High performance IPMSM drives without rotational position sensors using reduced order EKF Energy Conversion, IEEE Transactions on 868 IEEE Transactions on Energy Conversion, Vol 14, No 4, December 199[.]
IEEE Transactions on Energy Conversion, Vol 14, No 4, December 1999 868 High Performance IPMSM Drives without Rotational Position Sensors Using Reduced-Order EKF Yoon-Ho Kim, Member, IEEE Yoon-Sang Kook Dept of Electrical Engineering, Chung-Ang University 22 HukSuk-Dong, DongJak-Ku, Seoul, 156-756 Korea Phone : +82-2-820-5290 Fax : +82-2-812-1407 Email : yhkim@cau.ac.kr Keywords : IPMSM(1nterior Permanent Magnet Synchronous Motor), EKF(Extended Kalman Filter) Absfracf - An extended Kalman filter(EKF) based approach for position sensor elimination in interior permanent magnet synchronous motor(1PMSM) drives is presented in this paper The EKF is capable of estimating system parameters and state variables for the IPMSM by eliminating virtually all influences of structural noises in the vector control scheme This paper presents a design method of a reduced-order EKF Position and angular speed of the rotor are obtained through the reducedorder EKF only by measuring stator currents Also, due to an angle modification scheme with error tracking, the sensorless drive system is robust to parameter variations Simulation and experimental results are provided to verify the proposed approach based on the reduced-order EKF I , INTRODUCTION Recently, motor drive systems without electromechanical sensors, so called ‘sensorless drives’, have gained increasing popularity in industrial applications because of inherent drawbacks of electromechanical sensors In general, electromechanical sensors are used to obtain speed or position information of motors A drawback of these sensors is performance degradation due to vibration or humidity [I-21 Thus, variable control strategies for PMSM drives without electromechanical sensors have been presented by many authors in drives such as BDCM [I], SPMSM [2] and IPMSM [3] Especially, the IPMSM has found wide application on high performance motor drives because of high speed operation These are high power density machines capable of operating at high speed and inverter efficiencies over wide speed ranges, including considerable ranges of PE-019-EC-0-10-1998 A paper recommended and approved by the iEEE ElectricMachineryCommittee of the ,EEE Power Engineering ~ i in ~ the fiEEE ~ ti^^^ on E~~~~ ,~ for aubiication Conversion Manuscript submitted January 22, 1998: made availabie for printing November IO, 1998 ~ ~~ constant power operation The rotor magnetic circuit saliency preferentially increases the q-axis inductance and introduces a reluctance torque term into IPMSM torque equations [4] Previous studies about sensorless drives for the IPMSM can be divided into three categories In the first one, the idea is to manipulate the motor equations in order to express rotor position and speed as functions of terminal quantities [2] In this scheme, the sensitivity to motor parameters is a major drawback In the second scheme, sensorless drives have been developed on the basis of state observers The major disadvantage of this scheme is that linearization of the nonlinear equations describing system behaviors along the nominal state trajectoly does not guarantee the overall stability [ ] In the last scheme, thanks to their ability to perform state estimation for nonlinear systems, EKF is adopted to estimate the rotor angle and speed The Kalman filter based on minimization of the estimation error covariance is suitable for obtaining high-accuracy estimates of state variables and model parameters and eliminating measurement noise So far, however, computation requirements, parameter sensitivity, and initial conditions have unfavorably characterized this approach As for computation load, however, the reduced-order EKF can easily bear real-time calculation using floating-point DSPs In this paper, the reduced-order EKF to estimate backEMF and the angular speed as state variables only by measuring stator currents for IPMSM drives is proposed Furthermore, using the rotor position and speed estimation strategy, the actual rotor position, as well as the motor speed, can be estimated with little error even in the transient state such as speed fluctuation or sudden load variation Also, since the angle error is newly modified with error tracking at each estimation step, the overall control system is robust to parameter variations Therefore, regardless of detuned parameters, the estimated angle always keeps track of the actual angle To verify the feasibility of the proposed method, simulation and experiments are performed Main features of the proposed algorithm are : Mechanical parameters are not required, and problems of parameter sensitivity are partially overcome 0885-8969/99/$10.00 1998 IEEE The control algorithm ensures that speed and position estimation are performed properly at start-up, even if the actual rotor initial position is unknown and different from the initial value assumed by the algorithm Measurements of the voltages applied to motors are no longer required (4) 10 01 The state equations of IPMSM are expressed as II MATHEMATICAL MODEL OF THE IPMSM The voltage equation of the IPMSM, based on the d-q reference frame theory, is expressed in matrix fonn as : X(f) = Ax(t) t Bu(t) (9 (6) y(t) = C.