Research Article A Two-stage Kalman Filter for Sensorless Direct Torque Controlled PM Synchronous Motor Drive

Transcription

1 Mathematical Problems in Engineering Volume 213, Article ID , 12 pages Research Article A Two-stage Kalman Filter for Sensorless Direct Torque Controlled PM Synchronous Motor Drive Boyu Yi, Longyun Kang, Kai Jiang, and Yujian Lin Electric Power College, South China University of Technology, Guangzhou, Guangdong 5164, China Correspondence should be addressed to Boyu Yi; boyuyi@gmail.com Received 5 December 212; Accepted 15 March 213 Academic Editor: Jun-Juh Yan Copyright 213 Boyu Yi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper presents an optimal two-stage extended Kalman filter (OTSEKF) for closed-loop flux, torque, and speed estimation of a permanent magnet synchronous motor (PMSM) to achieve sensorless DTC-SVPWM operation of drive system. The novel observer is obtained by using the same transformation as in a linear Kalman observer, which is proposed by C.-S. Hsieh and F.-C. Chen in The OTSEKF is an effective implementation of the extended Kalman filter (EKF) and provides a recursive optimum state estimation for PMSMs using terminal signals that may be polluted by noise. Compared to a conventional EKF, the OTSEKF reduces the number of arithmetic operations. Simulation and experimental results verify the effectiveness of the proposed OTSEKF observer for DTC of PMSMs. 1. Introduction Owing to their characteristics of high efficiency, high power density, and reliability, AC machines and more recently especiallypermanentmagnetsynchronousmachines(pmsms) have obtained dominance 1. To control the torque and flux levels of PMSM rotor flux oriented control is seen as an industry standard 2, 3. Alternatives however exist, and direct torque control (DTC) can provide accurate fast torque control. Moreover, DTC uses no current controller and no motor parameters other than the stator resistance, which yields a faster torque response and lower parameter dependence than with field oriented control 4 6. To implement the idea of the DTC on PMSM, the controller must obtain some real-time parameters of PMSM, including stator flux, position, and rotor speed. So the mechanical sensor which can measure rotor position and speed is necessary for the realization of DTC. But the application of mechanical sensor increases the complexity of the control system and degrades the performance of control system when encountering adverse environmental conditions. Moreover, it increases the system cost and maintenance requirements. So sensorless DTC technique has become the hot issue in research and drawn many researchers and engineer s attention. Several parameter estimation techniques have been reportedintheliterature7 13. One of major methods is basedonthelinearornonlinearstateobservers,suchas extended Luenberger observer and extended Kalman filter (EKF). Some works 1 13 have shown the applicability of the EKF to estimate parameters for electric machines. These observers have been shown to be the best computer algorithms for processing noisy discrete measurements and obtaining high-accuracy estimates of dynamic system states (stator flux, rotor position, and speed). The EKF-based algorithms described in 1 13 have good performance, but they appear to be very complex because of the high order of the mathematical models. The applicability of the extended Kalman filter to real-time signal processing problems is generally limited by the complex mathematical operations. Therefore, this paper presents an optimal two-stage extended Kalman filter (OTSEKF) for flux, position, and speed estimation of a direct torque controlled (DTC) PMSM drive. The proposed observer is an effective implementation of extended Kalman filters. Following the same

