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					                                                     International Journal of Advances in Science and Technology,
                                                                                                Vol. 2, No.6, 2011


Speed Control of Brushless DC Motors Using Optimal
        PID Controller with LQR Approach
                            VK.Chakravarthy Naik1 and G.Venkata Subba Reddy2
  1
      Department of Electrical and Electronics Engineering, G.Pulla Reddy Engineering College. Kurnool,
                                            Andhra Pradesh, India

                                           coolchakri97@gmail.com
  2
      Department of Electrical and Electronics Engineering, G.Pulla Reddy Engineering College. Kurnool,
                                            Andhra Pradesh, India

                                             gvsr802@yahoo.co.in

                                                    Abstract

        BLDC motor due to its simpler construction and inherent stability is well suited for speed control.
Generally PID controllers are used to control the speed. Earlier PID tuning has been done using Ziegler Nichols,
Cohen coon, genetic algorithms; all these methods take lot of iterations and expertise. Few PID tunings are trial
and error methods and it will take a lot of time to get the optimal parameters of the PID controller. So a novel
optimal PID tuning algorithm has be developed which is quite reliable and simpler.

      Methodology of linear quadratic regulator is utilized to search the optimal parameters of the PID controller.
The augmented state vector of performance measure involves output signals and the weighting functions are
determined through poles assignment and the existence criteria of the optimal PID controller are derived. The
new PID tuning algorithm is applied to the speed control of BLDC motors. Computer simulation output shows
that the performance of the optimal PID controller is better than that of traditional PID controller.

     A MATLAB/SIMULINK Models has been developed for BLDC with all its characteristics and speed
control.

Keywords: Optimal PID control, linear-quadratic regulator, brushless DC motor.


1 Introduction
The most common type of actuator that converts electrical energy into mechanical motion is the electric motor. The
speed of permanent magnet DC motors is easy to be controlled for the torque-speed curve being almost hear 111.
Brushless DC (BLDC) motors develop from permanent magnet DC motors. BLDC motors have the advantages like
permanent magnet DC motors, such as good response and easy control. In addition, BLDC motors use Hall effect
sensors in place of mechanical commutations and brushes. Therefore, BLDC motors possess other benefits that
permanent magnet DC motors lack for, such as low noise, no mechanical loss, and etc .BLDC motors have been
widely applied in many control systems--the spindle motor in compact disc drives with the goal of a low moment of
inertia for example. Proportional-Integral-Derivative (PID) controllers are still implemented in industry
enormously. There have been a lot of approaches to search the parameters of PID controllers, including time
response tuning [2], time domain optimization [3], frequency domain shaping [4] and genetic algorithms [5].
However, most of them require substantial experiences and several iterations. Since landing the optimal parameters
of PID controllers is not an easy task, it is important to establish a systemic design procedure. Linear-Quadratic




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Regulator (LQR) optimal control problems have been widely investigated in the literature. The performance
measure is a quadratic function composed of state vector and control input. If the linear time-invariant system is
controllable, the optimal control law will be obtained via solving the algebraic Riccati equation [6]. In reference [7],
an optimal PI-lead controller is designed using LQ technique. However, the augmented state vector involves
differential states such that the optimal controller will take full state signals. The drawbacks of the design result in

the requirement of observers. The PI/PID controller tuning via LQR approach is presented in reference [8]. The PI
controller is designed for a 6rst order system only. The PID tuning formula is derived under the constraint of pole-
zero cancellations. One of the controller zeros must be equal to the larger pole of the system. In this paper, an
optimal PID controller for a general second- order system is developed using LQR manner. There is no constraint
in the searching space of the optimal PID parameters. The new PID tuning algorithm is applied to the speed control
of BLDC motors. The performance measure to be minimized contains output error signals and differential control
energy. The optimal PID controller receives error signals only, and it doesn‟t need to feedback full states. Besides, a
systematic procedure for the selection of weighting functions is proposed in this context. The Q matrix can be
determined from the roots of the characteristic equation. Once the poles of the closed-loop system are assigned, the
existence criteria of the optimal PID controller are derived. Both computer simulations and experiment results
demonstrate that the effectiveness of the optimal PID designs.

