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State Estimation for Sensorless Control of BLDC Machine with
Particle Filter Algorithm978-1-5386-9254-7/19/$31.00 ©2019
IEEE

State Estimation for Sensorless Control of BLDC Machine with Particle Filter Algorithm

Yaser Chulaee 1, Hossein Abootorabi Zarchi 2, Seyyed Iman Hosseini Sabzevari 3

1,2,3 Faculty of Engineering

Ferdowsi University of Mashhad

yaser.chulaee@mail.um.ac.ir, abootorabi@um.ac.ir, imanhosseini@mail.um.ac.ir

Abstract – This paper presents a technique in order to estimate the rotor speed and position of the BLDC machine. In the proposed method, particle filter (PF) is employed to estimate state variables of the machine using measured currents and line voltages. PF is a type of stochastic filters that has wide applications in state estimation of non-linear systems. The main aim of this paper is to utilize particle filter in the sensorless control of BLDC machine and investigate proposed PF algorithm performance. In addition, effective parameters in estimation accuracy and transient response of the filter are discussed. The simulation is performed in MATLAB/SIMULINK environment and results denote that proposed sensorless drive have good accuracy in wide speed range and load torque variation. Also, the algorithm performance is not influenced by the incorrect initial position. Index Terms - Brushless DC Machine (BLDCM), Particle Filter, Sensorless Drive

I. INTRODUCTION

Recently, BLDC machines have gained interest in HVAC, EVs and home appliances, etc. because of their advantages such as high torque-inertia ratio, compact size, ease of maintenance, and low audible noise due to the absence of brushes. In practice, our goal is to utilize these machines effectively. For this purpose, various control strategies have been used, almost all of them need the information regarding position and/or speed of machine. As in most BLDC applications position information is needed only for every 60 degrees, hall-effect sensors have been employed [1, 2]. However, in a high-performance application, the exact position and speed are required and, as a result, encoders or resolvers have been used. Adopting these sensors in the drive control system increases cost and complexity, in addition to decreasing reliability. Over the last two decades, many research endeavors have been conducted in order to eliminate these sensors from the drive system, which leads to the introduction of sensorless drives [3, 4]. Some works in the literature obtain position and speed information by using open-loop techniques. In these methods, motor position is identified by terminal voltages measuring [5]. One of the well-known open-loop approaches uses motional back emf voltages as in [6]. This paper shows that inadequate back emf amplitude at low speed makes position detection problematic. This is the main drawback of all motional emf schemes. Another disadvantage that is common in all open-loop approaches is parameters variation sensitivity due to lack of

internal improving mechanism [5]. Closed-loop methods, for example, sliding-mode observer (SMO) and stochastic filtering, had been introduced and developed to overcome these disadvantages. As shown in [7], SMO suffers from the chattering phenomenon. Traditionally, one of the stochastic filters that had been used as an observer is Kalman filter (KF). In KF, all state variables are estimated via measuring the states in a noisy environment. The simple KF is only appropriate for linear stochastic systems, and the extended Kalman filter (EKF) derived from KF is suitable for nonlinear systems. Both of them are based on the least square variance estimation. References [8, 9, and 10] employed EKF to estimate position and speed of the machine using measured stator line voltages and currents. In [11], a novel EKF with the help of Active Learning Method (ALM) was proposed. EKF algorithm uses a linearized model of machine that is done by Jacobian matrix calculation, and, this procedure, besides adding more complexity and hardware implementation problem, is not suitable for highly nonlinear model systems. Reference [12] uses the unscented Kalman filter (UKF) for sensorless control of the PMSM. UKF is an improved type of KF that simplifies its algorithm via eliminating Jacobian matrix calculation step. Application of stochastic filters mentioned above (EKF and UKF) is based on the assumption that the process and measurement noise distributions are Gaussian. One solution for highly nonlinear systems at the presence of non-Gaussian noises is the particle filter (PF). PF algorithm was invented to numerically implement the Bayesian estimator. The PF has some similarity with the UKF but, in contrast to it, theoretically, the estimation error in PF can converge to zero [13]. In this paper, a sensorless drive system of the BLDC machine was investigated. The position and speed of the machine was estimated by means of measuring machine terminal variables using the PF algorithm that employed an effective resampling method. Estimated states fed into the drive system as input signals, and the speed of the machine is controlled effectively.

