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Research on fuzzy PID control strategy for brushless direct current motor.

1. Introduction

The trapezoidal wave or square wave brushless DC motor (BLDC) is a kind of brushless DC motor which is recognized universally (Guo, 2008; Abreu, 2015). The brushless DC motor has a huge potential for development and application in defense, industrial, office automation, automotive electronics, household and other areas, for its AC and DC advantages. For the reason of wide application of the brushless DC motor and strong market demand, the response to the national energy conservation and the call of the green, as well as the protection to the motor supplies' safety, stability and efficiency, the research of high-precision control of brushless DC motor system, to improve product performance, have important theoretical significance and market value. The fact is that the traditional PID control technology for multi-variable, strong coupling and nonlinear brushless DC motor control system can't meet the demand. To solve these problems, the paper uses fuzzy PID control strategy. It is the combination of the virtues of fuzzy control technology and PID control technology, possessing both excellent suitability for nonlinear and time-varying systems, but also to eliminate the steady-state error, and the excellent characteristics of eliminating steady-state error.

2. Model of Brushless DC Motor and Speed Control Method

2.1. Mathematical Model of Brushless DC Motor

All In the premise of no influences of control performance, Experimental assumptions are as follows (Hemati, 1990): stator of a brushless DC motor is completely symmetrical, three-phase windings use Y-connected style, each phase winding difference at 120[??], and each winding's resistance and inductance are the same. At the same time, magnetic saturation and eddy current losses are ignored, winding are distributed evenly in the smooth surface of the stator, rotor damping effect are Ignored, the wave of rotor magnetic field is square wave, the wave of anti-voltage is trapezoidal wave. Without considering armature reaction effect of stator windings and the mutual inductance between the armature windings, symmetrical three-phase winding and the rotor reluctance do not change with the position of the rotor. Its equivalent circuit and drive circuit (Junhyuk, 2004) as shown in Fig 1.


As we can see from the figure, A, B, C are the three-phase equivalent circuits of the motor, V1-V6 are the six power switches, using two-two conduction mode. Use the sampled position signal to determine which two phases conduct, then supply the power through the inverter, forms the rotating magnetic field, so as to drive the rotor to rotate. The three-phase brushless DC motor stator voltage equation expression as follows:


In the formula(1), [u.sub.a], [u.sub.b], [u.sub.c] are terminal voltage of three-phase stator motor (V); [e.sub.a], [e.sub.b],[e.sub.c] are counter electromotive force (V) in a three-phase motor winding; R = [R.sub.a] = [R.sub.b] = [R.sub.c] are three-phase motor winding resistance ([OMEGA]); [i.sub.a], [i.sub.b], [i.sub.c] are three phase current of stator motor(A); L = [L.sub.a] = [L.sub.b] = [L.sub.c] are the inductance of three-phase windings of the motor (H); M = [L.sub.ab] = [] = [L.sub.bc] = [] = [] = [L.sub.cb] are mutual inductance between Three-phase motor stator winding(H). Ignore the power switch operation transition process and the inductance of armature winding. Electromagnetic power output of the motor is as followed:

[P.sub.e] = [e.sub.a] [i.sub.a] + [e.sub.b] [i.sub.b] + [e.sub.c][i.sub.c] (2)

Ignore other losses of the motor rotor; the electromagnetic power can be completely converted into kinetic energy of the rotor:

[P.sub.e] = [T.sub.e][omega] (3)

The electromagnetic torque equation can be obtained by formula (2), (3).

[T.sub.e] = ([e.sub.a][i.sub.a] + [e.sub.b][i.sub.b] + [e.sub.c][i.sub.c])/[omega] (4)

In the formula, [P.sub.e] is motor electromagnetic power; [T.sub.e] is motor electromagnetic torque (N * m); [omega] is motor machinery angular velocity (rad/s). To construct a complete motor mathematical model, the equations of motion of the motor are also needed:

J d[omega]/dt = [T.sub.e] - [T.sub.L] - B [omega] (5)

Where J is rotary inertia of the motor, [T.sub.L] is load torque (N * m), and B is damping coefficient ([s.sup.-1]).

2.2. Mathematical Model of Brushless DC Motor

According to the theory of DC motor, brushless DC motor speed can be expressed as:

n = [U.sub.d][I.sub.d] - [I.sup.2.sub.D][R.sub.[SIGMA]] - 2[DELTA]U[I.sub.D]/2[C.sub.e][T.sub.D]/[C.sub.T] (6)

In the formula, [U.sub.d] is terminal voltage two-phase conducting windings. [R.sub.[SIGMA]] is total resistance of the stator armature circuit. [I.sub.D] is the stator armature current. [[DELTA].sub.U] is voltage drop of the power tube. [C.sub.e] is internal voltage constant of brushless [[DELTA].sub.C] motor. [C.sub.T] is internal torque constant of brushless DC motor. [T.sub.D] is motor torque.

