DC Model-based IMC Method for Brushless DC Motor ...

Journal of Automation and Control Engineering Vol. 4, No. 2, April 2016

DC Model-based IMC Method for Brushless DC Motor Speed Control Hon H. Trinh and Thinh D. Le Faculty of Electrical and Electronic Engineering, Ho Chi Minh City University of Technology, Vietnam Email: {trinhhoanghon09, thinh1113}@gmail.com

leads to the concept of robustness control [9]. There were some researches about this issue such as using model reference adaptive backstepping approach [10], auto tunning algorithm [11], and neural net-based [12]. The internal model control (IMC) which was introduced by Garcia and Morari in 1980’s [13] and widely applied in chemical process control is another typical robustness control method. IMC method is very easily designed based on system model, well working with linear system, controlling output response through a simple low-pass filter, and very robust with wide ranges changing of parameters. However, it suffers some drawbacks as problems with non-linear and multi-input multi-output (MIMO) systems, especially large amount of calculation [14, 15]. There are researches which simplify the IMC with phase-lock loop assisting for BLDC speed control [16, 17]. However, it seems very rare report about solving the major weakness of IMC method which is large amount calculating. Hence, this paper describes a solution to reduce IMC method calculation for BLDC speed control by using DC model-based instead of BLDC’s. To compare between the two model-based IMC systems, a BLDC motor with 3 Hall sensors and sinusoidal backEMF is worked with 3-phase sinusoidal inputs voltage. In addition, the BLDC motor’s parameters and coefficients are copied from reference [18]. The remaining parts of this paper are arranged as follows. First, the modeling of BLDC motor is based on its mathematical equations is expressed in Section II. Then, in Section III, the detail construction of IMC method for controlling speed of BLDC motor based on BLDC model is clearly presented. After that, the Section IV describes the construction of DC model-based IMC method for BLDC speed control. Next, the comparison of robustness between those 2 IMC model-based systems is shown in Section V. Finally, some conclusions are claimed in Section VI.

Abstract—This paper focuses on a novel idea of using DC motor model-based to build Forward and Inverse Model of internal model control (IMC) method for brushless DC motor (BLDC) speed control. This contribution reduces large amount of calculation compared to using BLDC motor model-based but it still achieves all positive characteristics of IMC method. In detail, the robustness of IMC method for BLDC speed control is compared between the traditional BLDC model-based and the novel DC model-based on MatLAB Simulink environment. Index Terms—brushless DC (BLDC), internal model control (IMC), speed control, direct current (DC) motor modelbased, robustness control

I.

INTRODUCTION

Nowadays, Brushless DC motors are used in very large range of application such as industrial automation, aerospace, automotive, instrumentation and appliance. As the name described, BLDC motors have no brush like DC’s, it used electronic commutation instead. Moreover, there is no slip in BLDC motor because it is a kind of permanent magnet synchronous motor. With these advantages, it leads to several other benefits of BLDC motors such as long life and noiseless operation, high dynamic response and efficiency, large speed range and so on. Because of electronic commutation, the input voltage supply is relative to the rotor position which can be achieved by 2 common methods. Those are using Hall sensors [1]-[3] and using back electromotive force (backEMF) [4], [5] methods. Since the latter has no sensors, it is also called sensorless method. About back-EMF signals, there are some different experiments on its waveform as sinusoidal [5], [6] and trapezoidal [2], [7], [8]. There are many BLDC motors applications which require the angular speed to be stable such as computer’s hard disk drivers and helicopter robots, etc. However, the motor behaviors may be changed with time for example, load torque increase or motor parameters change during operation like higher resistance when higher temperature. Furthermore, there are usually some incomplete information of motors. Therefore, the motor output speed is needed to be stable under those uncertainties, which

II.