II. MATHEMATICAL MODEL OF THE BLDC

The electrical equations of BLDC machines are [14] = −R i − λ f (θ )ω + V + V (1)

2019 10th International Power Electronics, Drive Systems and Technologies Conference (PEDSTC) 12-14 February, Shiraz University, Iran

172

= −R i − λ f (θ )ω − V + V (2)

= −R i − λ f (θ )ω − V − V (3)

T = λ f (θ )i + λ f (θ )i + λ f (θ )i (4)

where ωm, θ , R , L , λ are mechanical rotor speed in rad/sec, mechanical rotor position in radian, line-to-line resistance (), inductance (H), and flux linkage in V/(rad/sec), respectively. f , f , f have the same shape as the back EMFs with a maximum amplitude of ±1 and are specified in Table I. The mechanical equations are as follows: = T − T − T (5)

= ω (6) T = B ω (7)

where J , B are the moment of inertia in kg.m2, friction coefficient in Nm/(rad/sec), and electromagnetic, load and friction torque are denoted as T , T , T , respectively [15]. Finally, by combining equations (1)-(7), the BLDC machine can be described in state space form as shown in (8), (9) and (10) at the bottom of the page.

III. THE PF ALGORITHM

In state estimation, based on the Bayesian filtering approach, one solution is constructing the posterior probability density function (pdf) by using all available information, including measurement data. Since this pdf contains all statistical information, optimal state estimation and its accuracy can be derived from it [16]. By using Bayes’s rule a posterior pdf is obtained as [13]: p(x |y : ) = ( | ) ( | : )( | : ) (11)

In (11): p(x |y : ) = p(x |x : )p(x |y : ) dx : (12)

TABLE I Functions f , f and f

0 ≤ θ 6π θ −1 1

≤ θ 1 −1 1 − 6π (θ − π6) ≤ θ 1 −1 + 6π (θ − π2) −1

≤ θ 1 − 6π (θ − 5π6 ) 1 −1

≤ θ −1 1 −1 + 6π (θ − 7π6 ) ≤ θ −1 1 − 6π (θ − 9π6 ) 1

≤ θ 2π −1 + 6π (θ − 11π6 ) −1 1 p(y |y : ) = p(y |x )p(x |y : ) dx (13)

Except for some cases (e.g. linear Gaussian model), analytical solutions to these equations are not available. For other models, (11) needs to be evaluated approximately. The particle filter is represented to approximate posterior pdf by using a set of random weighted particles {(x , w ), n =1:N }, (14) [16]. p(x |y : ) ≈ ∑ w δ(x − x ) (14)

where x is a state value of mth particle, w is its weight, δ(. ) denotes the Dirac delta function, and N is the number of particles. Actually, the weight of a particle demonstrates the probability of the particle which is represented by the equation (15): ω ( ) ∝ ω ( ) ( | ) ( | )( | ) (15)

State estimation with the help of the particle filter is a recursive procedure. After several iterations, most particles will have small weights. This leads to the allocation of a large computational effort to update particles that have a small role in approximating the pdf. This phenomenon is called degeneracy, which is a common problem with PF. To prevent degeneracy, one of the most well-known methods is

x = f(x)x + Bu

ωθ = 0 0 ( ) 00 0 ( ) 00 0 ( ) 0( ) ( ) ( ) 0 0 0 0 1 0

iiiωθ + 0000 00 0 0 VVT

y = Hx = 1 0 0 0 00 1 0 0 00 0 1 0 0 iiiωθ

(9)

(10)

(8)

173

BLDC

Speed Controller

Figure 1

resampling [16]. In the resampling process, small weight particles are replicated with those that have large weight. Consider a nonlinear system described by state equations as follows: x (t) = f(x(t))x(t) + Bu(t) + σ(t) (16) y(t) = h(x(t)) + ν(t) (17)

where σ(t) and ν(t) are zero-means white Gaussian noises with covariance Q(t) and R(t), respectively. Equations (16) and (17) are in continuous time domain, but the PF algorithm is implemented in discrete domain. These equations are discretized as: x = (I + f(x )t)x + Btu + w (18) y = h(x ) + ν (19)

where I is the identity matrix, and t is the sampling period. Note that in these equations, process and measurement noises pdf are assumed to be known. At the first iteration, based on the initial assumed pdf p(x ), N initial particles (x . (i = 1. … . N)) are generated randomly. Choosing parameter N is an important issue in particle filter that affects estimation accuracy and computational burden. In order to obtain prior particles x . , initial particles are propagated in the system discrete equation (18). In the next step, relative weight must be assigned to each prior particle with respect to the measurement data. This is done by evaluating the pdf p(y |x ) , with calculating the likelihood of particles as follows [13]: q = p(y |x . ) ~ ( ) | | exp ( ( ) ( ) ) (20)

where m is related to the output matrix dimension and y is measured data, respectively. According to equations (15) and (20), the weight of each particle is directly proportional to their likelihood. After that, for choosing the large-weighted particles and constructing posterior particles x . , resampling on the basis of normalized likelihood is utilized. Now, the posterior pdf p(x |y : ), is obtained. Finally, the estimation of the states is obtained by calculating the average of approximated pdf p(x |y : ). Note that these posterior particles are propagated at the next iteration in order to generate prior particles.

