEW PAPER                                                               International J. of Recent Trends in Engineerin...
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Adaptive control of saturated induction motor with uncertain load torque

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Adaptive control of saturated induction motor with uncertain load torque

  1. 1. REVIEW PAPER International J. of Recent Trends in Engineering and Technology, Vol. 3, No. 4, May 2010 Adaptive Control of Saturated Induction Motor with Uncertain Load Torque T.S.L.V.Ayyarao1, G.I.Kishore2, and M.Rambabu3 1 GMR Institute of Technology/EEE, Rajam, India Email:ayyarao.tummala@gmail.com 2 GMR Institute of Technology/EEE, Rajam, India Email: gi_kishore@yahoo.co.in 3 GMR Institute of Technology/EEE, Rajam, India Email:m.rambabu@gmail.comAbstract—The most popular method for high motor and then field oriented model. An adaptive controlperformance control of induction motor is field of induction motor with saturation is developed whichoriented control. Variation of load torque cause input- estimates a time varying load torque.output coupling, steady state tracking errors anddeteriorated transient response in this control. A II.INDUCTION MOTOR MODELnonlinear adaptive controller is designed for inductionmotor with magnetic saturation which includes both A.Model of Induction Motorelectrical and mechanical dynamics. The control An induction motor is made by three stator windingsalgorithm contains a nonlinear identification scheme and three rotor windings.which asymptotically tracks the true value of theunknown time varying load torque. Once this dψ sbparameter is identified, the two control goals ofregulating rotor speed and rotor flux amplitude are Rsisb + = usbdecoupled, so that power efficiency can be improvedwithout affecting speed regulation. dtIndex Terms—Adaptive control, Induction motor with d ψ rd saturation, Matlab/Simulink R r i rd / + =0 dt I. INTRODUCTION dψ sa Induction motor is known for constant speedapplications. It is maintenance free, simple in operation, Rsisa + = usarugged and generally less expensive than either DC or dtsynchronous motors. On the other hand its model is morecomplicated than other machines and because of this, it isconsidered as ‘the benchmark problem in nonlinear d ψ rq R r i rq + =0 (1)control’. There are many approaches to induction motor dtcontrol. Field oriented control of induction motor was Where R, i, ψ, us denote resistance, current, flux linkage,developed by Blaschke. Field oriented control achieves and stator voltage input to the machine; the subscripts sinput-output decoupling. Applications clearly indicate and r stand for stator and rotor, (a,b) denote thethat even though nonlinearities may be exactly modeled, components of a vector with respect to a fixed statorthe physical parameters involved are most often not reference frame, (d’,q’) denote the components of aprecisely known. This motivated further studies on vector with respect to a frame rotatingadaptive versions of feedback linearization and input- We now transform the vectors (ird’, irq’), (ψrd’, ψrq’) in theoutput linearization [1], since cancellations of rotating frame (d’,q’) into vectors (ira, irb), (ψra, ψrb) in the stationary frame (a,b) bynonlinearities containing parameters are required. Loadtorque and rotor resistance are estimated. The common ⎡ira ⎤ ⎡cosδ − sin δ ⎤ ⎡ird / ⎤assumption made in these control laws is the linearity of ⎢i ⎥ = ⎢ sin δ ⎢ ⎥ cosδ ⎥ ⎢irq / ⎥the magnetic circuit of the machine. In many variable ⎣ rb ⎦ ⎣ ⎦⎣ ⎦torque applications, it is desirable to operate the machine (2)in the magnetic saturation region to allow the machine todevelop higher torque. The paper is organized as follows.We shall first describe a fifth model [2] of induction 84© 2010 ACEEEDOI: 01.IJRTET.03.04.197
  2. 2. REVIEW PAPER International J. of Recent Trends in Engineering and Technology, Vol. 3, No. 4, May 2010⎡ψ ra ⎤ ⎡cosδ − sin δ ⎤ ⎡ψ rd / ⎤ ⎛ψ d ⎞ ⎡ cos ρ sin ρ ⎤⎛ψ a ⎞ ⎜ ⎟=⎢⎢ψ ⎥ = ⎢ sin δ ⎢ ⎥ ⎜ψ q ⎟ − sin ρ cos ρ ⎥⎜ψ ⎟ ⎜ ⎟ cosδ ⎥ ⎢ψ rq/ ⎥ (7)⎣ rb ⎦ ⎣ ⎦⎣ ⎦ ⎝ ⎠ ⎣ ⎦⎝ b ⎠(3) Since cos ρ = (ψ a / |ψ| ), sin ρ = (ψb/|ψ|),The system (1) becomes with | ψ | = ψa2 + ψb2. ψ a i a + ψ b ibdw n p M id = = (ψ a ib − ψ a i a ) − T L ψdt JL r Jdψ a R R ψ a ib − ψ b i a = − r ψ a + r Mi a − n p ωψ b iq = dt Lr Lr ψdψ b R R = − r ψ b + r Mib + n p ωψ a ψ d = ψ a 2 +ψ b 2 = ψ dt Lr Lr ψ q = 0. (8) We now interpret field oriented model as state feedback di a MR r npM = ψa + wψ b transformation (involving state space change of dt σL s L r 2 σL s L r coordinates and nonlinear state feedback) into a control system of simpler structure. Defining the state space change of coordinates ⎛ M 2 Rr + Lr 2 Rs ⎞ −⎜ ⎟ia + 1 ua ω =ω ⎜ ⎟ σL ⎝ σLs Lr 2 ⎠ s ψ d = ψ a 2 +ψ b 2di b MR r npM ψb = ψb − wψ a ρ = arctandt σLs Lr 2 σLs Lr ψa ψ a i a + ψ b ib id = ψ ⎛ M 2 Rr + Lr 2 Rs ⎞ −⎜ ⎟ib + 1 ub ψ a ib − ψ b i a ⎜ σL L 2 ⎟ σL (4) iq = (9) ⎝ s r ⎠ s ψ Substituting the above equations (10) in (9) the systemWhere i, Ψ, us denote current, flux linkage and stator yields the field oriented modelvoltage input to the machine; the subscripts s and r stand dω Tfor stator and rotor; (a,b) denote the components of a = μψ d i q − Lvector with respect to a fixed stator reference frame and dt Jσ = 1-(M2/LsLr).Let u = (ua,ub)T be the control vector. Let α = RrN/Lr, ψ dd β = (M/ σLsLr), γ = (M2RrN/ σLsLr)+(Rs/ σLs),μ = (npM/JLr). =αψα d − d+ M System in (4) can be written in compact form asx = f ( x ) + u a g a + u b g b + p1 f 1 + p 2 f 2 ( x )& dt 2(5) did iq 1 = −γid + αβψ d + n pωiq + αM + udB. Field Oriented Model dt ψ d σLsIt involves the transformation of vectors (ia, ib), (ψ a, di qψ b) in the fixed stator frame (a, b) into vectors in a frame = − γ i q − β n p ωψ d(d, q) which rotate along with the flux vector (ψ a, ψ b); dtdefining iq id 1ρ = arctan (Ψb/Ψa) (6) − n p ω id − α M + uq the transformations are ψd σ Ls⎛ id ⎞ ⎡ cos ρ sin ρ ⎤⎛ ia ⎞ dρ iq⎜ ⎟=⎢ = n pω + αM .⎜ iq ⎟ − sin ρ cos ρ ⎥⎜ i ⎟ (10) ⎜ ⎟ dt ψd⎝ ⎠ ⎣ ⎦⎝ b ⎠ 85© 2010 ACEEEDOI: 01.IJRTET.03.04.197