2 2 Mathematical Problems in Engineering approachasgivenin14, the two-stage structure employed in OTSEKF can decouple the conventional EKF into two parallel observers, which are called the full order filter and the augmented state filter. Compared to the conventional EKF, OTSEKF can reduce the computational burden. To facilitate the understanding, the complete equations of this filter are presented and compared to a straight implementation of the conventional EKF equations. The paper is organized in six sections. In Section 2, the continuous and the discrete model of the PMSM are recalled. In Section 3, the DTC-SVPWM strategy of PMSMs is introduced briefly. In Section 4, based on a conventional EKF algorithm, optimal two-stage extended Kalman filter equations are developed, and theoretical operation requirements of both OTSEKF and EKF are calculated. In Section 5, experimental and simulation results are discussed. Finally, a conclusion wraps up the paper. 2. The Model of PMSM As elaborated in 1, themachineequationsin therotor (dq) reference frame are as follows: with R s A c (t) = ω r ω r R s, B u c (t) =cos θ r sin θ r, sin θ r cos θ r L q G c (t) = 1, BΘ c cos θ r C (t) = sin θ r (t) = ψ f, sin θ r L q. cos θ r L q For digital implementation of the observer, the discretized machine equations are required, provided that the input vector U is nearly constant during a sampling period T s.theseequationscanbeobtainedfrom(2): X k+1 =A k X k +B Θ k Θ k +B u k U k, Θ k+1 =G k Θ k, Y k =C k X k. (3) (4) dψ sd dt = R s ψ sd +ω r ψ sq + R s ψ f +u α cos θ r +u β sin θ r, Tolerating a small discretization error, a first-order series expansion of the matrix exponential is used: dψ sq dt = ω r ψ sd R s L q ψ sq u α sin θ r +u β cos θ r, e A ct A k =I+A c T, e G ct G k =I+G c T, (5) dω r dt =, dθ r dt =ω r, (1) with A 1 c (e A ct I)B c B k =TB c, where u α, u β, i α,andi β are the stator voltages and currents in the (αβ) reference frame, ψ sd, ψ sq are stator flux linkages in the (dq) reference frame, and L q are the machine dq axes inductances, R s is the stator winding resistance, and ψ f isthefluxproducedbythemagnets.theangularvelocityω r is measured in electrical radians per second. θ r is the electrical position. In order to remove constant term in (1), the state vector is chosen to be X(t) = x 1 x 2 T = ψ sd ψ f ψ sq T, estimated parameter vector to be Θ=ω r θ r T,inputvector to be U = u α u β T,andoutputtobeY(t) = i α i β T. The PMSM model is described by the general nonlinear statespace model 15, 16: A k = 1 R s T L s ω r T s d ω r T s 1 R s, G k = 1 T T s 1, L s q B u k =cos θ r T s sin θ r T s, B Θ sin θ r T s cos θ r T k = s ψ f T s, cos θ r C k = sin θ r sin θ r L q. cos θ r L q (6) d dt X (t) =A c (t) X (t) +B u c (t) U (t) +BΘ c d dt Θ (t) =G c (t) Θ (t), Y (t) =C(t) X (t), (t) Θ (t), (2) 3. Principle of Sensorless DTC-SVPWM The basic idea of DTC technique is to calculate and control stator flux linkage and torque of PMSM directly to achieve high dynamic performance. DTC technique involves statorflux vector, torque estimators, hysteresis controllers, and switching tables in order to determine directly an inverter switching state 17, 18. The hysteresis controller minimizes

3 Mathematical Problems in Engineering 3 R s i s T s ωr + Δω r T + ΔT Δω + s ω PI e e s PI + ω r ψr RFVC ψ s ψ s SVPWM VSI θ s u α,u β T e Flux and torque estimator i α i β Clarke transformation i a i b Sensor M Figure 1: System diagram of the SVPWM-DTC scheme. the flux and torque errors17, 18. However, in the conventional DTC system, the switchover among the basic vectors is discontinuous because the universal voltage inverter has only eight available basic space vectors. In a control period, only one voltage space vector can be selected so the flux and torque ripples are unavoidable. To reduce the ripples of the electromagnetic torque andfluxlinkageinpmsmdrives,amodifieddtcusing Space Vector Pulse Width Modulation (SVPWM) method (called DTC-SVPWM) is proposed in this paper. SVPWM techniques have several advantages that are offering better DC bus utilization, lower torque ripples, lower total harmonic distortion in the AC motor current, and lower switching loss. The same flux and torque estimators as for basic DTC are also used in the DTC-SVPWM method. The main difference is that DTC-SVPWM has two PI controllers and a Reference Flux Vector Calculator (RFVC) instead