TABLE II Specifications of the BLDC motor used in simulation

Parameter Value

Number of poles 8

Flux linkage 0.175 Wb

Rated speed 1500 Rpm

Friction factor 0.001 Nm/rad/sec

As mentioned earlier, resampling is a critical step, hence various techniques had been proposed in many papers. These methods have some differences in features such as computational complexity, runtime, and the number of random numbers used [17]. One of the effective methods is stratified resampling that is used in the proposed PF algorithm. Stratified resampling divides the whole population of particles into subpopulations [18]. For more information about the structure of the stratified method and also other resampling strategies, you can see [17].

IV. SENSORLESS DRIVE SCHEME

Fig. 1 shows a block diagram of the sensorless speed control of the BLDC machine. Drive structure consists of two main parts: Power stage and control section. Power stage includes a dc power supply, an inverter, and the BLDC machine. In the control section, the proportional integral (PI) controller is used as a speed regulator that compares reference of the speed with the estimated speed of the machine and generates proper currents command. As demonstrated in Fig. 1, particle filter algorithm estimates speed and position of the rotor by means of measured currents and voltages. Then, these estimated parameters are fed into the control section as feedback signals. As a result, this structure leads to the elimination of speed and position sensors, hence called sensorless.

V. SIMULATION RESULTS

The sensorless speed control of the BLDC motor using particle filter that act as state estimator, has been simulated in MATLAB/SIMULINK software. The specification of the BLDC motor used in the simulation is presented in Table II.

PF algorithm has been implemented in a MATLAB function block that operates similar to a digital processor. The measured currents and line voltages are inputs of the PF block. At each time step, after the PF algorithm process explained in section III, estimated speed and position have been used as feedback signals for the drive system. In the real-time system, any measurement data consist of noise and inaccuracy. Hence, to simulate measurement noise, a white Gaussian noise with known mean and variance was added to the measured variables.

Fig. 1. The proposed sensorless speed control drive structure

174

Fig. 3. (a) Position and (b) speed estimation error at speed reference of 200 to

1200 rpm

A critical step in PF algorithm implementation is choice elements of P0, Q and R matrices. Choosing the proper value of these parameters, results in a better performance and lower convergence time of the filter. Also, transient specifications of the filter can be directly modified with these matrices where matrix P0 and Q affects peak of the transient and transient time, respectively. Diagonal elements of the matrix P0 determine variances of initial particles of each state that construct initial pdf p(x ). The amount of system noise and model parameters uncertainty are presented by the matrix Q. In the simulation environment, any variations in system parameters are not considered, which means elements of Q must be small. Typically, P0 and Q are chosen by a trial-and-error procedure with regard to stability and transient characteristics. Matrix R indicates statistical properties of measurement noise. In this paper, measured data includes only phase currents of the motor, and hence, in practice, elements of the R are determined with

Fig. 4. Details of (a) position and (b) speed estimation error between = 3

and = 6 sec.

the help of current senor manufacturer datasheet, and based on added noise variance in simulation. Covariance matrices P0, Q and R used in simulated PF algorithm are given below:

P = 0.05 0 0 0 00 0.05 0 0 00 0 0.05 0 00 0 0 10 00 0 0 0 0.01 (21)

Q = 0.2 0 0 0 00 0.2 0 0 00 0 0.2 0 00 0 0 1 00 0 0 0 0.01 (22)

R = 5 0 00 5 00 0 5 (23)

E rr

o r

o f

P o

si ti

o n

-0.09

-0.06

-0.03

0

0.03

0.06

/)

Fig. 2. Estimation results of the proposed PF algorithm at = 0 for various speed references. Actual and estimated (a) mechanical speed (b) position and (c)

line current of the motor.

ω (r

ad /s

A)

175

Fig. 5. Transient behaviour of the proposed PF algorithm at electrical initial

position, =15 rad.

Fig. 6. Transient behaviour of the proposed PF algorithm at electrical initial

position, =5 rad.