A. Mathematical Equations of BLDC Motor in Laplace Domain [8] The 3 phase currents ia, ib and ic are calculated as follows:

Manuscript received October 7, 2014; revised May 15, 2015. ©2016 Journal of Automation and Control Engineering doi: 10.12720/joace.4.2.104-110

MODELING BLDC MOTOR

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Va  Ea  Ria  ia  Ls  Vb  Eb  Rib  ib  Ls  Vc  Ec  Ric  ic  Ls 

Te  TL  Jsm  m (1)

where J motor moment inertia; β friction coefficient; ωm rotor angular speed; TL load torque. And the mechanical angular speed (ωm) is calculates from the mechanical angular distance (θm):

where Va, Vb, Vc stator phase voltage; R stator winding phase resistance; L stator winding phase effective inductance; Ea, Eb, Ec phase back electromotive force (back-EMFs); s Laplace factor. Additionally, the back-EMFs are given as:   Ea  K e m F (e )  K e m Fa  2  )  K e m Fb  Eb  K e m F (e  3  4  Ec  K e m F (e  )  K e m Fc  3 

(5)

m  sm

(6)

where the mechanical angular distance (θm) is relative to the electrical angular distance (θe) by number pair of poles (PP):

e  PP.m

(7)

(2) B. Modeling BLDC Motor on MatLAB Simulink The detail construction of a BLDC motor is shown in Fig. 2 with the Current I_abc block is built from (1). Similarly, the Torque Te, Omega, Theta_e and Back EMF E_abc blocks are based on (3), (6), (7), and (2) relatively. About the F_abc block, it is based on its definition as mentioned in Fig. 1.

where Ke back-EMF constant, voltage constant; ωm rotor mechanical angular speed; θe rotor electrical position; Fa, Fb, Fc commutation functions of electrical rotor position. Since the back-EMFs are sinusoidal, the commutation functions Fa, Fb, Fc are also sinusoidal (see Fig. 1), and these 3 functions are 1200 out phase of each other.

Figure 2. BLDC motor model construction in simulink.

III.

To control the rotor speed of BLDC motor, an IMC system based on BLDC model is structured as described in Fig. 3 where the BLDC motor is built in Section II. Importantly, the Forward ( Pˆ ) and Inverse (Q) are based on BLDC model. In detail, the Forward Model is kept the same as the motor model except 3 things. First, parameters and coefficients are added a subscript M which is stands for Model to distinguish from the practical one’s. Second and third, load and friction are eliminated (TLM=0, βM=0). Similar to the Forward Model, the Inverse Model also has those 3 different things. Furthermore, in the Inverse Model, there are some differential components which may cause extremely high overshoot at initial. This phenomenon is known as kickstarter which can be prevented by using a low-pass filter (with very small TdM coefficient) at each differential component as presented in (8).

Figure 1. Commutation functions relative to rotor position.

Next, the electrical torque (Te) is the sum of each phase torque:

Te  Ta  Tb  Tc

(3)

With Kt is torque constant, each phase torque component Ta ,Tb ,Tc are defined as:

Ta  K t ia Fa  Tb  K t ib Fb T  K i F t c c  c

(4)

And the relationship between electrical torque, load torque, and angular speed is indicated in (5):

©2016 Journal of Automation and Control Engineering

MODELING BLDC MODEL-BASEL IMC SYSTEM FOR BLDC MOTOR SPEED CONTROL

d s  dt TdM s  1 105

(8)


Now, the 3-phase voltage and electrical torque are calculated as (9), (10) relatively, and these 2 equations are used in Inverse Model.  1 LM sia  Ea Va  RM ia  T dM s  1   1 LM sib  Eb Vb  RM ib  T dM s  1   1 LM sic  Ec Vc  RM ic  T dM s  1 

Te 

1 J M sm TdM s  1

experiments, a mid-value 0.05 is set to be the standard of Tf. TABLE I.

NO.