Experiment with brushless DC motor uses the constant torque speed mode, internal motor torque is constant, Motor power and speed are linear with each other. The speed is changed by the change of internal motor armature voltage. Electromagnetic torque of the motor does not change with the change in the motor speed.

Operating power of the motor varies with the change of the motor speed linearly at the runtime. This speed control method has a high steady state performance, with a wide range of speed applications. This method can realize step less speed regulation, having a small motor energy loss during the process of controlling speed. It can be widely applied to the motor under the requirements of high-quality case.

3. Fuzzy PID Brushless DC Motor Control Strategy

3.1. Principle and Structure of Fuzzy PID Controller

The fuzzy PID controller of Brushless DC motor control system mainly focuses on its characteristics of nonlinear and strong coupling. According to the system running condition, use the fuzzy control thought to achieve the online nonlinear adjustment of PID parameters timely. Always keep the good match of controller parameters and controlled object, to obtain higher stability and quicker response of the system. These virtues make it possible to meet the quality control requirements of motor supplies.

The basic principle of fuzzy PID controller is to input PID controller error e and error change rate [e.sub.c] simultaneously to the fuzzy controller (Krishnan, 2009; Luo, 2009; Li, 2016), and to find out the fuzzy relationship between e, [e.sub.c] and PID parameters. The aim is to achieve the real-time online adjustment of three PID parameters and amend three parameters [k.sub.p], [k.sub.i] and [k.sub.d] which are obtained by the fuzzy controller into the PID controller, to meet the dynamic requirements of controller parameters in different e and [e.sub.c]. Thus make it possible to achieve real-time control of the controller, which will enable make the controlled object BLDCM to obtain a good dynamic and static performance (Wang, 2003; Xia, 2009). Fuzzy PID controller structure is shown as the dashed box in Fig 2. r (t) is a speed set value of brushless DC motor fuzzy PID control system, y (t) is a output value of the BLDC fuzzy PID control system, n (t) is a the speed feedback detection output value of the system, e (t) is the fuzzy PID controller input value, [e.sub.c] (t) is the rate of change of speed deviation, u (t) is output value of fuzzy PID controller.


3.2. Design of Fuzzy PID Controller

As can be seen from the fuzzy PID control structure, its control process can be divided into fuzzy control and PID process control processes (Xu, 2011). Fuzzy PID control process and control process can be seen as a serial process in each cycle. Fuzzy control process is forward. PID control process is in the post. Their functions and objectives are different. Therefore, the fuzzy PID controller can be designed according to the following two processes.

Fuzzy control process: firstly, the fuzzy controller input values speed deviation e and error change rate [e.sub.c] converse and blur of quantized factor [K.sub.e], [], the corresponding results are E and EC. Use three correction parameters [DELTA][k.sub.p], [DELTA][k.sub.i] and [DELTA][k.sub.d] as output values of fuzzy logic controller, [DELTA][K.sub.P], [DELTA][K.sub.I] and [DELTA][K.sub.D] are corresponding fuzzy values before clarity. To facilitate the design, take seven fuzzy factors to cover the fuzzy domain. They are NB (Negative Big), NM (Negative Medium), NS (Negative Small), Z (Zero), PS (Positive Small), PM (Positive Middle) and PB (Positive Big). The fuzzy domain of E and EC is [-3, 3], and the fuzzy domain of [DELTA][K.sub.P], [DELTA][K.sub.I] and [DELTA][K.sub.D] are [-1, 1]. The type of membership function of fuzzy subsets was taken as Z-type, S-type and triangle-type.

Secondly, according to three different time control parameters and their interactions, and practical experience and knowledge of operators and experts, we can get control rules of fuzzy PID control parameters [DELTA][K.sub.P], [DELTA][K.sub.P] and [DELTA][K.sub.D]. When [absolute value of E] and [absolute value of EC] of brushless DC vary, the rules are shown in Table 1.

Then according to the control rule, make [DELTA][K.sub.P], [DELTA][K.sub.I], [DELTA][K.sub.D] to be cleared, which are results decided by Mamdani fuzzy inference engine. Use the maximum membership degree average method(mom). Those above results are converted into a clear value [DELTA][K.sub.P], [DELTA][K.sub.I] and [DELTA][K.sub.D]. Then obtain PID corrected parameters [DELTA][k.sub.p], [k.sub.i] and [DELTA][k.sub.d] by scaling factor [K.sub.up], [K.sub.ui] and [K.sub.ud].