Note that all simulation tests performed at closed-loop control with the number of the particles N, equal to 15. The estimated speed, position and motor line current by simulated PF algorithm and actual ones that had been measured in sensoreless drive mode are shown in Fig. 2(a), (b) and (c), respectively. Additionally, in order to evaluate sensorless drive performance, the speed of machine when the drive system utilized sensors data is shown in Fig. 2(a) (labeled as “actual sensored drive”). In this simulation experiment, the mechanical speed reference of the motor changes from 200 rpm (20.94 rad/sec) to 1200 (125.66 rad/sec) rpm at time t = 3 seconds and reduced to 200 rpm at t = 6 seconds. As it can be seen in this figure, simulation results indicate that the proposed PF algorithm has a good performance in a wide speed range. Fig. 3 shows the speed and position estimation error and details error in 3 seconds time duration are shown in Fig. 4. As demonstrated in Fig. 4, the maximum estimated position and speed error are 0.065 radian and 6.4 rad/sec, respectively. An important issue in estimation by stochastic filters is behavior at the startup procedure when the assumed initial position value θ , differs from the actual one that may lead to wrong convergence. Hence, a simulation experiment with the incorrect initial position is performed to investigate the proposed filter performance. Fig. 5 and 6 illustrate results of the simulation with two non-zero initial positions of the rotor. As shown in Fig. 6 the PF algorithm has a good operation in this condition, and the filter converges to the actual position in less than 25 ms. For evaluating the persistence of the proposed sensorless drive under load torque variation a simulation test is performed. As shown in Fig. 7(b), at = 1 seconds, a 2 N.m load torque is suddenly applied to the machine. Fig. 7(a) shows that speed estimation has a good accuracy even at load torque change.

Fig. 7. (a) Estimated and actual speed of the motor (b) mechanical torque at load torque variation and = 1200 rpm.

VI. CONCLUSION In this paper, a sensorless control of the BLDC machine using PF algorithm as state estimator is presented. Particle filter due to the elimination of model linearization step and propagation of N particles in the system model has suitable performance even for state estimation in highly non-linear systems at the presence of non-Gaussian noises. The number of the particles N, is an important parameter in PF algorithm, and increasing this parameter leads to the enhancement of filter accuracy in exchange for increasing computational burden, and, a trade-off between these two factors is needed. Resampling is such a vital step in particle filter algorithm that adopting convenient resampling strategy can prevent from divergence. For simplifying practical implementation of the PF algorithm, straightforward equations and a step-by-step procedure are presented. Simulation results indicate that the proposed PF algorithm is capable of estimating states of the BLDC machine at various speeds, load torque, and different initial positions with good accuracy. In addition, based on the simulation, the presented sensorless drive system has a good transient and steady state performance.

REFERENCES

[1] P. Vas, Sensorless vector and direct torque control, Oxford, Oxford University Press, 1998.

[2] A. Khazaee, H. Abootorabi Zarchi, and G. Arab Markadeh, "Loss model based efficiency optimized control of brushless DC motor drive," ISA Transactions, 2018.

[3] D. Lenine, B. Rami Reddy, and S. Vijay Kumar, "Estimation of speed and rotor position of BLDC motor using extended Kalman filter," IET-UK International Conference on Information and Communication Technology in Electrical Sciences (ICTES 2007), 2007, pp. 433–440.

[4] P. Tomei and C. Verrelli, "Observer-Based Speed Tracking Control for Sensorless Permanent Magnet Synchronous Motors With Unknown Load Torque," IEEE Transactions on Automatic Control, vol. 56, no. 6, pp. 1484-1488, June 2011.

[5] R. Bojoi, M. Pastorelli, J. Bottomley, P. Giangrande, and C. Gerada, "Sensorless control of PM motor drives - A technology status review," 2013 IEEE Workshop on Electrical Machines Design, Control and Diagnosis (WEMDCD), 2013, pp. 168-182.

[6] T. de Almeida, G. de Paula, A. de Castro, W. Pereira, and J. Monteiro, "Sensorless vector control for BLDC machine," 2017 Brazilian Power Electronics Conference (COBEP), 2017.

[7] A. Deenadayalan and G. Saravana Ilango, "Position sensorless sliding mode observer with sigmoid function for Brushless DC motor," 2012 International Conference on Advances in Power Conversion and Energy

θ (r

( N

.m )

176

Technologies (APCET), 2012, pp. 1-6. [8] B. Terzic and M. Jadric, "Design and implementation of the extended

Kalman filter for the speed and rotor position estimation of brushless DC motor," IEEE Transactions on Industrial Electronics, vol. 48, no. 6, pp. 1065-1073, Dec. 2001.

