Unscented Kalman Filter Based Observer for Vector Controlled Induction Motor

Unscented Kalman Filter Based Observer for Vector Controlled Induction Motor

Prashant Bhopale1, Swati Dhobaley2, D. R. Mehta3

1,2,3 Dept. of Electrical Engg. Veermata Jijabai Technological Institute, Mumbai-19., India.

AbstractThe extended kalman filter(EKF) suffers from 1st order approximation error which causes deviation in the mean and covariance of estimation while Unscented kalman Filter(UKF) uses the nonlinearity in the system without linearization to overcome this errors. The nonlinear approach of unscented transform has been used for rotor flux estimation in the presence of white Gaussian noise and results inferred.

KeywordsExtended Kalman Filter; Unscented Kalman Filter; nonlinear;white gaussian noise.

1. INTRODUCTION

   The induction motor drives are widely used in industrial application due to the advantages in terms of robustness and prize, hence the motor control industry has become a strong and aggressive sector. During the last few decades the field of controlled electrical drives has undergone rapid expansion hence better estimation of the control parameter and states of control vectors are necessary. The Vector control induction motor consists of controlling the stator currents represented by a vector. The different methods of speed and flux estimation needs model which is sensitive and based on machine parameters and also require apriori knowledge of electrical and sometimes mechanical characteristics.

   Some type of speed estimation is essential for speed control of induction motor. Magnetic saliencies depending speed estimation like rotor slotting [1], variation of leakage reactance [2] or rotor asymmetries [3] which are considered as true measurement and independent of machine parameters. Due to reduced range of frequency they are not widely used. Although machine model dependent estimations which uses open loop speed calculators[4,5],Model reference adaptive system(MRAS)[6,7,8,9] and Extended Kalman Filter[10]. Also deriving machine flux used methods are from machine models are integration of back emf [4,5], flux observer

   [22]Julier showing nonlinear estimation gives better performance than linearized strategies like EKF.

   This paper is organized as follows. The fundamental model of vector control induction machine is described in Section II together with the theoretical basis for FOC strategy. The UKF algorithm is presented in Section III. In section IV MATLAB/Simulink simulation results obtained with the proposed algorithm applied on an ideal model of the machine is provided. In Section VI conclusions are drawn.

2. SYSTEM DESCRIPTION.

   1. Vector Controlled Induction Motor Drive

      First, The block diagram of the proposed system is shown in Fig.1 The continuous-time mathematical model of the induction machine can be described in state-space form, with rotor speed treated as a time-varying parameter,

      = , + + () (1)

      = 1,2, 4

      x is state variable and the input voltage vector u are given as x = [ids iqs dr qr ] where s and r stands for stator and rotor and d and q are vectors of respective fixed stator frame and ()is process noise.

   2. Mathematical Modelling of Vector Controlled Induction Motor Drive

      The four first order differential equations resulting from the expansion of eqn. 1 are:

      [6,7,8,9, 11], Extended Luenberger Observer [12], Monitoring local saturation effect [13] and extended kalman filter

      R

      s

      L

      x 1 =

      m

      Rr L2

      m 1

      + x +

      r

      L2 L

      Rr Lm

      r

      L2 L

      x3 +

      wr Lm

      r

      L2 L

      x4 +

      uds (2)

      1

      L

      [13,14,15,16,17]. But the above mentioned methods which are

      model dependent have considered the linearized model of

      x = Rs + Rr L2 x

      r

      1. L L2 L 2

         • wr Lm x

           r

           L2 L 3

           + Rr Lm x

           r

           L2 L 4

           + 1 u

           L qs

           (3)

           induction motor, although where 1st order approximation

           using EKF can introduce mean and covariance errors.

           x = Rr Lm x Rr x w x

      2. Lr 1 Lr 3 r 4

         (4)

         In practice, the rotor currents or fluxes are not easily available and measurable. Therefore an observer is required to estimate the unknown states and can also provide better estimates of known current states that are contaminated by noise in some circumstances,. In this paper we propose the UKF based observer for flux estimation of vector controlled induction motor which takes into account the nonlinearity of the model using nonlinear approach of unscented transform

         x = Rr Lm x

      3. Lr 2

   + wr x3

   • Rr x

   Lr 4

   (5)

   = = 1

   = 1,2,3, 2 (11)

   Where,

   2( +)

   = 2 + (12)

   Fig.1 Indirect Vector Control Drive

   Where L is linkage inductance i.e.