  3. 3. REVIEW PAPER International J. of Recent Trends in Engineering and Technology, Vol. 3, No. 4, May 2010 ψ III.INDUCTION MOTOR CONTROL dd A. Control Strategy Let ω r (t) and ψ r (t) be the smooth bounded =αψα dr − d+Mi reference signals for the output variables to be controlled which are speed ω and flux modulus ψ . The control dt strategy is shown in Figure (1). The goal is to design dρ i qref = n pω + αM . control inputs u d and u q so that for any unknown time dt ψd varying TL (t), we obtain (14) lim [ ω ( t ) − ω r ( t )] = 0 t → ∞ Now i qref and i dref are the inputs and these inputs are lim [ψ d ( t ) − ψ r ( t )] = 0 used to control ω and ψ . Different models are t → ∞ (11) available to incorporate flux saturation. Let’s use the A standard approach for simplifying the dynamics for the model presented in [3] as given below. dψd design of the speed and flux loops is to use proportional = M (−f −1(ψd ) +idref) α plus integral control loops to generate current command. That is, choose the inputs as t u d = − k d 1 ( i d − i dref ) − k d 2 ∫ ( i d (τ ) − i dref ) d τ . dt 0 dρ i qref (12) = n pω + αM . (15) t dt ψd u q = − k q 1 ( i q − i qref ) − k q 2 ∫ ( i q (τ ) − i qref (τ )) d τ . 0 Where M and α are at their nominal values. In steady (13) state ψ d = f ( i d ), which represents the saturation Proper choice of the gain results tracking the reference curve for induction motor. In the absence of saturation currents so that current dynamics can be ignored. ψ = Mi d and f −1 (ψ d ) = ψ d / M . Therefore (11) Therefore the system can then be replaced by the d following reduced order. reduces to linear case. Defining the error variables ~ ω = ω − ωr ~ ψ = ψ −ψ r (16) Using the reduced motor equations and substituting T L (t ) = c o + c1 (t − t o ) as in [4]. Where co and c1 are constant coefficients of the first order polynomial used to estimate the time varying parameter TL (t). We deduce the error equations. ~ & c o ( t ) c1 ( t ) ω = μψ d i q − − (t − t o ) − ω r & J J ~& ψ = − M α ( f −1 (ψ d ) + i d ) − ψ r & (17) Figure 1. Field Oriented Control of Induction motor. Choosing the Lyapunov function ~ 1 ~ 2 ~2 ~2 dω V L= t ) (γ 1ω 2 + ψ d + γ 2 c o + γ 3 c1 ) T ( (18) Mr.T.S.L.V.Ayyarao is an Assistant Professor = μψ d i qref − 2at GMR Institute of Technology, Rajam. dt J 86 © 2010 ACEEE DOI: 01.IJRTET.03.04.197
  4. 4. REVIEW PAPER International J. of Recent Trends in Engineering and Technology, Vol. 3, No. 4, May 2010Where γ 1 , γ 2 ,γ 3 are positive design parameters 350 ~ ˆ ~chosen by the designer. c o = c o − c o , c1 = c1 − c1 are ˆ 300 ˆ ˆthe coefficient estimation errors and c o , c 1 areestimates of the coefficients. 250 reference speed (rad/s) 200The desired currents inputs are chosen as follows 150 1 ~ cˆ ˆ ci qref = [ − k w ω 2 + o + (t − t o ) + ω r ] & μψ d 100 J J (19) 1 50i dref = [ Mαf −1 (ψ d ) + ψ r − kψ d ] & Mα 0 0 1 2 3 4 5 6 7 8 time(s)Where kw and k ψ d are positive design parameters. Figure 2. Reference SpeedThe adaptation laws are given by 1.4 1 ~ λ1ω&c0 =ˆ (− ) 1.35 λ2 J 1.3 1 ~ λ 1ω ( t − t o )&c1 =ˆ (− ) 1.25 λ3 referenc e flux (w b) J 1.2(20) 1.15 1.1 With & ~ ~ 2V = −λ1 k wω 2 − kψdψ d 1.05 1(21) 0.95 0.9 0 1 2 3 4 5 6 7 8 IV. SIMULATION time(s) Figure 3. Reference Flux The proposed algorithm has been simulated inMatlab/Simulink for a 15 KW motor, with rated torque70 Nm and rated speed 220 rad/s, whose data are listed in Simulation results for f −1 (ψ d ) = ψ d / M are shown in the Appendix. The simulation test involves the [9]. Simulation results for f (ψ d ) = ψ d / M −1 3following operating sequences: the unloaded motor isrequired to reach the rated speed and rated value of 1.3 (saturation) are shown in Figure.4 & 5. The controllerWb for rotor flux