[9] M. Preindl and S. Bolognani, "Model Predictive Direct Speed Control with Finite Control Set of PMSM Drive Systems," IEEE Transactions on Power Electronics, vol. 28, no. 2, pp. 1007-1015, Feb. 2013.

[10] A. Bado, S. Bolognani, and M. Zigliotto, "Effective estimation of speed and rotor position of a PM synchronous motor drive by a Kalman filtering technique," PESC `92 Record. 23rd Annual IEEE Power Electronics Specialists Conference, 1992, pp. 951-957.

[11] A. Ejlali and J. Soleimani, "Sensorless vector control of 3-phase BLDC motor using a novel Extended Kalman," 2012 International Conference on Advances in Power Conversion and Energy Technologies (APCET), 2012, pp. 1-6.

[12] D. Janiszewski, "Sensorless control of Permanent Magnet Synchronous Motor based on Unscented Kalman Filter," 2011 International Conference on Power Engineering, Energy and Electrical Drives, 2011, pp. 1-6.

[13] D. Simon, Optimal state estimation - Kalman, H infinity, and Nonlinear Approaches, Hoboken (N.J.), Wiley-Interscience, 2006.

[14] B. K. Lee and M. Ehsani, "Advanced Simulation Model for Brushless DC Motor Drives," Electric Power Components and Systems, vol. 31, no. 9, pp. 841-868, Sep. 2003.

[15] R. Krishnan, Permanent magnet synchronous and brushless DC motor drives, Boca Raton, CRC, 2010.

[16] M. Arulampalam, S. Maskell, N. Gordon, and T. Clapp, "A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking," IEEE Transactions on Signal Processing, vol. 50, no. 2, pp. 174-188, Feb. 2002.

[17] T. Li, M. Bolic and P. Djuric, "Resampling Methods for Particle Filtering: Classification, implementation, and strategies," IEEE Signal Processing Magazine, vol. 32, no. 3, pp. 70-86, May 2015.

[18] G. Kitagawa, "Monte Carlo Filter and Smoother for Non-Gaussian Nonlinear State Space Models," Journal of Computational and Graphical Statistics, vol. 5, no. 1, pp. 1-25, March 1996.

177

2019-04-22T15:45:30-0400

State Estimation for Sensorless Control of BLDC Machine with Particle Filter Algorithm

Yaser Chulaee 1, Hossein Abootorabi Zarchi 2, Seyyed Iman Hosseini Sabzevari 3

1,2,3 Faculty of Engineering

Ferdowsi University of Mashhad

yaser.chulaee@mail.um.ac.ir, abootorabi@um.ac.ir, imanhosseini@mail.um.ac.ir

Abstract – This paper presents a technique in order to estimate the rotor speed and position of the BLDC machine. In the proposed method, particle filter (PF) is employed to estimate state variables of the machine using measured currents and line voltages. PF is a type of stochastic filters that has wide applications in state estimation of non-linear systems. The main aim of this paper is to utilize particle filter in the sensorless control of BLDC machine and investigate proposed PF algorithm performance. In addition, effective parameters in estimation accuracy and transient response of the filter are discussed. The simulation is performed in MATLAB/SIMULINK environment and results denote that proposed sensorless drive have good accuracy in wide speed range and load torque variation. Also, the algorithm performance is not influenced by the incorrect initial position. Index Terms - Brushless DC Machine (BLDCM), Particle Filter, Sensorless Drive

I. INTRODUCTION

Recently, BLDC machines have gained interest in HVAC, EVs and home appliances, etc. because of their advantages such as high torque-inertia ratio, compact size, ease of maintenance, and low audible noise due to the absence of brushes. In practice, our goal is to utilize these machines effectively. For this purpose, various control strategies have been used, almost all of them need the information regarding position and/or speed of machine. As in most BLDC applications position information is needed only for every 60 degrees, hall-effect sensors have been employed [1, 2]. However, in a high-performance application, the exact position and speed are required and, as a result, encoders or resolvers have been used. Adopting these sensors in the drive control system increases cost and complexity, in addition to decreasing reliability. Over the last two decades, many research endeavors have been conducted in order to eliminate these sensors from the drive system, which leads to the introduction of sensorless drives [3, 4]. Some works in the literature obtain position and speed information by using open-loop techniques. In these methods, motor position is identified by terminal voltages measuring [5]. One of the well-known open-loop approaches uses motional back emf voltages as in [6]. This paper shows that inadequate back emf amplitude at low speed makes position detection problematic. This is the main drawback of all motional emf schemes. Another disadvantage that is common in all open-loop approaches is parameters variation sensitivity due to lack of