of hysteresis controllers and the switching table The system structure ofdtc-svpwmcanbebuiltandshowninfigure1. This system uses three-way closed-loop control of speed, flux linkage, and torque. Adopting speed deviation Δω r as input value,outerloopproportionalintegral(pi)controlleroutputs torque reference input T e. Then taking torque deviation ΔT e as input value, torque loop PI controller outputs Δω s,which is deviation of rotating speed between stator flux linkage and rotor flux linkage. Therefore, the reference speed of stator flux ω s canbeobtainedbyaddingtherotorfluxlinkagespeedsω r and Δω s. RFVC is used to determine the reference stator flux linkage vector ψ s for the next sample time. The ψ r inputted to RFVC is amplitude of given value of the stator flux linkage. The vector relationships between stator flux linkage vector ψ s androtorfluxlinkageψ f can be drawn in the rotor flux (dq),statorflux(xy),andstationary(αβ) frames as shown in Figure 2. θ s,whichisthephaseangleofstatorfluxlinkage, can be obtained by the flux estimator. The increment of stator flux angle Δθ s inthenextsamplingtimecanbeacquiredby ω s T s. y q β δ θ r Δθ s θ s ψ s ψ s ψ f Figure 2: The vector diagram of PMSM. Define the flux deviations between ψ s and ψ s as Δψ s ; then Δψ sα = ψ s cos (θ s +Δθ s ) ψ s cos θ s, Δψ sβ = ψ s sin (θ s +Δθ s ) ψ s sin θ s, where Δψ sα and Δψ sβ are components of Δψ s in the (αβ) reference frame. ψ s and ψ s are amplitudes of ψ s and ψ s. In order to make up for flux deviations Δψ sα and Δψ sβ, reference stator voltages u α and u β should be applied on the motorwhichcanbecalculatedby u α = Δψ sα T s +R s i α, u β = Δψ sβ T s +R s i β. α x d (7) (8)

4 4 Mathematical Problems in Engineering Substituting (7)into(8), (9) can be acquired: u α = ( ψ s u β = ( ψ s cos (θ s +Δθ s ) ψ s cos θ s) T s +R s i α, sin (θ s +Δθ s ) ψ s sin θ s) T s +R s i β. Based on stator voltage components u α and u β,voltage vector selection, duration time, and switch signal of inverter canbeobtainedthroughsvpwmmodule. 4. Design of OTSEKF Observer To enhance the reliability and robustness of the overall system, the sensorless control of DTC-SVPWM is desirable. So a state observer is necessary to achieve sensorless operation of a PMSM drive. The linear algorithm proposed by Hsieh and Chen 14, which is named the optimal two-stage Kalman estimator (OTSKE), is extended to the nonlinear estimation case. Then, the optimal two-stage extended Kalman filter (OTSEKF) is proposed to estimate the stator flux, position and rotor speed needed for DTC-SVPWM. This algorithm can effectively save computation cost compared to conventional EKF Conventional EKF. Basedondiscretizedmachineequations, an EKF is constructed to estimate the stator flux, torque, andspeedofthepmsm.thestatevectorischosentobe X a k = X k Θ k T. u α, u β and i α, i β are chosen as input and output vectors because these quantities can be easily obtained from measurements of stator phase currents and voltage construction using DC link voltage and switching status. Considering the noise and parameter errors, the state space model in the rotor (dq) reference frame is described by with A k = X a k+1 = A kx a k + B ku k +W k, Y k+1 = C k X a k +η k, A k B Θ k G k B u k, B k =, C k =C k, W k = W x k W Θ k, (9) (1) (11) where W k and η k are zero-mean noise with covariances Q and R, respectively, and are independent from the system state X a k. The system noise W k takes into account system disturbance and model inaccuracies, while η k represents the measurement noise. The noise covariance matrices are defined as follows: E(W x k (Wx k τ )T )=Q x k δ τ, E(W Θ k (WΘ k τ )T )=Q Θ k δ τ, E(W x k (WΘ k τ )T )=Q xθ k δ τ, E(η k (η k τ ) T )=R k δ τ, E(W x k (η k τ) T )=, E(W Θ k (η k τ) T )=. (12) The overall structure of the EKF is well known by employing a two-step prediction and correction algorithm 1. Hence, the filter is given by with F k 1 = X a k k 1 = A k 1X a k 1 k 1 + B k 1U k 1, P k k 1 = F k 1 P k 1 k 1 F T k 1 +Q k 1, K k =P k k 1 H T k (H kp k k 1 H T k +R) 1, X a k k =Xα k k 1 +K k (Y k C k X a k k 1 ), P k k =P k k 1 K k H k P k k 1, E k 1 F k 1 = F k 1, G k 1 H k =H 1 k H 2 k, K k = K x k K Θ k, X (A k 1X k 1 +B u k 1 U k 1 +B Θ k 1 Θ k 1), F k 1 = 1 R s T L s ω r T s d ω r T s 1 R s, Y k = I α, T Iβ L s q E k 1 = Θ (A k 1X k 1 +B u k 1 U k 1 +B Θ k 1 Θ k 1), x E k 1 = 2 T s U q T s, x 1 T s ψ f T s U d T s H 1 k = X (C k 1X k 1 ), (13)