Finally, accumulate them with three initial parameters of PID [k.sub.p0], [k.sub.i0] and [k.sub.d0]. The ultimate outputs of fuzzy logic controller are current values of parameter required for conventional PID. The formulas are as follows:

[k.sub.p] = [k.sub.p0] + [K.sub.up] [DELTA][K.sub.p] = [k.sub.p0] + [DELTA][k.sub.p] (7)

[k.sub.i] = [k.sub.i0] + [K.sub.ui] [DELTA][K.sub.i] = [k.sub.i0] + [DELTA][k.sub.i] (8)

[k.sub.d] = [k.sub.d0] + [K.sub.ud] [DELTA][K.sub.d] = [k.sub.d0] + [DELTA][k.sub.d] (9)

PID control process: the procedure uses the incremental PID control. After the sampling period and the response factors are determined, simply enter the three measurement bias during the process, the incremental amount of control. It represents the actuator's position change during two sampling time intervals. Compared with the position PID control, incremental mode has much simpler computational algorithm, smaller amount of calculation and it is easy to achieve by software programming, which is widely used in the actual production. More specific implementation process of PID controller is based on the speed error e and error change rate ec, and three fuzzy controller's output parameters [DELTA]kp, ki and [DELTA]kd generated by fuzzy inference. Finally obtain u is the control amount of Fuzzy PID control system, through the proportional, integral, and cumulative control processes. Note that during the PID control process, proportional, integral and derivative control process may be not only a serial process, but also do parallel processing. The premise is that it must be completed before the accumulation control process.

4. Modeling and Simulation of BLDC Fuzzy PID Control System

4.1. The Workflow of Model-Based Design

Model-Based Design (MBD) is innovating the way for engineers and scientists working. In Model-Based Design, a system model is right on the center of the development process, from requirements development, through design, implementation, and testing. Modelbased design gives us a complete product from idea to generate the code development process. In MBD, a system model is right in the center around the development process, from requirement's development, through design, implementation, and testing. This workflow is elaborated to create hardware and software partitioning, automatically create hardware and software implementation code, and verify the hardware and software implementations in the context of the complete system. It is not only overcoming the defects of low efficiency and difficulty of meeting the requirements in traditional methods of progress, but also to meet time-to-market and cost objectives. MBD workflow is illustrated in Fig 3.

4.2. Overall System Simulation Model

In MATLAB / Simulink simulation software environment, according to the brushless DC motor mathematical model and fuzzy PID controller structure, use a modular system modeling idea to design and establish a general simulation model of fuzzy PID brushless DC motor speed control system. As shown in Fig 4.

Fuzzy PID brushless DC motor speed control system simulation model includes two parts. One is the brushless DC motor fuzzy closed-loop control system; the other is performance evaluation system model. The brushless DC motor fuzzy closed-loop control system includes three parts, including Speed given module n*, speed control subsystem module F-PID and brushless DC motor model. The brushless DC motor model mainly consists of DC power supply model DC, inverters the Universal Bridge, permanent magnet synchronous motors and PWM pulse width modulator PWM Generator, pulse control unit controller. Performance evaluation system is established by IST2E, which represent the controller's performance of the control deviation according to "Optimal Control". Its empirical formula is:

[J.sub.2] = [[integral].sup.[infinity].sub.0] [t.sup.4][e.sup.2] (t) dt (10)



4.3. Fuzzy PID Controller Simulation Model

The fuzzy PID controller is the core of the whole control system. According to the design idea of the fuzzy PID controller of the brushless DC motor, set up fuzzy PID controller model F-PID, in the work environment of Simulink, as shown in Fig 5. The whole controller is mainly composed of two parts, the fuzzy controller module Fuzzy and other arithmetic units of PID.


At the same time, fuzzy controller module Fuzzy is the core of the whole fuzzy PID controller. Its main function is to provide three required parameters kp, ki, kd for the fuzzy PID controller. The simulation model is shown in Fig 6.


4.4. Parameter Settings

Brushless DC motor control system simulation model parameter settings are shown in Table 2.

Use the traditional empirical method to obtain initial PID three parameters [k.sub.p0]=40, [k.sub.i0] = 1, [k.sub.d0] = 0.0101, according to the principles of fuzzy PID controller aforementioned. To realize the fuzzy control, calculate the input variables e and ec quantization factor Ke = 0.002, Kec = 0.000007 according to the basic domain to fuzzy domain conversion formula. The scale factors of output variables [DELTA][K.sub.p], [DELTA][K.sub.i], [DELTA][K.sub.d] are that [] = 0.065, [K.sub.ui] = 1.15, [K.sub.ud] = -0.00015.

4.5. Simulation Results

To verify the feasibility and effectiveness of the application of brushless DC motor speed control of fuzzy PID control strategy in brushless DC motor speed control, apply directly to the brushless DC motor speed control system. Make the necessary settings and model parameters and embed the corresponding program file, make sure the system is running, and record the results.