amplitude. At t = 2s a load torque parameters are chosen as kw=12000, k ψ d =1000,40Nm, which is unknown to the controller, is applied. At kd1=46000, kd2=88000, kq1=196000, kq2=188000. Figure.4t = 5s., the speed shows the actual speed and it tracks the reference speedis required to reach 300 rad/s., well above the nominal even though load torque changes as a function of time.value, and rotor flux amplitude is weakened. The The flux response tracks the reference flux as shown inreference signals for flux amplitude and speed consists of Figure.5.The simulation results show the superiortep functions, smoothed by means of second order behavior of the control algorithm.polynomials. A small time delay at the beginning of thespeed reference trajectory is introduced in order to avoidoverlapping between flux and speed transients.Figure 2 & 3 shows the reference speed and flux. 87© 2010 ACEEEDOI: 01.IJRTET.03.04.197
  5. 5. REVIEW PAPER International J. of Recent Trends in Engineering and Technology, Vol. 3, No. 4, May 2010 Speed vs time measurement of noise, simplifying modeling assumptions 350 and unmodeled dynamics. The authors can conclude on the topic discussed and proposed. Future enhancement 300 can also be briefed here. 250 Appendix A Induction motor Data speed(rad/s) 200 Rs 0.18 Ω 150 Rr 0.15 Ω ψr 1.3 Wb rated 100 ω 220 rad/sec rated np 1 50 J 0.0586 Kgm2 M 0.068 H Lr 0.0699 H 0 0 1 2 3 4 5 6 7 8 Ls 0.0699 H time(s) TL 70 Nm Figure 4. Actual Speed T Rated power 15 Kw 1.4 REFERENCES 1.35 [1] R.Marino, Peresada and Valigi, “Adaptive input-output Linearizing control of Induction motor”, IEEE transactions 1.3 on Automatic control, vol. 38, pp. 209-222. 1.25 [2] P.C.Krause, ‘Analysis of Electric Machinery’, New York: 1.2 McGraw-Hill, 1986. [3] G.Heinemann and W.Leonhard, “Self-tuning field oriented flux(wb) 1.15 control of an induction motor Drive”, in Proc. Int. Power 1.1 Electro. Conf., Tokyo, Japan, April 1990. [4] Ebrahim, Murphy, “Adaptive Control of induction motor 1.05 with magnetic saturation under time varying load torque 1 uncertainty”, IEEE transactions on systems theory, Macon, March 2007. 0.95 [5] Alberto Isidori, Nonlinear Control, Springer-Verlag, 0.9 0 1 2 3 4 5 6 7 8 London:1990. time(s) [6] J.J.E.Slotine and Weiping Li, Applied Nonlinear Control, Figure 5. Actual Flux. Prentice-Hall, 1991. [7] W.Leonhard, Control of Electric Drives, Berlin:Springer- Verlag, 1985. [8] I.Kanellakopoulos, P.V.Kokotovic, and R.Marino, “An CONCLUTION extended direct scheme for robust adaptive nonlinear In this paper we propose a detailed nonlinear model control,” Automatica, vol. 27, no.2, pp. 247-255, 1981. [9] T.S.L.V.Ayyarao, A.Apparao and P.Devendra, “Control ofof an induction motor, an adaptive input-output Induction motor by adaptive input-output linearization,”decoupling control which has some advantages over the NADCAP, Hyderabad, 2009.classical scheme of field oriented control. With acomparable complexity exact decoupling between speedand flux regulation is achieved and critical parameter isidentified. The main drawback of the proposed control isthe requirement of flux measurements. Additionalresearch should analyze the influence of sampling error, 88© 2010 ACEEEDOI: 01.IJRTET.03.04.197

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