5 Mathematical Problems in Engineering 5 H 1 k = cos θ r sin θ r H 2 k = sin θ r L q, C cos θ r k =C k L q Θ (C k 1X k 1 ), Using the two transformation matrices defined above, overlined expressions that correspond to vectors and matrices inthenewbasecanbeobtained: X a k k 1 =T(M k) X a k k 1, P k k 1 =T(M k ) P k k 1 T(M k ) T, x 1 sin θ r H 2 k = x 1 cos θ r x 2 cos θ r L q x 2 sin θ r L q P x P xθ P ( ) =, P xθ T P Θ, where X a k k =T(N k) X a k k, K k =T(N k ) K k, P k k =T(N k ) P k k T(N k ) T, (17) Q x Q ( ) = Q xθ T Q xθ. Q Θ (14) X k X a k =, K k = Θ k K x k K Θ k. (18) Considering characteristic of matrix T( ),(17)become The EKF mentioned above takes into account the system and measurement noise and exhibits excellent robustness to model inaccuracies, measurement noise, and system uncertainties. But it is obviously that four-order matrix operations are necessary to complete EKF computational operation. Considering Pulse Width Modulation (PWM) period is very small, only high performance DSP or FPGA can qualify for this work. X a k k 1 =T( M k)x a k k 1, P k k 1 =T( M k )P k k 1 T( M k ) T, X a k k =T( N k)x a k k, K k =T( N k )K k, (19) 4.2. OTSEKF Algorithm. Using the same processing method as in 14, the OTSEKF can be obtained by making coordinate transformation. So it is necessary to define a transformation matrix T( );the T( ) is specified as follows: I J T (J) =. (15) I P k k =T( N k )P k k T( N k ) T. To decouple the EKF into a full order filter and an augmentedstatefilter,thefollowingtwo-stepiterativesubstitution method is used. Step 1. Substituting (13)into(19), we have The main advantage of using the T transformation is that the inverse transformation T 1 (M) = T( M) involves only a change of sign. Two blending matrices M k and N k are defined, respectively, by M k =P xθ k k 1 (PΘ k k 1 ) 1 and N k = P xθ k k (PΘ k k ) 1. The transformation operation can be achieved by two transformation matrices T(M k ) and T(N k ) so that the variance-covariance matrices P( ) in new base are block diagonal: P ( ) = Px ( ) P Θ. (16) ( ) X a k k 1 =T( M k) A k 1 X a k 1 k 1 +T( M k) B k 1 U k 1, P k k 1 =T( M k ) F k 1 P k 1 k 1 F T k 1 T( M k) T +T( M k )Q k 1 T( M k ) T, K k =T( N k )P k k 1 H T k (H kp k k 1 H T k +R) 1, X a k k =T( N k)(x α k k 1 +K k (Y k C k X a k k 1 )), P k k =T( N k )P k k 1 T( N k ) T T( N k )K k H k P k k 1 T( N k ) T. (2)

6 6 Mathematical Problems in Engineering Step 2. Substituting (17)intotheright-handsideof(2), the following equations are written: From (22), we have P x k k 1 =F k 1P x k 1 k 1 FT k 1 +(F k 1N k 1 +E k 1 M k G k 1 ) X a k k 1 =T( M k) A k 1 T(N k 1 ) X a k 1 k 1 +T( M k ) B k 1 U k 1, P k k 1 =T( M k ) F k 1 T(N k 1 ) P k 1 k 1 (21) P Θ k 1 k 1 (F k 1N k 1 +E k 1 M k G k 1 ) T +Q x k M k(q xθ k )T Q xθ k MT k +M kq Θ k MT k. (34) T(N k 1 ) T F T k 1 T( M k) T +T( M k )Q k 1 T( M k ) T, K k =T(M k N k ) P k k 1 T(M k ) T H T k (H k T(M k ) P k k 1 T(M k ) T H T k +R) 1, (22) (23) X a k k =T(M k N k ) X a k k 1 + K k (Y k C k T(M k ) X a k k 1 ), (24) P k k =T(M k N k ) P k k 1 T(M k N k ) T K k H k T(M k ) P k k 1 T(M k N k ) T. (25) Supposing variance-covariance matrices P(M) are block diagonal, the following relations are obtained by using (22) and (25): where =M k G k 1 P Θ k 1 k 1 GT k 1 +QxΘ k 1 The above equations lead to M k (G k 1 P Θ k 1 k 1 GT k 1 +QΘ k 1 ) =M k N k K x k S k, (26) M k =(F k 1 N k 1 +E k 1 )G 1 k 1, (27) S k =H 1 k M k +H 2 k. (28) M k = M k +(Q xθ k M k Q Θ k )(PΘ k k 1 ) 1, (29) N k =M k K x k S k. (3) The equations of full order filter are acquired by the next steps. From (21), we give the prediction equations of X k and Θ k as where X k k 1 =A k 1 X k 1 k 1 +B u k 1 U k 1 +u k 1, (31) Θ k k 1 =G k 1 Θ k 1 k 1, (32) u