Design Implementation of Speed Controller Using Extended Kalman Filter for PMSM

Design Implementation of Speed Controller Using Extended Kalman Filter for PMSM

Mamatha Gowda,

Dept. of ECE,

Prairie View A&M University, Prairie View, TX 77446

USA

Warsame H. Ali,

Dept. of ECE

Prairie View A&M University, Prairie View, TX 77446

USA

Penrose Cofie,

Dept. of ECE,

Prairie View A&M University, Prairie View, TX 77446

USA

John Fuller,

Dept. of ECE,

Prairie View A&M University, Prairie View, TX 77446

USA

Abstract A novel design impl1ementation of proportional integraldifferential equivalent controller using state observer based Extended Kalman Filter (EKF) for a Permanent Magnet Synchronous Motor (PMSM) is proposed. The EKF is constructed to achieve a precise estimation of the speed and current from the noisy measurement. Then, a proportional integral derivative (PID) controller is developed based on Linear Quadratic Regulator (LQR) to achieve speed command tracking performance. In the present method, the speed and q-axis current are estimated accurately using the EKF algorithm. The steady state and transient state response of the overall system is greatly enhanced and also the speed control is achieved effectively under load disturbance. The experimental results for the speed response and q-axis current as well as the control signal variations when the PMSM is subjected to the load disturbance are presented. The results verify the effectiveness of the proposed method.

Keywords – Permanent Magnet Synchronous Motor, Extended Kalman Filter, PID, Vector Control.

1. INTRODUCTION

   The most utilized motor in the field of variable speed electric drive applications is the Permanent Magnet synchronous Motor (PMSM). The reason for this use is due to its higher full load efficiency and power factor, high torque/inertia ratio, and its wide range of speed control [1-4]. In PMSM drives, the vector control theory is applied in the d-q reference frame where the flux and torque are controlled independently [5-7].

   To measure the shaft position of the PMSM the optimal encoder is used. Speed signal is obtained through the discrete differentiation of the encoder position; this signal is very noisy and has an inherent delay. Thus, it adversely impacts the

   1

   performance of the system. To solve this problem, the speed can be estimated through the EKF algorithm. The estimated speed is very accurate and much better than that obtained using the encoder. The Extended Kalman Filter produces the optimal estimation of the states

   based on the least square method. Here, a feedback system control is utilized to estimate the process. The EKF provides good tolerance for the mathematical model error and noises in the measurement.

   Many research works have concentrated on speed and position estimation of PMSM using sliding mode method, high frequency signal injection method, adaptive control theory, fuzzy control, state observer, and the EKF approach

   [8-16]. In all, the EKF method is more attractive as well as popular and is continuously being used in research and applications because it delivers rapid, precise, and accurate estimation. Also, in many applications, the EKF method is implemented because of its low-pass filter characteristics. The feedback gain used in EKF achieves quick convergence and provides stability for the observer.

   In this paper a vector control method is developed and implemented by means of the Extended Kalman Filter algorithm to provide the speed control. Here, state space representation of the model and observer is obtained. Utilizing the state space model, a PID controller is designed using Linear Quadratic Regulator (LQR) approach [17]. The accurate estimation of states is very essential to achieve better control and performance of the PMSM drives. Here the EKF is utilized for the precise

   estimation of the rotor speed and stator q-axis current. Since the For the speed controller design, the system output is

   drive speed and drive current measured directly from the machine terminals contain noise, they are not precise for speed

   y 0

   1isq

   . (9)

   control. In the proposed implementation, the speed and q-axis current are estimated accurately by introducing EKF algorithm theory. The proposed method yields a smooth and quick speed tracking. It also reduces the actual disturbance applied to the

   mech

   If the motor output is considered as the drive current, then the motor output is given by:

   system, and provides better control of the control signal variation. The overall system performance under load condition

   y' 1

   0isq

   . (10)

   is greatly enhanced.