2 Brushless DC motors:
Permanent magnet DC motors use mechanical commentators‟ and brushes to achieve the commutation. However,
BLDC motors adopt Hall effect sensors in place of mechanical commutators and brushes [9]. The stators of BLDC
motors are the coils, and the rotors are the permanent magnets. The stators develop the magnetic fields to make the
rotor rotating. Hall effect sensors detect the rotor position as the commutating signals. Therefore, BLDC motors use
permanent magnets instead of coils in the armature and so do not need brushes. In this paper, a three-phase and two-
pole motor BLDC motor is studied. The main controller of the BLDC motor is implemented through the
TMS320F240 of the Texas Instrument. The speed of the BLDC motor is controlled by means of a three-phase and
half-bridge pulse-width modulation (PWM) inverter. Figure 1 shows the open loop control of the BLDC motor.




                         Fig. 1. The open loop control of the BLDC motor
The dynamic characteristics of BLDC motors are similar to permanent magnet DC motors. The characteristic
equations of BLDC motors can be represented as

vapp (t) =L      + R . i(t) + Vemf(t)              (1)

Vemf = Kb . W(t)                                  (2)
T(t) = Kt . i(t)                                  (3)

T(t) = J           + D . W(t)                     (4)
Where vapp(t) is the applied voltage, W(t) is the motor speed, L is the inductance of the stator, I(t) is the current of
the circuit, R is the resistance of the stator, is the back electromotive force, T is the torque of motor, D is the viscous




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coefficient, J is the moment of inertia, K, is the motor torque constant, and Kb is the back electromotive force
constant. Figure 2 shows the block diagram of the BLDC motor. From the characteristic equations of the BLDC
motor, the transfer function is obtained
            =                                 (5)




                               Fig. 2. The block diagram of the BLDC motor

3 Optimal PID controller design
Figure 3 shows the block diagram of optimal PID
The transfer function of the BLDC motor is defined control for the BLDC motor.
As

G(s) =                  =                                (6)


Where y(s) is the speed of the motor, U(s) is the PWM control signal. The mathematical equation of a PID
controller in frequency domain is
GPID(s) = Kp +        + Kd S                              (7)
Where kp , is the proportional constant, ki , is the integral constant, and kd is the derivative constant. The control
objective is to design an optimal PID controller such that the tracking error will approach zero. The tracking error is
defined as
e(t) = r – y(t)                                             (8)
Where r is the command, y is the output signal. The control input signal is
u(s) = e(s) (     + Kp + Kds)                               (9)
that is, the control energy needed to achieve error regulation will be
u(t) = Ki              Kp e(t) + Kd              )                (10)
To secure the optimal value of kp, , ki, and kd , introduce the augmented state vector


Z(t)                                                       (11)

From equation (1 l), the differential equation of z(t) is follow,




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ż(t) =               = Ā Z(t) +    ů(t)                      (12)

Where

ė (t) =         -ỳ (t)                               (13)
ё (t) =         -ÿ (t)                               (14)
 e (t) = - y (t)                                     (15)
From equation (6), the following equation is obtained

S2 y(s) + asy (s) + by (s) = cu(s) (16)




                                     Fig.3. Optimal PID control.
Therefore
S3 y(s) + as2 y(s) + bsy(s) = csu(s)                  (17)
Taking the inverse Laplace transform of equation (17)
Gets
 y(t) + aÿ(t) + bỳ(t) = ců(t)                       (18)
Substituting equations (13)-(15) into equation (18) yields
 e(t)   = -aё(t) – bė(t) – ců(t)                    (19)
Thus, the matrices of the augmented system are derived


Ā=                                                   (20)


  =                                                 (21)

Let the performance measure be

Jp =        Zr (t) QZ(t) + ůT(t) Rů(t))dt           (22)




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where Q and R are symmetric matrices, and they are positive semi-definite and positive definite respectively. To
minimize the performance measure under the dynamic constraints (12), the weighting functions Q and R should be
appropriately designed. The optimal differential control input is obtained
ů(t) = -R-1              T
                             Sz(t)
           = -Kz(t)
           = -[K1 K2 K3]z(t)                                            (23)
Where K is the Kalman gain, S is the positive definite, symmetric matrix to satisfy the algebraic Riccati equation