k 1 =(A k 1 N k 1 +B Θ k 1 M kg k 1 ) Θ k 1 k 1. (33) Then using (27), (29), and (32), the above equation can be written as where P x k k 1 =F k 1P x k 1 k 1 FT k 1 + Qx k, (35) Q x k =Qx k QxΘ k MT k M k(q xθ k M k Q Θ k )T. (36) From (23)and(28), we have Then K x k (H1 k Px k k 1 (H1 k )T +R k +S k P Θ k k 1 ST k ) (37) = P x k k 1 (H1 k )T + K x k S kp Θ k k 1 ST k. K x k = Px k k 1 (H1 k )T (H 1 k Px k k 1 (H1 k )T +R k ) 1. (38) From (24), (28), and (3), we have Then X k k = X k k 1 +(M k N k ) Θ k k 1 + K x k (Y k C k X k k 1 S k Θ k k 1 ). (39) X k k = X k k 1 + K x k (Y k C k X k k 1 +(S k S k ) Θ k k 1 ), (4) where From (25), (28), and (3), we have S k =C k M k. (41) P x k k = Px k k 1 +(M k N k ) P Θ k k 1 (MT k NT k ) (42) (K x k H1 k Px k k 1 + Kx k S kp Θ k k 1 (MT k NT k )). Then P x k k = Px k k 1 Kx k H1 k Px k k 1. (43) Besides prediction function (32), the rest of augmented state filter is obtained by expanding (21) (25)and using(26) (28): P Θ k k 1 =G k 1P Θ k 1 k 1 GT k 1 +QΘ k, Θ k k = Θ k k 1 + K Θ k (Y k C k X k k 1 S k Θ k k 1 ),

7 Mathematical Problems in Engineering 7 K Θ k = PΘ k k 1 ST k (H1 k Px k k 1 (H1 k )T +R k +S k P Θ k k 1 ST k ) 1, P Θ k k = PΘ k k 1 KΘ k S kp Θ k k 1, (44) The initial conditions of this OTSEKF are established with the initial conditions of a conventional EKF ( X, Θ, P x, P Θ, PxΘ ), so that where X k k 1 and P x k k 1 can be calculated by (31)and(35). Finally, based on the above analysis, the OTSEKF algorithm can be organized by the next two parts 15, 16. The first part of OTSEKF for state and parameter prediction is as follows: P Θ k k 1 =G k 1P Θ k 1 k 1 GT k 1 +QΘ k, N =P xθ (PΘ ) 1, X = X N Θ, Θ = Θ, P x =Px N P Θ NT, (48) E k = Θ Θ k k (A k (X k +N k Θ k )+B Θ k Θ k +B u k U k), M k =(F k 1 N k 1 +E k 1 )(G k 1 ) 1, M k = M k +(Q xθ k M k Q Θ k )(PΘ k k 1 ) 1, Q x k =Qx k QxΘ k MT k M k(q xθ k M k Q Θ k )T, P x k k 1 =F k 1P x k 1 k 1 FT k 1 + Qx k, Θ k k 1 =G k 1 Θ k 1 k 1, (45) P Θ =PΘ. According to variables of full order filter (x 1 and x 2 ), stator flux linkage in machine rotor (dq) reference frame are obtained as follows: ψ sd =x 1 +ψ f, ψ sq =x 2. (49) The estimated flux linkages ψ sd, ψ sq are transformed into the stationary reference frame via u k 1 =(A k 1 N k 1 +B Θ k 1 M kg k 1 ) Θ k 1 k 1, ψ sα= cos θ r ψ sβ sin θ r sin θ r ψ sd. (5) cos θ r ψ sq X k k 1 =A k 1 X k 1 k 1 +B u k 1 U k 1 +u k 1. The second part for state and parameter correction is as follows: S k =C k M k, S k =H 1 k M k +H 2 k, B k =H 1 k Px k k 1 (H1 k )T +R k, K Θ k = PΘ k k 1 ST k (B k +S k P Θ k k 1 ST k ) 1, P Θ k k = PΘ k k 1 KΘ k S kp Θ k k 1, Θ k k = Θ k k 1 + K Θ k (Y k C k X k k 1 S k Θ k k 1 ), K x k = Px k k 1 (H1 k )T (B k ) 1, P x k k = Px k k 1 Kx k H1 k Px k k 1, X k k = X k k 1 + K x k (Y k C k X k k 1 +(S S) Θ k k 1 ), N k =M k K x k S k. (46) Using (17), the original state X canbeobtainedbythesumof the state X with the augmented state Θ: X k k 1 = X k k 1 +M k Θ k k 1, (47) X k k = X k k +N k Θ k k. The stator flux linkage and electromagnetic torque estimators are then given by ψ s = ψ2 sα +ψ2 sβ, θ s = arctan ( ψ sβ ψ sα ), T e = 3 2 P n (ψ sα i sβ ψ sβ i sα ), (51) where T e is the electromagnetic torque and P n is the motor pole pairs. The estimated speed ω r obtained from the OTSEKF observer is used to close the speed loop to achieve sensorless operation. So the sensorless DTC- SVPWM scheme is illustrated in Figure Algorithm Complexity. Tables 1 and 2 show the number of arithmetic operations required at each time sample by the standard EKF algorithm and the OTSEKF, respectively (where rough matrix-based implementation is used). The state vector X k, measurement Y k, input vector U k, and parameter Θ k dimensions are, respectively, n, m, q, andp. TocomputestandardEKF,thetotalnumberofmultiplications is 358, and the total number of additions is 289, whereas the number of arithmetic operations required by OTSEKF algorithm is 284 multiplications and 215 additions. This means that the number of multiplications is reduced by 2.67% and the number of additions is reduced by 25.61%.