2. DESCRIPTION OF MOTOR MODEL

   The stator voltage and stator flux linkages equations in the rotor references frame are [5]:

   vsq (t)

   mech

   Ls s Rs

   1

   TL

   1

   Js B

   p

   isq (t)

   2 fd

   vsd

   Rs isd

   • L d i

   s dt sd

   • m

     Ls isq

     , (1)

     fd

     2

     p

     v R i

     • L d i

       (L i

       ) , (2)

       sq s sq

       s dt sq

       m s sd fd

       sd Ls isd fd , (3)

       Fig1. Q-axis subsystem of a simplified PMSM

       sq

       Ls isq

       (4)

3. PID PARAMETERS TUNING WITH STATE- FEEDBACK AND STATE-FEEDFORWARD LQR

   where

   vsd , vsq , isd , isq , sd , sq are d-axis and q-axis voltages,

   The controller is designed as a single-inputsingle-output (SISO)

   currents and flux linkages respectively. Rs and Ls are the stator winding resistance and inductance respectively.

   The electromagnetic torque generated and the acceleration is

   system. The following discussion is based on the q-axis controller/observer design [18].

   The state space model of the simplified PMSM, G1(s), with reference to the q-axis is:

   given by:

   x1 (t) A1 x1 (t) B1u1 (t) ,

   y1 (t) C1 x1 (t)

   x1 (0) x10

   (11a)

   (11b)

   T p

   i , (5)

   where, x (t) R21 , u (t) R11 , y (t) R11 , and A1,B1 andC1 are

   em 2

   d

   fd sq

   T T B

   1 1 2

   constant matrices.

   dt mech

   em L

   J

   J mech

   , (6)

   The entire system output is the motor output plus the load disturbance and this is given by:

   p

   m 2

   mech

   (7)

   y(t) y1 (t) d(t)

   where y(t) R11 , d(t) R11 .

   (12)

   where mech

   poles.

   is the speed of the motor and p is the number of

   The state space model of the speed controller, G2(s), can be written as:

   Since the torque developed by the motor is directly proportional to the motor drive current, it is difficult to control

   x 2 (t) A2x2 (t) B2 u2 (t),

   x2 (0) x20

   , (13a)

   the drive current directly. Therefore the drive current is

   y2 (t) C2x2 (t) u1(t) , (13b)

   indirectly controlled through the input voltage. The simplified

   u2 (t) y(t) Ec r(t)

   (13c)

   model of the PMSM q-axis subsystem is shown in Fig.1. In

   2 2 2 2 2

   vector control, assuming ids=0, the state-space model of the

   where x (t) R 21 , u (t) R11 , y (t) R11 , r(t) R11 , and A , B

   , C , E are constant matrices.

   PMSM q-axis subsystem is derived as [18]: 2 c

   R p

   In order to convert the PID tuning problem to an optimal

   sq

   s

   Ls

   ( fd )

   Ls 2 isq

   1

   L v

   (8)

   design, we modify the closed loop cascade system into an

   augmented system with d(t)=0. The result is the following

   p

   B

   s sq

   equation:

   mech

   ( fd )

   e

   2 J

   J

   mech

   0

   x (t) A x (t) B u (t) E r(t) , (14a)

   e

   e

   e

   e

   1

   ye (t) y1 (t) Ce xe (t)

   where

   (14b)

   A A1 0 , B B1 , E 0

   , x x1 (t) , C C

   0.

4. STATE OBSERVER BASED EXTENDED KALMAN

   e B C A

   e 0 e B E e x (t) e 1

   FILTER

   2 1 2

   2 c

   2

   The resulting state-feedback LQR for the augmented system is:

   The state space model of the simplified PMSM given by (11a)

   u (t) K x (t) K x (t) K x (t)

   (15)

   and (11b) is the non-linear system. To apply the EKF algorithm,

   1 e e

   1

   2

   where K R12 , K

   1 1 2 2

   R12 .

   the system needs be discretized and linearized [19]. The discrete approximated equation is given by:

   The quadratic cost function J for the system is given as:

   xk (I AT)xk1 BTuk , (21a)

   T

   J 0 [xe

   (t)Q x (t) u T (t)Ru (t)]dt

   (16)

   yk cxk . (21b)

   e 1 1

   where Q 0 is the state variation and R 0 is the control energy consumption.