With load T[L.sub.3]N-m, white Noise and 0.05s sudden load torque disturbance, the waveform of brushless DC motor system traditional PID control and fuzzy PID control are shown in Fig 7. In traditional PID control, brushless DC motor speed overshoot 12.3%, the adjusting time is 5ms. In Fuzzy PID control of brushless DC motor, speed doesn't overshoot, settling time is 4ms. After 0.1s, change the given speed, it changes to i000r / min rapidly. Fuzzy PID's ability is significantly better than PID regulator's. After 0.05s, give the sudden disturbance torque, the former is slightly better than the latter, but not very obvious.


Compare the performance of Fuzzy PID controller and conventional PID controller, simulation results are shown in Fig 8.


As can be seen from Fig 8, fuzzy PID controller's optimize performance is significantly better than conventional PID controller's in the control of BLDC control system. It indicates that the control system using the former method has a faster response, a better ability to inhibit the larger deviations. At the same time, the transition time is shorter, and the controller parameters have better selectivity.

In summary, compared with the conventional PID control, Fuzzy PID controller achieve a fast speed tracking control of brushless DC motor in the case of variable speed, improving responsiveness and immunity of the overall system.

5. Conclusions

Brushless DC motor is actually a multivariable, strong coupling time varying nonlinear system. The traditional PID control is only an approximate control within a certain range, with low accuracy and limited performance. It is only adapted to cases which have low control accuracy and performance requirements. To optimize motor performance, we need to improve motor control accuracy further. According to the characteristics of brushless DC motor, this article combine both advantages of fuzzy control and PID control technology and design a brushless DC motor fuzzy PID control strategy. Using MBD method, we can quickly build a fuzzy PID system simulation model, and accomplish efficiently the system test, simulation and implementation. The simulation results show the feasibility of fuzzy PID controller for brushless DC motor speed control system and improvement in its performance, meeting the high accuracy requirements of brushless DC motor. At the same time, this strategy is easy to adjust the policy improvements for on-line control of different conditions, with good promotional value.

Recebido/Submission: 19/03/2016

Aceitacao/Acceptance: 4/07/2016


We would like to thank the members of the project team for the great efforts of scientific research projects. Moreover, we greatly appreciate the reviewers' comments that lead to an improved presentation of the results. This work was supported by Research Program of science and technology at Universities of Inner Mongolia Autonomous Region (NJZZ14288).


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Lili Jing (1), Hua Ge (1), Yang Nie (1,2)


(1) Department of Physics, Jining Normal University, Inner Mongolia, China

(2) Digital Engineering Center, Communication University of China, Beijing, China
Table 1--The Fuzzy Control Rules of [DELTA][K.sub.P],
[DELTA][K.sub.I], [DELTA][K.sub.D]

               [absolute value of EC]

[absolute         NB         NM         NS         ZO
value of E]

NB             PB/NB/PS   PB/NB/NS   PM/NM/NB   PM/NM/NB
NM             PB/NB/PS   PB/NB/NS   PM/NM/NB   PS/NS/NM
NS             PM/NB/ZO   PM/NM/NS   PM/NS/NM   PS/NS/NM
ZO             PM/NM/ZO   PM/NM/NS   PS/NS/NS   ZO/ZO/NS
PS             PS/NM/ZO   PS/NS/ZO   ZO/ZO/ZO   NS/PS/ZO
PM             PS/ZO/PB   ZO/ZO/NS   NS/PS/PS   NM/PS/PS
PB             ZO/ZO/PB   ZO/ZO/PM   NM/PS/PM   NM/PM/PM

               [absolute value of EC]

[absolute         PS         PM         PB
value of E]

NB             PS/NS/NB   ZO/ZO/NM   ZO/ZO/NS
NM             PS/NS/NM   ZO/ZO/NS   NS/ZO/ZO
NS             ZO/ZO/NS   NS/PS/NS   NS/PS/ZO
ZO             NS/PS/NS   NM/PM/NS   NM/PB/ZO
PS             NS/PS/ZO   NM/PM/ZO   NM/PB/ZO
PM             NM/PM/PS   NM/PB/PS   NB/PB/PB
PB             NM/PM/PS   NB/PB/PS   NB/PB/PB

Table 2--Parameter List of the BLDCM Control System

             Stator resistance   The magnetic field
             Rs=4.76 [OMEGA]     flux 0.1848Wb

Brushless    Stator inductance   Field width 120o
DC Motor     Ls=0.0085H

             Moment of inertia   Number of pole
             J= 0.000i05ikg*m2   pairs p=2
                                 rated power P=5.5KW

rated        Ud=300V
power DC

Torque       TL=3N*m

Amplifier    G1=0.1              Simulation Algorithm
Gainl                            ode45
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Author:Jing, Lili; Ge, Hua; Nie, Yang
Publication:RISTI (Revista Iberica de Sistemas e Tecnologias de Informacao)
Date:Jul 1, 2016
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