8 8 Mathematical Problems in Engineering R s i s T s ωr + Δω r T Δω e + ΔT s ω e + s PI PI + ω r ψ r RFVC ψ s ψ s SVPWM VSI T e θ s Flux and torque estimator u α,u β ψ α ψ β i α i β OTSEKF i α i β Clarke transformation i a i b M Figure 3: Block diagram of the proposed sensorless DTC-SVPWM scheme. Table 2: OTSEKF arithmetic operation requirements. Closed loop Controller OTSEKF M A k, C k, E k, H 1 k, H 2 k, Bu k, Number of multiplications (n =4, m=2, q=2, p=2) Number of additions (n =4, m=2, q=2, p=2) Open loop Conventional EKF Figure 4: Open-loop and closed-loop simulation structure. Table 1: EKF arithmetic operation requirements. Number of multiplications (n =4, m=2, q=2) Number of additions (n =4, m=2, q=2) A k, B k, C k Function of system (22) Function of system (5) X a k k 1 n 2 +nq(24) n 2 +nq n(2) P k k 1 2n 3 (128) 2n 3 n 2 (112) X a k k 1 2nm (16) 2nm (16) K k n 2 m+2nm 2 +m 3 (72) n 2 m+2nm 2 +m 3 2nm(56) P k k n 2 m+n 3 (96) n 2 m+n 3 n 2 (8) Total Simulation and Experimental Results 5.1. Simulation Results. To verify validity and feasibility of OTSEKF algorithm, the sensorless DTC-SVPWM technique described in the previous sections is implemented in MATLAB/SIMULINK environment. As stated in 14, the B Θ k Function of system (2) Function of system (3) P Θ k k 1 2p 3 (16) 2p 3 p 2 (12) M k 1 n 2 p+np 2 (16) n 2 p+np 2 np(12) M k 1 2np 2 (16) 2np 2 (16) Q x k 1 2n 2 p+np 2 (24) 2n 2 p+np 2 (24) P x k k 1 2n 3 (16) 2n 3 n 2 (12) Θ k k 1 p 2 (4) p 2 p(2) u k 1 n 2 p+np 2 +np(2) n 2 p+np 2 +np n(18) X k k 1 n 2 +nq(8) n 2 +nq(8) S k mnp (8) nmp (8) S k mnp (8) mnp mp(4) B k 2mn 2 (16) 2mn 2 m 2 (12) K Θ k k 3p 2 m+pm 2 (32) 3p 2 m+pm 2 +m 2 4pm (2) P Θ k k p 3 +p 2 m (16) p 3 +p 2 m p 2 (12) Θ k k 2mp + nm (12) 2mp + nm (12) K x k k n 2 m+nm 2 (16) n 2 m+nm 2 2nm(8) P x k k n 2 m+n 3 (16) n 2 m+n 3 n 2 (12) X k k 2nm + mp (12) 2nm + 2mp (16) N k npm (8) np (4) Total EKF and OTSEKF are mathematically equivalent without requiring system constraints. So a comparison is made

9 Mathematical Problems in Engineering 9 6 Real speed and estimated speed (OTSEKF) 4 OTSEKF speed error (rpm) (rpm) Real Estimated (a) (b).4 Difference of speed estimation 8 Real position and estimated position (OTSEKF) (rpm).1 (rad) Real Estimated (c) (d).25 OTSEKF position error Difference of position estimation (rad).1 (rad) (e) (f) Figure 5: Continued.