(9)

(10)

Moreover, inside the Inverse Model block, the component phase currents are calculated from each phase torque. The difficulty is calculating 3 torque components Ta, Tb, Tc from total torque Te via just (3). Therefore, the phase currents are assumed to be ideal as sinusoidal with Io amplitude as indicated in (11) since input voltage and back-EMF are both sinusoidal.

ia  I o Fa  ib  I o Fb i  I F o c c

(11)

PARAMETERS AND COEFFICIENTS OF MOTOR IN SIMULATION [18]

Parameters and Coefficients (sign)

Value (unit)

1

IMC filter (Tf)

0.05

2

Differential filter (TdM)

0.001

3

Load torque (TL)

4

Reference angular speed (𝜔ref)

5

Moment inertia (J)

6.5 × 10-5(Kg.m2)

6

Phase resistance (R)

0.1(Ω)

7

Phase inductance (L)

0.5 (mH)

8

Phase back-EMF constant (Ke)

9

Phase torque constant (Kt)

10

DC voltage (VDC)

11

Friction (β)

12

Simulation time (Tsim)

13

Pairs of pole (PP)

0.03 (Nm) 1400 (RPM)

0.03 (V/(rad/s)) 0.03 (Nm/A) 24 (V) 5 × 10-6(Nm/(rad/s)) 3 (s) 2

Hence, combined (11) with (3) and (4) the electrical torque now becomes:

Te  Kt I o sin 2 (e )  sin 2 (e  2 / 3)  sin 2 (e  4 / 3)  Te  1.5Kt I o

(12)

Equation (12) proves that the BLDC motor with sinusoidal back-EMF can be equivalent to a DC motor which has 1.5 times of torque constant (Kt). This is an important conclusion for using DC model-based instead of BLDC’s in IMC method, which is clearly described in Section IV. In addition, the inverter block (in Fig. 3) can generate sinusoidal 3-phase voltage supply to the motor depending on the rotor position. Since input voltage and back-EMF are both sinusoidal and concern with rotor position, they are almost the same as indicated in Fig. 4. Fig. 3 also illustrates the IMC filter which is used to control the system output response and overshoot. In this project, a first order low-pass filter is used as IMC filter (see Fig. 5). There are some experiments with several values of IMC filter coefficient (Tf), these experiments are done with

Figure 3. BLDC model-based IMC system for BLDC motor speed control.

(a)

3-phase input voltage

the motor parameters in the Table I. The Fig. 6 presents 2 output responses when Tf =0.005 (upper of Fig. 6(a)) and when Tf =0.5 (upper of Fig. 6(b)). Comparing these results with the IMC filter response (lower graphs of Fig. 6), the IMC filter is concluded that it can be used to control the output response. In other words, the response of the IMC filter is the expected output of the system. This is a huge advantage of IMC method since the filter is very simple. For other ©2016 Journal of Automation and Control Engineering

(b) 3-phase back-EMF Figure 4. 3-phase voltage and back-EMF at transient state.

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Figure 5. IMC filter block in MatLAB Simulink.

(b) Output response with TdM=0.01 Figure 7. Experiment with several TdM values and BLDC model-based, IMC filter response (lower graphs).

IV. (a)

MODELING DC MODEL-BASEL IMC SYSTEM FOR BLDC MOTOR SPEED CONTROL

The DC motor model is described by just only 4 equations, which are much less than BLDC’s. Since the DC model is used in the Forward and Inverse Model (see Fig. 8), all parameters and coefficients are also with subscript M and load, friction are simplified. The DC current is: V  Ea  RM i (13) i

Output response with Tf=0.005

LM s

With the armature back-EMF Ea:

Ea  KeM m

As mentioned in the 12, Section III. The torque is calculated as: (15) Te  KtDC i  1.5KtM i

(b) Output response with Tf=0.5 Figure 6. Experiment with several Tf values and BLDC model-based, IMC filter response (lower graphs).