ĀT S + S Ā - S R-1 r S + Q = 0                                         (24)
Taking integration on both sides of (23), the optimal
PID controller for the BLDC motor is obtained
U(t) = -Ki               (t)dt – K2 e(t) – K3 ė(t)                   (25)
that is,
[Ki        Kp           Kd] = -[K1 K2               K3]              (26)
To achieve the desired response, the chosen of weighting functions Q and R is not easy virtually. In the next section,
the guide lime of selecting weighting is offered for the poles of the closed-loop transfer function having been
assigned. Furthermore, the existing criteria of an optimal PID controller are derived to meet the desired performance
Theorem Let a, a2 and a3 be the desired roots of the characteristic equation of the closed loop system. If the
following conditions are held
       2        2             2        2        2     2
   1           2    +        1        3        2    3     b2    (27)
      2              2            2        2
   1       +        2    +       3             – 2b              (28)
then exists an optimal PID controller such that the design specification is satisfied.

Proof: Let Q be the weighting function and S be the solution of the algebraic Riccati equation in the form of


Q=                                                               (29)



S=                                                               (30)

Substituting the weighting matrix into equation (24)
yield
q1 = R-1c2s213                                                  (31)
q2 = 2bs23 + R-1 c2 s132 – 2as13- 2R-1c2s13s33                                  (32)
q3 = 2as33+R-1 c2 s233-2s23                                      (33)
From equations (23) and (26), the optimal PD controller is obtained.

Ќ i = R-1 cs13                                                    (34)



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Ќp = R-1 cs23                                                                                         (35)
Ќ d = R-1 cs33                                                                                          (36)
The closed-loop transfer function is

          = G(s) GPID(s) 1+G(s)GPID (s)

          = ckds2 + ckps + cki s3 + (a +ckd)s3 + (b + ckp)s +cki                                                                                      (37)
Let
S3 + (a+ckd)s2 +(b +ckp)s + cki
= (s-          1)(s-            2)(s-               3)                                         (38)
After comparing the coefficients, the relationships between the desired roots and the PID controller are
Ki =           1    2        3                                                                         (39)
Kp =           1    2                   1       3               2       3 -b                           (40)
Kd =            1            2+             3 -a            c                                          (41)


If the PID controller is optimal, equations (39)-(41) should be equivalent to (34)-(36). Therefore,
                                                         2
S13 =                   1        2          3                                                          (42)
                                                                                                2
S23 =                   1        2                  1       3               2    3   -b)               (43)
S33 =                    1                  2+          3 –a)               c2                         (44)


Substituting above equations into equations (31)-(33), the guideline of selecting weighting is derived
                     2           2              2
q1 = R              1           2           3       c2                                                 (45)
                            2           2                2           2               2     2
q2 = R(                  1           2                  1           3            2       3     -b2)      2
                                                                                                                                                 (46)
                        2                 2                 2
q3 =                1                    2 +             3       –a2) c2                              (47)
Furthermore, the weighting matrix Q
is positive semi- definite. Thus
   2        2                2              2                2          2
  1        2             1              3                2          3            b2                  (48)
      2          2                  2
  1             2 +             3       –2b                                                          (49)
4 Computer simulations
The specifications of the BLDC motor are shown in table 1. From table 1 and equation (S), the transfer
function of the BLDC motor is obtained

G(s) = 275577.36/s2 + 417.7s + 43567.5                                                                   (50)




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To control the speed of the BLDC motor at 1000 rpm, a traditional PID controller is designed using trial and error
method first. After several iterations, the values of ki, kp, and kd are chosen as 15, 30 and 0.001 respectively. The
speed response and applied voltage of the traditional PID controller are shown in figure 4 and 5 The roots of the
characteristic equation are - 346.3922861.9 I and 4.4974.




To design the optimal PID Controller, calculate the augmented system matrices

Ā=                                                    (51)


  =                                                   (52)

The weighting matrices Q and R all selected as

Q=                                                     (53)

      R=1                                               (54)
After solving the algebraic Riccati equation and substituting into equation (23), the optimal values of Ki, Kp and Kd
are obtained as 10, 70.556 and 0.0212 respectively. The speed response and applied voltage of the optimal PID
controller as shown in fig 5 and 7.The roots of the characteristic equation are -3124.9 +-3117.9 I and -0.14142.
Table 2 lists 2 the performance of the two definite controllers.