10 1 Mathematical Problems in Engineering 4 Estimated torque.1 Estimated stator flux linkage (N m) Stator flux β component (Wb) (g) Stator flux α component (Wb) (h) Figure 5: Simulation results for parameters estimation. (a) Actual rotor speed and estimated rotor speed (OTSEKF). (b) OTSEKF speed error. (c) Difference of rotor speed estimation between EKF and OTSEKF. (d) Actual rotor position and estimated rotor position (OTSEKF). (e) OTSEKF position error. (f) Difference of rotor position estimation between EKF and OTSEKF. (g) Estimated torque. (h) Estimated stator flux linkage. between outputs of two algorithms. Real-time parameter estimations observed by OTSEKF are used to formulate the closed loop, such as stator flux, rotor speed, and position. The estimations obtained by conventional EKF algorithm are not included in the DTC-SVPWM and only evaluated in open loop. The block diagram of evaluation module is shown in Figure 4. Figure 5 is the simulation results of parameter estimation based on OTSEKF. The speed dynamic response of the sensorless drive is shown in Figure 5(a). The machine is accelerated from rpm to 6 rpm. During transients, OTSEKF has good tracking performance (speed and position). In Figures 5(g) and 5(h), the torque and flux responses show that ripples are significantly suppressed due to the SVPWM modulation scheme. In Figures 5(c) and 5(f), difference in speed and position estimations between the two observers is very little. It proves equivalence of both observers. The difference between two observers is caused by accuracy loss in OTSEKF, which uses more calculation steps Experimental Results. The drive system consists of a three-phase, eight-pole, 1.2 kw, PMSM driven from a threephase Insulated Gate Bipolar Transistor (IGBT) inverter. The PMSM is mechanically coupled to a magnetic clutch, which provides rated torque, even at very low speeds. The drive system is implemented by an Expert 3 control system of Myway Company. The core DSP processor in Expert 3 control system is TMS32C6713 with a clock speed of 225 MHz. The actual rotor position and speed are obtained from an incremental encoder with 1 pulses per revolution. The inverter space-vector PWM control, current regulator, and observer algorithm operate with a 1 us sampling/integration time step.thecurrentsflowinginthestatorwindingsaremeasured with two hall-effect current sensors. The experiment test is conducted to testify the tracking performance of OTSEKF and to demonstrate that the OTSEKF is mathematically equivalent to EKF. Figure 6 presents the corresponding experimental results. Figures 6(a) and 6(b) show the rotor speed and angle dynamic response of OTSEKF and the errors between the estimated and real values when the command speed is 6 rpm. We can observe that the OTSEKF tracks the actual speed and angle very well, and the steady state errors are very small. Figure 6(e) presents trajectory of the estimated stator flux components, and the stator flux trajectory is almost circular. Figure 6(f) shows the response of torque is fast and smooth. The differences between the two observers are represented in Figures 6(c) and 6(d). As we can expect, the differences in speed and position estimations are still small, andthisprovesthatthetwoobserversaremathematically equivalent. 6. Conclusion The real-time digital implementation of extended Kalman filters algorithm for senseless DTC-SVPWM requires a very fast signal processor specialized and optimized to perform complex mathematical calculations and manipulate a large amount of data. This is generally considered as a major disadvantage to apply Kalman filter algorithm to industrial field. So this paper proposes a novel extended Kalman filter called optimal two-stage extended Kalman filter. OTSEKF can reduce computational cost effectively compared to conventional EKF. The simulation and experimental results show that OTSEKF has good tracking performance. Equivalence is verified by little difference in estimations between OTSEKF and EKF. The OTSEKF algorithm provides a new idea to solve computational

11 Mathematical Problems in Engineering 11 8 Speed estimation Difference of speed estimation 6.1 (rpm) 4 2 (rpm) Real Estimated Error (a) (b) 7 Position estimation Difference of position estimation (rad) (rpm) Real Estimated (c) Estimated torque Error (d) Estimated stator flux linkage (N m) (e) Stator flux β component (Wb) Stator flux α component (Wb) (f) Figure 6: Experimental results for parameters estimation. (a) Real speed, estimated speed (OTSEKF), and estimated speed error. (b) Real rotor position, estimated rotor position (OTSEKF), and estimated position error. (c) Difference of speed estimation between EKF and OTSEKF. (d) Difference of rotor position estimation between EKF and OTSEKF. (e) Estimated torque. (f) Estimated stator flux linkage.