And rotor angular speed (ωm) is:

Similarly to the IMC filter, the differential low-pass filter which is mentioned from (8) to (10) is tested with some coefficient values of TdM. Comparing the results when TdM=0.0001 and TdM=0.01 with the expected output which are shown in Fig. 7, it is seen that the differential filter does not affect much on the output system. However, the simulation becomes failed on MatLAB Simulink if the filter coefficient TdM is zero. Therefore, standard value of TdM is set to be 0.001 for later experiments.

m 

V  RM i 

1 LM si  Ea TM s  1

1 J M sm TM s  1

Figure 8. DC model-based IMC system for BLDC motor speed control.

TdM=0.0001


Te JM s

(16)

The Forward Model based on DC motor is built from those 4 equations above. Similar to the BLDC model-based, the Inverse Model also needs differential filter as calculating of voltage (17) and torque (18):

Te 

(a) Output response with

(14)

107

(17) (18)


It is clear that the number of equations is much less than so the calculating amount is much reduced by using DC model-based instead of BLDC’s in Forward and Inverse Model. With the same motor parameters as in Section III, two experiments with IMC filter coefficient Tf are done and the results are presented in Fig. 9. Comparing this figure with Fig. 6, it can be easily seen that the IMC filter in the DC model-based system plays the same role as in BLDC model-based system. Hence, the output response is simply controlled by the IMC filter coefficient in both model-based IMC systems. In addition, the experiments with differential filter coefficient TdM (see Fig. 10) with DC model-based system give the same results as the IMC system based on BLDC (see Fig. 7). That means, the TdM value should be considerable small but not zero, otherwise it causes failed of the simulation on MatLAB Simulink environment.

(b) Output response with TdM=0.01 Figure 10. Experiment with several TdM values and DC model-based, IMC filter response (lower graphs).

V.

ROBUSTNESS EXPERIMENT OF BLDC AND DC MODEL-BASED IMC SYSTEM

To know how good the performance of the new improved system, the classical IMC for BLDC speed control with BLDC model-based is compared with the novel system based on DC motor model in this Section V. Robust control can keep the system remain stable of differences between practical and model parameter values. Let denoted all model parameters with the subscript M. Hence, those robustness experiments are made as expressed in Table II with the external disturbance as load change and reference output response as shown in Fig. 11. Under those situations, in this project, the system is considered stable if its output is remained within +/5% (upper and lower dash line in the lower graph of Fig. 11) of the expected output value which is 1400 round per minutes (RPM). In addition, the experiment results are from Fig. 12 to Fig. 19.

(a) Output response with Tf =0.005

TABLE II.

NO.

(b) Output response with Tf=0.5 Figure 9. Experiment with several Tf values and DC model-based, IMC filter response (lower graphs).

(a)

Parameters and Coefficients (sign)

Upper Limit

Lower Limit

1

Moment inertia (J)

2JM

0.5JM

2

Phase resistance (R)

2RM

0.5RM

3

Phase inductance (L)

1.5LM

0.5LM

4

Phase back-EMF constant (Ke)

1.2KeM

0.8KeM

5

Phase torque constant (Kt)

1.2KtM

0.8KtM

6

Friction (β)

2β

0.5β

7

DC voltage (VDC)

8

Load Torque (TL)

-20% +20%

According to the Table II, the order of robustness experiments and comparison between 2 IMC modelbased systems are as follows. First, the output responses of those 2 systems when practical and model parameters are matched are described in Fig. 12. Second, the comparison of robustness with J when practical moment inertia are double and a half of its model value in presented in Fig. 13. Third, the Fig. 14 illustrates the results of 2 model-based systems output response when R=2RM and R=0.5RM. Next, the sensitive parameters L is

Output response with TdM=0.0001


PARAMETERS AND COEFFICIENTS RANGES OF ROBUSTNESS EXPERIMENTS

108


supposed to be increased 50% and decrease 50% around its model value, the 2 systems output responses are clearly indicated in Fig. 15. After that, the 2 systems robustness are compared when back-EMF constant is 20% higher and lower than its model value, this leads to the involving of torque constant Kt in the experiment because the two constant are equal, and their results are seen in Fig. 16. (a) BLDC model-based system

(b) DC model-based system

Figure 16. Output response when Ke=1.2KeM, Kt=1.2KtM and Ke=0.8KeM, Kt=0.8KtM.