(t) where x ( t ) = [iqd ids E," where superscript 's' denotes stationary reference frame, and 'p' means differential operator In this paper, by choosing back-EMF as a state variable, state equations are simplified First, we assume that generally the mechanical time constant is much larger than the electrical time constant Here, the time constant is a measure of rise time Therefore, if one time interval of a current control loop is set smaller than a mechanical rise time, it can be assumed that the angular speed is constant during one estimation interval In this paper, by introducing this assumption, an algorithm estimating the back-EMF parameters is designed The components of the back-EMF are defined as : Then, from the assumption that the speed of a motor is constant within one estimation interval, the derivative of back-EMF is expressed in matrix form as : (3) back-EMF Eq', Ed' as state variables, is given as follows : U ( t ) = [vq,q Vd"] In order to perform computer simulations and implement the Kalman Filter algorithm, continuous state equations can be transformed into discrete state equations as follows x ( k t l ) = F(k)x(k)+G(k)u(k) (7) $4 (8) = H(k)x(k) where F ( k ) = I + A T,, G ( k )= B I ; , H ( k ) = C Ill THE REDUCED-ORDER EXTENDED KALMAN FILTER ALGORITHM The EKF is an optimal recursive estimation algorithm based on the least-square sense for estimating the states of dynamic nonlinear systems That is, it is an optimal estimator for computing the conditional mean and covariance of the probability distribution of the state of a nonlinear stochastic system with uncorrelated Gaussian process and measurement noise Since the state models are nonlinear, the EKF can be applied to estimate state variables In this case, the back-EMF is considered as a state variable Nonlinear discrete models with white noise are given as follows : x(k t 1) =f(x(k),u(k)) z(k)=h(x(k))tv(k) Therefore, a dynamic model of IPMSM in the stationary reference frame, by choosing stator currents i,,', id' and Et,"]' , T 4k) (9) (10) where w(k) and v(k) are zero-mean noise with covariance Q and R respectively and are independent from the system state x(k) The system noise w ( k ) takes into account system disturbances and model inaccuracies, while v(k) represents the measurement noise The initial state vector i(0) is a Gaussian random vector with mean "(0) and covariance matrix P(O), and u(k) is the deterministic input vector For linearization process in the model, the partial derivative is introduced and discrete state models are 870 From the dynamic model given from (9) to (12), the hack-EMF can be estimated by the following EKF algorithm 1) estimation of an error covariance matrix P-(k t 1) = r(k)P(k)rT(k)+ 2) computation of a Kalmanfilter gain (13) K ( k + 1) = P-(k + ])A7 (k)[A(k)P-(k + l)AT(k) + RI-' ( I 4) ) update of a error covariance matrix P(k + 1) = [I - K(k + I)A(k)]P-(kt 1) (15) 4) state estimation P(k + 1) = P(k) + K(k + #Z(k + 1)- h(i(k +I))] (16) where P - ( k ) is a priori error covariance matrix The rotor speed w, is supposed to he constant within an estimation period Then, in order to estimate the angular speed and rotor position, the reduced-order EKF model is rearranged by choosing hack-EMF and angular speed as state variables instead of dq-axis stator currents As a result, the dimension of EKF model in (4) is reduced as shown in (17), which in turn saves computation time The rearranged system and measurement models with noise are given as follows : Once the back-EMF is estimated, the angular speed and the rotor position are also easily obtained The choice of initial values for matrices R , Q and P(0) is very important Generally, P(0) determines the initial transient characteristics of the filter On the contrary, R and Q represent the dynamic charateristics during the transient-state and steady-state In this paper, to obtain the coefficients of the covariance matrix, a Gaussian noise generator is used Matrices are given as follows : P(0) = :j 0.1 O (21) 0.001 The initial state vector x(0) can be considered as a null vector N.SENSORLESS CONTROL STRATEGY A Rotor Angle Estimation Strategy [I, ( k t 1) - 6] I , ( k ) - Go,]= [F,, O ] ~ ~ ~ ; $t' v( ] k ) ( 8) The models are denoted by the following equations To achieve high performance vector control, the accurate rotor angle, that is, the angle transformed in a synchronous reference frame is required But when the angular speed is close to zero, a lot of enor is included in the back-EMF Therefore, the estimated angle is largely fluctuating and the system doesn't converge to the steady state In this paper, the trigonometric function of rotor angle is not used directly Instead, the arc-tangent function of the back-EMF is used The reason for this is that stability of the system is guaranteed in the low speed or start-up The rotor angle is obtained from (2) as follows : I In the case of reverse speed direction, 180 degree phase difference is generated For this, it is necessary to compensate the difference Therefore, the rotor angle is given as follows : 871 if w, > 0, 8, " = 0, ~1 ' (23) controlled to obtain the maximum torque The relationship between the magnitude I , and phase angle p of armature current to obtain the maximum torque per armature current is derived as follows B Speed Estimation Strategy The information of the angular speed can be obtained from (2) The angular speed can be derived as follows : p = sin-' (27) In the maximum torque control, the current phase angle is actively controlled according to (27), which is a function of I, The speed attainable at the maximum torque is limited by the available maximum output voltage of the inverter As the power factor is improved by the maximum torque control, the limited speed also increases As a result, the maximum output power becomes large, and power capability is greatly expanded Fig shows the block diagram of a high performance speed control system with maximum torque control The magnitude command I,' of the armature current is determined through the speed controller The current phase command p is calculated according to (27) based on motor parameters The current commands and I