internal improving mechanism [5]. Closed-loop methods, for example, sliding-mode observer (SMO) and stochastic filtering, had been introduced and developed to overcome these disadvantages. As shown in [7], SMO suffers from the chattering phenomenon. Traditionally, one of the stochastic filters that had been used as an observer is Kalman filter (KF). In KF, all state variables are estimated via measuring the states in a noisy environment. The simple KF is only appropriate for linear stochastic systems, and the extended Kalman filter (EKF) derived from KF is suitable for nonlinear systems. Both of them are based on the least square variance estimation. References [8, 9, and 10] employed EKF to estimate position and speed of the machine using measured stator line voltages and currents. In [11], a novel EKF with the help of Active Learning Method (ALM) was proposed. EKF algorithm uses a linearized model of machine that is done by Jacobian matrix calculation, and, this procedure, besides adding more complexity and hardware implementation problem, is not suitable for highly nonlinear model systems. Reference [12] uses the unscented Kalman filter (UKF) for sensorless control of the PMSM. UKF is an improved type of KF that simplifies its algorithm via eliminating Jacobian matrix calculation step. Application of stochastic filters mentioned above (EKF and UKF) is based on the assumption that the process and measurement noise distributions are Gaussian. One solution for highly nonlinear systems at the presence of non-Gaussian noises is the particle filter (PF). PF algorithm was invented to numerically implement the Bayesian estimator. The PF has some similarity with the UKF but, in contrast to it, theoretically, the estimation error in PF can converge to zero [13]. In this paper, a sensorless drive system of the BLDC machine was investigated. The position and speed of the machine was estimated by means of measuring machine terminal variables using the PF algorithm that employed an effective resampling method. Estimated states fed into the drive system as input signals, and the speed of the machine is controlled effectively.

II. MATHEMATICAL MODEL OF THE BLDC

The electrical equations of BLDC machines are [14] = −R i − λ f (θ )ω + V + V (1)

2019 10th International Power Electronics, Drive Systems and Technologies Conference (PEDSTC) 12-14 February, Shiraz University, Iran

172

= −R i − λ f (θ )ω − V + V (2)

= −R i − λ f (θ )ω − V − V (3)

T = λ f (θ )i + λ f (θ )i + λ f (θ )i (4)

where ωm, θ , R , L , λ are mechanical rotor speed in rad/sec, mechanical rotor position in radian, line-to-line resistance (), inductance (H), and flux linkage in V/(rad/sec), respectively. f , f , f have the same shape as the back EMFs with a maximum amplitude of ±1 and are specified in Table I. The mechanical equations are as follows: = T − T − T (5)

= ω (6) T = B ω (7)

where J , B are the moment of inertia in kg.m2, friction coefficient in Nm/(rad/sec), and electromagnetic, load and friction torque are denoted as T , T , T , respectively [15]. Finally, by combining equations (1)-(7), the BLDC machine can be described in state space form as shown in (8), (9) and (10) at the bottom of the page.

III. THE PF ALGORITHM

In state estimation, based on the Bayesian filtering approach, one solution is constructing the posterior probability density function (pdf) by using all available information, including measurement data. Since this pdf contains all statistical information, optimal state estimation and its accuracy can be derived from it [16]. By using Bayes’s rule a posterior pdf is obtained as [13]: p(x |y : ) = ( | ) ( | : )( | : ) (11)

In (11): p(x |y : ) = p(x |x : )p(x |y : ) dx : (12)

TABLE I Functions f , f and f

0 ≤ θ 6π θ −1 1

≤ θ 1 −1 1 − 6π (θ − π6) ≤ θ 1 −1 + 6π (θ − π2) −1

≤ θ 1 − 6π (θ − 5π6 ) 1 −1

≤ θ −1 1 −1 + 6π (θ − 7π6 ) ≤ θ −1 1 − 6π (θ − 9π6 ) 1

≤ θ 2π −1 + 6π (θ − 11π6 ) −1 1 p(y |y : ) = p(y |x )p(x |y : ) dx (13)

Except for some cases (e.g. linear Gaussian model), analytical solutions to these equations are not available. For other models, (11) needs to be evaluated approximately. The particle filter is represented to approximate posterior pdf by using a set of random weighted particles {(x , w ), n =1:N }, (14) [16]. p(x |y : ) ≈ ∑ w δ(x − x ) (14)

where x is a state value of mth particle, w is its weight, δ(. ) denotes the Dirac delta function, and N is the number of particles. Actually, the weight of a particle demonstrates the probability of the particle which is represented by the equation (15): ω ( ) ∝ ω ( ) ( | ) ( | )( | ) (15)