Practically, friction is very hard to determine and easily changed depending on operating environment. Therefore, the 2 model-based IMC systems are assumed to work with double and a half friction for robustness test. The test results are shown in Fig. 17. In reality, a BLDC motor may work under bad conditions such as the battery is running out of power (suppose 20% DC voltage lower) or the load torque is higher than the nominal load (suppose 20% load increase). The results of 2 IMC systems robustness under these 2 bad situations are given in Fig. 18. Finally, let all parameters and coefficients in Table II are at their upper and lower values combined with the 2 bad working conditions to make the total robustness experiments of 2 IMC model-based systems, the results of output speed response are described in Fig. 19.

Figure 11. External load disturbance (upper graph) and reference output response (lower graph).

(a) BLDC model-based system


Figure 12. Output response (upper graphs) of 2 model-based IMC systems when practical and model parameters are equal.



Figure 17. Output response when double and half friction. (a) BLDC model-based system


Figure 13. Output response when J=2JM and J=0.5JM.




Figure 18. Output response when 20% DC voltage decrease and 20% load torque increase.


Figure 14. Output response when R=2RM and R=0.5RM.



Figure 19. Output response when 20% DC voltage decrease, (a) BLDC model-based system

20% load torque increase and parameters are at upper and lower limits.


Figure 15. Output response when L=1.5LM and L=0.5LM.


109


[8]

From those output angular speed results of 2 IMC systems from Fig. 12 to Fig. 19, it is clearly seen that the outputs always stay within +/- 5% criteria around the reference value under the disturbance external load even though many parameters are mismatched with their model values in very wide ranges. That means, the 2 IMC model based systems are equally well robust with those many parameters and working conditions. VI.

[9] [10]

[11]

CONCLUSION

With the novel improvement of using DC motor model-based in Forward and Inverse Model, the new IMC system for BLDC speed control is much less amount of calculation than the classical IMC model based on BLDC motor. The results show that all the advanced features of the traditional IMC model are considerably remained in the developed IMC system such as robustness with many parameters and coefficients (J, R, L, Ke, Kt, β, VDC, TL) changing in wide ranges. In detail, the ranges of parameter J, R, β can be from a haft to double of their model values, while the inductance L may vary within +/- 50% of its model value. The back-EMF and torque constants can be 20% different from their model values. Moreover, robustness of the improved IMC system is also tested under some bad operating conditions such as 20% decrease of DC voltage and 20% increase of load torque. These wide ranges are well satisfied the reasonable changing of practical parameter values when the motor is operating. These unavoidable changes may be caused from effect of temperature, electromagnetic field, measurement errors, etc. Additionally, the DC model-based IMC system also achieves benefits of IMC principle such as easily controlling output response by setting IMC filter coefficient, and simply designing steps. Notice that, the designing process is even much easier since using the simpler DC model. REFERENCES [1]

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Hon H. Trinh was born in Dong Nai, Vietnam, in 1973. He received B.E and M.E degrees from Electrical Electronic Engineering of Ho Chi Minh city University of Technology, Vietnam, in 1997 and 2002 respectively. He received Ph.D. degree in computer vision area at Electrical Engineering Department from University of Ulsan, Korea in 2008. He is currently a lecturer in Faculty of Electrical and Electronic Engineering, Ho Chi Minh city University of Technology, Vietnam. His research interests include computer vision, pattern recognition, understanding and reconstructing outdoor scenes, designing the outdoor mobile robot for civil and military applications, electrical machinery, controlling electrical machinery, robotics, Human Computer Interaction (HCI), pattern recognition.

Thinh D. Le was born in Hue city, Vietnam, in 1991. He is currently an undergraduate in Faculty of Electrical and Electronic Engineering, Ho Chi Minh city University of Technology, Vietnam. His research interests include power electronic, computer vision, and modern control.

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