   The optimal state-feedback control gain which minimizes the performance index is given by:

   The nonlinear stochastic equation is:

   xk f (xk1, uk ,0)

   yk h(x k ,0)

   L

   L

   2

   (1 TRs ) i (Tfd p)

   T v

   (22a)

   (22b)

   1 T

   (17)

   sq mech

   f (x k , u k ) s s

   sq

   s

   (23a)

   Ke R Be P

   (T p fd ) i

   L

   (1 TB

   where matrix P>0 is the solution of the Riccati equation,

   2 J sq

   J ) mech

   e e e e

   PA A TP PB R1B TP Q 0

   (18)

   h(xk ,0) H xk . (23b)

   e

   Let (Ae, Be) be the pair of the given open loop system and h>0 represents the prescribed degree of relative stability. Then the closed-loop system (Ae – BeR-1B TP) has all its eigenvalues lying

   The Jacobian matrices of partial derivative of f and h with respect to x are given by:

   in the left of the h vertical line in the complex s-plane.

   Ak

   f (xk ,uk ) | x

   x k

   x k

   , (24a)

   The solution of the revised Riccati equation is given by:

   H h(xk ) | x

   x

   , (24b)

   P(A

   • hI) (A hI)T P PB R1B TP Q 0

   (19)

   k x k k

   e e e e

   where P 0, h 0.

   From (23)

   1

   Ak

   TRs Ls

   • ( Tfd

   Ls

   • p )

   2

   (24c)

   Controller(G2(s))

   Motor(G1(s))

   (T p fd )

   1 TB

   2 2 J J

   Hk 1

   0. (24d)

   Fig.2. PID controller system

   The total control law is equivalent to a PID controller with

   The PMSM states x can be estimated by using EKF algorithm during each sampling time interval as follows:

   1. Time update step

      [K1x1(t)] acting as proportional and derivative controller and

      x pk f (x k1 , u,0)

      (25a)

      [K2] acting as integral controller.

      P A P

      A T W Q

      W T

      By choosing the desired values of h and the weighting

      pk k

      k1 k

      k k1

      k

      matrices Q and R, the control gain can be determined. The block diagram of the designed augmented system including the

   2. Measurement update step

   K P H T (H P H T V RV T )1

   controller is shown in Fig.2.

   k pk k

   k pk k

   k k

   (25b)

   x ck x pk K k (yk Hk (x pk ,0)

   The calculated control gains are given by:

   P (I K H )P

   ck k k pk

   K1 0.2241 0.0030 , (20a)

   K2 – 0.1333 . (20b)

   where Wk and Vk are zero- mean- white Gaussian process and measurement noise with covariance Q and R.

   The important and difficult part in the design of the EKF is choosing the proper values for the covariance matrices Q and R

   Variable	Physical Meaning	Value	Unit
   Rs	Armature Resistance	0.1127
   Ls	Armature Inductance	3.63e-4	H
   J	Moment of Inertia	1.267e-4	Kg m2
   B	Damping Coefficient	2.485e-4	N m / rad / s
   fd	Stator Flux Linkage	0.0131	V / rad / s
   p	poles	10
   TL0	Static Friction	0.0237	N m

   [20-21]. The change of values of covariance matrices affects both the dynamic and steady-state. By using trial and error method, a suitable set of values of Q and R are selected to insure better stability and convergence time.

   The chosen values of Q, R and P are:

   Fig.3. Block diagram of PMSM drive

   Table.1 PMSM parameter

   Q = 0.008 0

   0 1.5

   ; R= [0.02] (26a)

   The reference speed is changed from 0 to 100rad/sec and the speed, current and control signal responses are recorded under no

   P = 1 0 (26b)

   0 1

5. EXPERIMENTAL RESULTS

   Experiments are conducted in order to verify and validate the result obtained through simulation. The PMSM speed controller is implemented using DS1104 dSPACE board. The model implementation is achieved in the real time (RTI) in dSPACE board where the simulink code is converted directly into DSP code in the expansion box. The control-desk which acts as user interface is used to regulate the output of the system. The block diagram used in the implementation is as shown in the Fig.3.