State estimation with the help of the particle filter is a recursive procedure. After several iterations, most particles will have small weights. This leads to the allocation of a large computational effort to update particles that have a small role in approximating the pdf. This phenomenon is called degeneracy, which is a common problem with PF. To prevent degeneracy, one of the most well-known methods is

x = f(x)x + Bu

ωθ = 0 0 ( ) 00 0 ( ) 00 0 ( ) 0( ) ( ) ( ) 0 0 0 0 1 0

iiiωθ + 0000 00 0 0 VVT

y = Hx = 1 0 0 0 00 1 0 0 00 0 1 0 0 iiiωθ

(9)

(10)

(8)

173

BLDC

Speed Controller

Figure 1

resampling [16]. In the resampling process, small weight particles are replicated with those that have large weight. Consider a nonlinear system described by state equations as follows: x (t) = f(x(t))x(t) + Bu(t) + σ(t) (16) y(t) = h(x(t)) + ν(t) (17)

where σ(t) and ν(t) are zero-means white Gaussian noises with covariance Q(t) and R(t), respectively. Equations (16) and (17) are in continuous time domain, but the PF algorithm is implemented in discrete domain. These equations are discretized as: x = (I + f(x )t)x + Btu + w (18) y = h(x ) + ν (19)

where I is the identity matrix, and t is the sampling period. Note that in these equations, process and measurement noises pdf are assumed to be known. At the first iteration, based on the initial assumed pdf p(x ), N initial particles (x . (i = 1. … . N)) are generated randomly. Choosing parameter N is an important issue in particle filter that affects estimation accuracy and computational burden. In order to obtain prior particles x . , initial particles are propagated in the system discrete equation (18). In the next step, relative weight must be assigned to each prior particle with respect to the measurement data. This is done by evaluating the pdf p(y |x ) , with calculating the likelihood of particles as follows [13]: q = p(y |x . ) ~ ( ) | | exp ( ( ) ( ) ) (20)

where m is related to the output matrix dimension and y is measured data, respectively. According to equations (15) and (20), the weight of each particle is directly proportional to their likelihood. After that, for choosing the large-weighted particles and constructing posterior particles x . , resampling on the basis of normalized likelihood is utilized. Now, the posterior pdf p(x |y : ), is obtained. Finally, the estimation of the states is obtained by calculating the average of approximated pdf p(x |y : ). Note that these posterior particles are propagated at the next iteration in order to generate prior particles.

TABLE II Specifications of the BLDC motor used in simulation

Parameter Value

Number of poles 8

Flux linkage 0.175 Wb

Rated speed 1500 Rpm

Friction factor 0.001 Nm/rad/sec

As mentioned earlier, resampling is a critical step, hence various techniques had been proposed in many papers. These methods have some differences in features such as computational complexity, runtime, and the number of random numbers used [17]. One of the effective methods is stratified resampling that is used in the proposed PF algorithm. Stratified resampling divides the whole population of particles into subpopulations [18]. For more information about the structure of the stratified method and also other resampling strategies, you can see [17].

IV. SENSORLESS DRIVE SCHEME

Fig. 1 shows a block diagram of the sensorless speed control of the BLDC machine. Drive structure consists of two main parts: Power stage and control section. Power stage includes a dc power supply, an inverter, and the BLDC machine. In the control section, the proportional integral (PI) controller is used as a speed regulator that compares reference of the speed with the estimated speed of the machine and generates proper currents command. As demonstrated in Fig. 1, particle filter algorithm estimates speed and position of the rotor by means of measured currents and voltages. Then, these estimated parameters are fed into the control section as feedback signals. As a result, this structure leads to the elimination of speed and position sensors, hence called sensorless.

V. SIMULATION RESULTS

The sensorless speed control of the BLDC motor using particle filter that act as state estimator, has been simulated in MATLAB/SIMULINK software. The specification of the BLDC motor used in the simulation is presented in Table II.

PF algorithm has been implemented in a MATLAB function block that operates similar to a digital processor. The measured currents and line voltages are inputs of the PF block. At each time step, after the PF algorithm process explained in section III, estimated speed and position have been used as feedback signals for the drive system. In the real-time system, any measurement data consist of noise and inaccuracy. Hence, to simulate measurement noise, a white Gaussian noise with known mean and variance was added to the measured variables.