   L

   2

   L = Ls m (6)

   Lr

   The states x1 and x2 are the stator d-q axis currents, which are usually measurable. The statesx3 and x4 are the d-q axis rotor fluxes, which are not easily measurable and require estimation. Sometimes, the measured stator currents are contaminated by noise, and require to be estimated as well. In some circumstances the estimates can be more reliable.

   An output or observation equation is required by the state space model and for the practical case in which stator and rotor flux are chosen as measurements:

   = (()) (7)

   = [ dr qr ] (8)

3. ALGORITHM OF UNSCENTED KALMAN FILTER

   determines the spread of sigma points around and usually set to a small positive value. is a secondary scaling parameter which is usually set to 0 and is a parameter used to incorporate any priory knowledge about the distribution of

   . + is the th row or column of the matrix

   square root of + and is the weight which is associated with the th point. These sigma points are propagated through the function.

   B. Implementation of UKF

   For L dimension state vector, 2L+1 sigma points are related as follows,

   Step1: Creating Sigma Points

   1 = 1 1 + 1 1 1

   (13)

   Where, = + and = 2 +

   Step 2: Propagating Sigma Points or Prediction

   , = ,1 , = 1,2, . . ,2 + 1

   (14)

   Step 3: Calculating Mean & Covariance

   =0

   Unscented Kalman Filter (UKF) also known as Sigma Point Kalman Filter[22], priory needed to created sigma points which are selected by priory knowledge of previous state and

   1 = 2

   , (15)

   covariance matrices. 2L+1 Sigma points are more useful when

   = 2

   +

   creating sigma points for L dimensional state vector. Following are the steps for UKF algorithm.

   A. Sigma Points:

   =0

   (16)

   ,

   1

   ,

   1

   1

   The dimensional random variable with mean and co-variance is approximated by 2 + 1 weighted points given by,

   , = ,

   (17)

   =0

   = 2 ,

   (18)

   0 =

   = ± + = 1,2,3, 2 (9)

   = 2

   [_(, ) _]

   = , = + 2 + (10)

   =0

   (19)

   ,

   1

   0 +

   0 +

   = 2

   +

   (20)

   =0

   ,

   ,

   1

   Step 4: Correction

   1. CONCLUSIONS

      In this paper, we have implemented nonlinear model of

      =

      (21)

      1

      indirect vector control induction motor without linearization, and simulation results are presented. The UKF based flux observer is created and performance of observer is presented

      = 1 +

      (22)

      in the presence of noise. Result shows that the Stator and Rotor flux of Vector Control Induction Motor can be

      =

      (23)

      successfully estimated by observer designed using UKF.

   2. ACKNOWLEDGMENT

      Process noise and Measurement noise are

      characterized by

      = 0 & = 0

      The authors would sincerely like to acknowledge the support of Mr. Pratik K. Bajaria, VJTI, Matunga, Mumbai, India, for his valued contributions to this paper.

      (24)

      = 0 &

      APPENDIX

      Rating and Parameters of Induction Motor used for

      =

      (25)

      0

      simulation

      4 Poles,60 Hz,220 V,20 Hp

      Rotor Resistance( ) = 1.772 Stator Resistance( )=1.282

4. SIMULATION AND ESTIMATION

The rotor flux estimation is obtained by applying UKF algorithm. The UKF used the nonlinear model (2)-(5) The simulation results for UKF is shown in Fig.2 where 9 sigma points are created by considering 4 state including rotor and stator current along with rotor and stator flux as augmented states which are estimated by observer. Scaling parameters and process and measurement noise of UKF are selected as [23]. Sampling time taken 0.01sec and simulation carried out for 10 sec when motor accelerates from rest to 255 rpm.

Fig.2 Results obtained by UKF observer

Stator Inductance( )=0.282mH Rotor Inductance( )=0.101mH Mutual Inductance( )=0.161mH

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