   The PMSM parameters are given in Table 1. Based on optimal control theory, the desired control gain K1 and K2 are determined and the Kalman filter gain is obtained using EKF algorithm. The output drive current and speed are estimated through the EKF algorithm as previously discussed. The inputs are taken directly from the machine terminals. At the output, the motor response is checked and the speed control is observed.

   load condition with non-observer. Fig.4 (a) shows the Speed response, Fig.4 (b) shows the actual Isq current response, and Fig.4(c) shows the control signal Vsq response for the input reference change.

   Similarly speed response, actual Isq current response, and control signal Vsq response are observed and recorded under load variations. The experiment results in Fig. 5(a)(b)(c) are shown for when load increase and Fig. 6(a)(b)(c) are shown for when load decrease. It is observed from the results that the speed deviation is around 22rad/sec and control signal variation is nearly 1.26Volts with the transient time of 1.45s. It is also observed that the measured speed signal is too noisy and there is an over shoot in the control signal.

   120

   100

   Speed (rad/sec)

   80

   60

   40

   20

   0

   -20

   Actual

   Reference

   0 1 2 3 4 5

   Time (sec)

   Fig.4(a)

   Current Isq (Amps)

   10

   5

   0

   -5

   -10

   Actual Isq

   X: 1.439

   Y: 2.397

   0 1 2 3 4 5

   Time(sec)

   Fig.4 (b)

   10

   Current Isq (Amps)

   5

   0

   -5

   -10

   X: 2.378

   Y: 8.185

   10

   Actual Isq

   X: 1.142

   Y: 1.864

   0 1 2 3 4 5

   Time (sec)

   Fig.5 (b)

   10

   Control signal (Volts)

   8

   6

   4

   2

   Vsq

   0

   9

   Control signal (Volts)

   8

   X: 0.9218

   Y: 6.923

   7

   6

   5

   X: 1.069

   Y: 6.597

   Vsq

   0 1 2 3 4 5

   Time (sec)

   0 1 2 3 4 5

   Time (sec)

   Fig.4 (c)

   Fig.4. Experimental results for the input reference change under no load with Non-observer

   1. Speed response (b) Actual Isq current response

   (c) Control signal Vsq response

   140

   Fig.5 (c)

   Fig.5. Experimental results for the load increase with Non observer

   (a) Speed response (b) Actual Isq current response

   (c) Control signal Vsq response.

   X: 1.727

   Y: 121.7

   140

   Actual

   Speed (rad/sec)

   120

   100

   80

   60

   Actual

   Reference

   X: 1.087

   Y: 77.23

   0 1 2 3 4 5

   Time (sec)

   Fig.5 (a)

   120

   Speed (rad/sec)

   100

   80

   60

   Reference

   0 1 2 3 4 5

   Time (sec)

   Fig.6 (a)

   Current Isq (Amps)

   10

   5

   0

   -5

   -10

   Actual Isq

   X: 1.758

   Y: -1.447

   120

   100

   Speed (rad/sec)

   80

   60

   40

   20

   0 EKF speed

   0 1 2 3 4 5

   Time (sec)

   -20

   Reference speed

   0 1 2 3 4 5

   Time (sec)

   Fig.6 (b)

   Fig.7 (b)

   10

   10

   X: 1.704

   Y: 8.586

   Vsq

   Control signal (Volts)

   9

   X: 1.565

   Y: 8.13

   8

   X: 2.982

   Y: 6.889

   7

   6

   5

   Current Isq ( Amps)

   0

   -5

   -10

   Actual EKF

   0 1 2 3 4 5

   Time (sec)

   5

   0 1 2 3 4 5

   Time (sec) 8

   Fig.7 (c)

   Fig.6 (c)

   Fig.6. Experimental results for the load decrease with Non observer

   (a) Speed response (b) Actual Isq current response

   (c) Control signal Vsq response.

   With the reference speed of 100rad/s, the experimental results with the EKF are shown in Fig.7 under no load condition. It is observed that the motor speed tracks the reference speed quickly and smoothly. Fig.7 (a) shows the Speed response, Fig.7 (b) shows the estimated speed response Fig.7 (c) shows the actual Isq current response, and Fig.7 (d) shows the control signal Vsq response for the input reference change. The actual current and estimated current have same value. The control signal variation is also smooth without any overshoots. It is evident that the proposed method estimates the rotor speed and q-axis current exactly.