Fig. 1. The proposed sensorless speed control drive structure

174

Fig. 3. (a) Position and (b) speed estimation error at speed reference of 200 to

1200 rpm

A critical step in PF algorithm implementation is choice elements of P0, Q and R matrices. Choosing the proper value of these parameters, results in a better performance and lower convergence time of the filter. Also, transient specifications of the filter can be directly modified with these matrices where matrix P0 and Q affects peak of the transient and transient time, respectively. Diagonal elements of the matrix P0 determine variances of initial particles of each state that construct initial pdf p(x ). The amount of system noise and model parameters uncertainty are presented by the matrix Q. In the simulation environment, any variations in system parameters are not considered, which means elements of Q must be small. Typically, P0 and Q are chosen by a trial-and-error procedure with regard to stability and transient characteristics. Matrix R indicates statistical properties of measurement noise. In this paper, measured data includes only phase currents of the motor, and hence, in practice, elements of the R are determined with

Fig. 4. Details of (a) position and (b) speed estimation error between = 3

and = 6 sec.

the help of current senor manufacturer datasheet, and based on added noise variance in simulation. Covariance matrices P0, Q and R used in simulated PF algorithm are given below:

P = 0.05 0 0 0 00 0.05 0 0 00 0 0.05 0 00 0 0 10 00 0 0 0 0.01 (21)

Q = 0.2 0 0 0 00 0.2 0 0 00 0 0.2 0 00 0 0 1 00 0 0 0 0.01 (22)

R = 5 0 00 5 00 0 5 (23)

E rr

o r

o f

P o

si ti

o n

-0.09

-0.06

-0.03

0

0.03

0.06

/)

Fig. 2. Estimation results of the proposed PF algorithm at = 0 for various speed references. Actual and estimated (a) mechanical speed (b) position and (c)

line current of the motor.

ω (r

ad /s

A)

175

Fig. 5. Transient behaviour of the proposed PF algorithm at electrical initial

position, =15 rad.

Fig. 6. Transient behaviour of the proposed PF algorithm at electrical initial

position, =5 rad.

Note that all simulation tests performed at closed-loop control with the number of the particles N, equal to 15. The estimated speed, position and motor line current by simulated PF algorithm and actual ones that had been measured in sensoreless drive mode are shown in Fig. 2(a), (b) and (c), respectively. Additionally, in order to evaluate sensorless drive performance, the speed of machine when the drive system utilized sensors data is shown in Fig. 2(a) (labeled as “actual sensored drive”). In this simulation experiment, the mechanical speed reference of the motor changes from 200 rpm (20.94 rad/sec) to 1200 (125.66 rad/sec) rpm at time t = 3 seconds and reduced to 200 rpm at t = 6 seconds. As it can be seen in this figure, simulation results indicate that the proposed PF algorithm has a good performance in a wide speed range. Fig. 3 shows the speed and position estimation error and details error in 3 seconds time duration are shown in Fig. 4. As demonstrated in Fig. 4, the maximum estimated position and speed error are 0.065 radian and 6.4 rad/sec, respectively. An important issue in estimation by stochastic filters is behavior at the startup procedure when the assumed initial position value θ , differs from the actual one that may lead to wrong convergence. Hence, a simulation experiment with the incorrect initial position is performed to investigate the proposed filter performance. Fig. 5 and 6 illustrate results of the simulation with two non-zero initial positions of the rotor. As shown in Fig. 6 the PF algorithm has a good operation in this condition, and the filter converges to the actual position in less than 25 ms. For evaluating the persistence of the proposed sensorless drive under load torque variation a simulation test is performed. As shown in Fig. 7(b), at = 1 seconds, a 2 N.m load torque is suddenly applied to the machine. Fig. 7(a) shows that speed estimation has a good accuracy even at load torque change.

Fig. 7. (a) Estimated and actual speed of the motor (b) mechanical torque at load torque variation and = 1200 rpm.

VI. CONCLUSION In this paper, a sensorless control of the BLDC machine using PF algorithm as state estimator is presented. Particle filter due to the elimination of model linearization step and propagation of N particles in the system model has suitable performance even for state estimation in highly non-linear systems at the presence of non-Gaussian noises. The number of the particles N, is an important parameter in PF algorithm, and increasing this parameter leads to the enhancement of filter accuracy in exchange for increasing computational burden, and, a trade-off between these two factors is needed. Resampling is such a vital step in particle filter algorithm that adopting convenient resampling strategy can prevent from divergence. For simplifying practical implementation of the PF algorithm, straightforward equations and a step-by-step procedure are presented. Simulation results indicate that the proposed PF algorithm is capable of estimating states of the BLDC machine at various speeds, load torque, and different initial positions with good accuracy. In addition, based on the simulation, the presented sensorless drive system has a good transient and steady state performance.

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