12 12 Mathematical Problems in Engineering complexion and allows us to use cheaper DSP in practical application. References 1 T. Sebastian, G. R. Slemon, and M. A. Rahman, Modelling of permanent magnet synchronous motors, IEEE Transactions on Magnetics,vol.22,no.5,pp , M. W. Degner, J. M. Guerrero, and F. Briz, Slip-gain estimation in field-orientation-controlled induction machines using the system transient response, IEEE Transactions on Industry Applications,vol.42,no.3,pp ,26. 3 M. Rashed, P. F. A. MacConnell, A. F. Stronach, and P. Acarnley, Sensorless indirect-rotor-field-orientation speed control of a permanent-magnet synchronous motor with stator-resistance estimation, IEEE Transactions on Industrial Electronics,vol.54, no. 3, pp , L.Zhong,M.F.Rahman,W.Y.Hu,andK.W.Lim, Analysisof direct torque control in permanent magnet synchronous motor drives, IEEE Transactions on Power Electronics, vol. 12, no. 3, pp , L.Zhong,M.F.Rahman,W.Y.Hu,K.W.Lim,andM.A. Rahman, A direct torque controller for permanent magnet synchronous motor drives, IEEE Transactions on Energy Conversion,vol.14,no.3,pp , Y. Wang and Z. Deng, Improved stator flux estimation method for direct torque linear control of parallel hybrid excitation switched-flux generator, IEEE Transactions on Energy Conversion,vol.27,no.3,pp ,212. 7K.B.LeeandF.Blaabjerg, AnimprovedDTC-SVMmethod for sensorless matrix converter drives using an overmodulation strategy and a simple nonlinearity compensation, IEEE Transactions on Industrial Electronics, vol.54,no.6,pp , Y.Yan,J.G.Zhu,andY.G.Guo, Initialrotorpositionestimation and sensorless direct torque control of surface-mounted permanent magnet synchronous motors considering saturation saliency, IET Electric Power Applications,vol.2,no.1,pp.42 48, Y. Wang and Z.-Q. Deng, A position sensorless method for direct torque control with space vector modulation of hybrid excitation flux-switching generator, IEEE Transactions on Energy Conversion,vol.27,no.4,pp , T.J.Vyncke,R.K.Boel,andJ.A.A.Melkebeek, Onthestator flux linkage estimation of an PMSM with extended Kalman filters, in Proceedings of the 5th IET International Conference on Power Electronics, Machines and Drives (PEMD 1),pp.1 6, April G. Foo, S. Sayeef, and M. F. Rahman, SVM direct torque controlled interior permanent magnet synchronous motor drive using an extended Kalman filter, in Proceedings of the 4th IET International Conference on Power Electronics, Machines and Drives (PEMD 8), pp , April M. Barut and R. Demir, Bi input-extended Kalman filter based speed-sensorless direct torque control of IMs, in Proceedings of the 19th International Conference on Electrical Machines (ICEM 1),pp.1 5,September21. 13I.M.Alsofyani,N.Idris,T.Sutikno,andY.A.Alamri, An optimized Extended Kalman Filter for speed sensorless direct troque control of an induction motor, in Proceedings of the IEEE International Conference on Power and Energy (PECon 12),pp , December C.-S. Hsieh and F.-C. Chen, Optimal solution of the two-stage Kalman estimator, IEEE Transactions on Automatic Control, vol.44,no.1,pp , M. Hilairet, F. Auger, and E. Berthelot, Speed and rotor flux estimation of induction machines using a two-stage extended Kalman filter, Automatica, vol. 45, no. 8, pp , A. Akrad, M. Hilairet, and D. Diallo, A sensorless PMSM drive using a two stage extended Kalman estimator, in Proceedings of the 34th Annual Conference of the IEEE Industrial Electronics Society(IECON 8),pp ,November M. Depenbrock, Direct self-control (DSC) of inverter-fed induction machine, IEEE Transactions on Power Electronics, vol. 3, no. 4, pp , I. Takahashi and T. Noguchi, A new quick-response and high-efficiency control strategy of an induction motor, IEEE Transactions on Industry Applications,vol.IA-22,no.5,pp , D. Swierczynski, M. P. Kazmierkowski, and F. Blaabjerg, DSP based direct torque control of permanent magnet synchronous motor (PMSM) using space vector modulation (DTC-SVM), in Proceedings of the IEEE International Symposium on Industrial Electronics (ISIE 2),vol.3,pp ,22. 2 S. Sayeef and M. F. Rahman, Comparison of proportional+integral control and variable structure control of interior permanent magnet synchronous motor drives, in Proceedings of the 38th IEEE Annual Power Electronics Specialists Conference (PESC 7), pp , June J. Siahbalaee, S. Vaez-Zadeh, and F. Tahami, A new loss minimization approach with flux and torque ripples reduction of direct torque controlled permanent magnet synchronous motors, in Proceedings of the 13th European Conference on Power Electronics and Applications (EPE 9),pp.1 8,September 29.

13 Advances in Operations Research Volume 214 Advances in Decision Sciences Volume 214 Applied Mathematics Algebra Volume 214 Probability and Statistics Volume 214 The Scientific World Journal Volume 214 International Differential Equations Volume 214 Volume 214 Submit your manuscripts at International Advances in Combinatorics Mathematical Physics Volume 214 Complex Analysis Volume 214 International Mathematics and Mathematical Sciences Mathematical Problems in Engineering Mathematics Volume 214 Volume 214 Volume 214 Volume 214 Discrete Mathematics Volume 214 Discrete Dynamics in Nature and Society Function Spaces Abstract and Applied Analysis Volume 214 Volume 214 Volume 214 International Stochastic Analysis Optimization Volume 214 Volume 214