   120

   100

   Speed (rad/sec)

   80

   60

   40

   20

   EKF

   0 Actual

   6

   Control signal Vsq (Volts)

   4

   2

   0

   Vsq

   -2

   0 1 2 3 4 5

   Time (sec)

   Fig.7 (d)

   Fig.7. Experimental results for the input reference change under no load with EKF

   (a) Speed response (b) Estimated Speed response

   1. Actual and estimated Isq current response

   2. Control signal Vsq response

   The performance of the proposed technique under load disturbance is also verified. The load is varied from – 0.2Nm to 0.2Nm. The Speed response, estimated Speed response, actual Isq current response, and control signal Vsq response are observed and recorded under load variations. The experimental results in Fig. 8(a)(b)(c)(d) are shown for when load increase and Fig. 9(a)(b)(c)(d) are shown for when load decrease. From the results it is evident that speed deviation is reduced from 22 rad/sec to 10 rad/sec and it quickly follows the reference speed.

   -20

   Reference 0 1 2 3 4 5

   Time (sec)

   Fig.7 (a)

   The voltage is also reduced from 1.25v to 0.05v and with less transient time of around 1.2s. The measured speed signal is smooth and contains less noise compared to the non-observer.

   120

   Speed (rad/sec)

   110

   100

   EKF

   120

   Speed (rad/sec)

   110

   EKF

   Reference

   90

   X: 1.541

   Y: 89.18

   80

   0 1 2 3 4 5

   Time (sec)

   Fig. 8 (a)

   10

   Reference

   X: 1.644

   Y: 110.1

   100

   90

   80

   0 1 2 3 4 5

   Time (sec)

   Fig. 9 (a)

   X: 1.663

   Y: 2.325

   Current Isq ( Amps)

   5

   10

   X: 1.459

   Y: 2.353

   Current Isq ( Amps)

   0 Actual

   5

   -5

   Actual 0

   -10

   0 1 2 3 4 5

   Time (sec)

   Fig. 8 (b)

   -5

   -10

   0 1 2 3 4 5

   Time (sec)

   10

   Current Isq ( Amps)

   5

   0

   -5

   -10

   X: 1.663

   Y: 2.325

   10

   Current Isq ( Amps)

   5

   Actual 0

   EKF

   Fig. 9 (b)

   X: 1.458

   Y: 2.32

   Actual EKF

   0 1 2 3 4 5

   Time (sec)

   Fig. 8 (c)

   8

   X: 2.565

   Y: 6.589

   Control signal Vsq (Volts)

   Vsq

   X: 1.345

   Y: 6.529

   7

   -5

   -10

   8

   0 1 2 3 4 5

   Time (sec)

   Fig. 9 (c)

   X: 1.605 Vsq

   Y: 7.268

   Control signal Vsq (Volts)

   7

   6

   X: 1.404

   Y: 6.629

   X: 2.677

   Y: 6.589

   X: 1.527

   Y: 5.917

   6

   5

   4

   0 1 2 3 4 5

   Time (sec)

   Fig. 8 (d)

   Fig.8. Experimental results for the load increase with EKF

   1. Speed response (b) Actual Isq current response (c)Estimated Isq current response

   (d) Control signal Vsq response.

   5

   4

   0 1 2 3 4 5

   Time (sec)

   Fig. 9 (d)

   Fig.9. Experimental results for the load decrease with EKF

   (a) Speed response (b) Actual Isq current response

   1. Estimated Isq current response

   2. Control signal Vsq response.

   From the obtained result, it is observed that the speed tracking is smooth and quick. The disturbance is quickly rejected by the system and attains the actual speed. With the implementation of EKF algorithm, the transient time and control signal variation are reduced compare to the non- observer.

6. CONCLUSION

A novel implementation of proportionalintegraldifferential equivalent controller using state observer based Extended Kalman Filter (EKF) for a Permanent Magnet Synchronous Motor (PMSM) is proposed. The proposed method accurately estimates the speed and current. Also the proposed approach mitigates speed deviation caused by the load disturbance and attenuates the control signal variation. The steady and transient state responses are also improved. The EKF algorithm method attains good speed tracking. The system performance is enhanced significantly using the proposed method.

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