                         Fig. 4. Speed response of the traditional PID controller.




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                          Fig 5 Applied voltage of the Traditional PID controller




                              Fig. 6. Speed response of the optimal PID controller




                                Fig. 7 Applied voltage of the optimal PID controller.
 To illustrate the procedure of choosing weighting functions, different case studies are considered. Let the assigned
poles (a1,, a 2, a3,)b e (-100, -200, -300) and set R = 1 . According to equations (45)-(47), the weighting (q, , q
2, q 3 ) is (4.74*    2, 3 . 9 5 *     ,6 . 9 3 *10-7 ') and the optimal PID controller (ki , kp , k d ) is (2.17 x 10, 2.41 x
   -1         -4
10 ,7 x 10 ). The speed response and applied voltage are shown in figure 8 and 9. Similarly, let the assigned poles
( a1, , a2 , a3,) be (-500, -1000, -1500) and set R = 1. The corresponding weighting (q1 , q 2 , q 3 ) is ( 7 . 4 . 6*
106, 4 . 0 3 *10 , 4 . 4 9 *10-5') and the optimal PID controller (ki , kp , k d ) is (2.72*103', 9.82, 9 . 3 7 *10 -3). The
speed response and applied voltage are shown in figure 10 and 11. It follows that the rise time and settling time will
be reduced if the poles are assigned at more left side of the real axis, but the overshoot will be increased
simultaneously. To get a better performance, we assign the poles (a1,, a 2, a3,) to be (-0.2, -4000, -5000). The
corresponding weighting (q1 , q 2, q 3 ) is (2.1 x 102, 5.27 x 103, 5.39 x 10-4) and the optimal PID controller (ki , kp ,
k d ) is (1.45x10,7.24x10,3.1x10-2).The speed response and applied voltage are shown in figure 12 and 13.




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6 Conclusions
This paper has presented a novel tuning formula that uses the LQR methodology to seek the optimal PID
parameters. The main merits of the proposed scheme include (I) there is no need for the observer design (2) there is
no constraint in the PID parameters (3) the weighting functions could be determined through poles assignment. The
existence conditions of the optimal PID controller for desired response are also submitted. The whole design idea
has been successfully implemented on the speed control of a BLDC motor. Computer simulations and
experimental results demonstrate that the excellent performance of the optimal PID controller.

                                                   References
[l] 0. Moseler and R Isermann, “Application of Model-Based Fault Detection to a Brushless DC motor”, IEEE
Trans. ND Electron., vol. 35, No. 12, pp. 1015-1020, 2000.
[2] Hang C-C. K. J. Astrom and W. K. Ho, „„Refinements of the Ziegler-Nichols tuning formula”, IEE Proc. Part D,
vol. 138, No. 2, pp. 11 1-1 18, March 1991.
[3] M. Zhuang and D. P. Atherton, “Optimum Cascade PID Controller Design for SISO Systems”, IEE Control, vol.
1, pp. 606-611, 1994.
[4] Voda A. A. and I. D. Landaq “A method of the auto calibration of PID controllers”, Autonmlica, vol. 3 1, No.
1, pp. 41-53, 1995. [SI C.-L. Lin, H.-Y. Jan, and N.-C. Shieh, “GA-Based Multiobjective PID Control for a Linear
Brushless DC Motor”, IEEUASME Trans. Mechatronics, vol. 8, No. 1, pp. 56-65, 2003,
[5] G. Marro, D. Prattichizzo, and E. Zattoni, “Geometric Insight Into Discrete-Time Cheap and Singular Linear
Quadratic Riccati (LQR) Problems”,IEEE Trans. Automatic Control, vol. 47, No. I, pp. 102- 107, 2002.
[6] M.-R. Issa and E. Barbieri, “Optimal PI-Lead Controller Design”, IEEE Proc. System Theory, pp. 364- 368,
1996. [SI J.-B. He, Q.-G. Wang and T.-H. Lee, “PVPID Controller Tuning Via LQR Approach”, IEEE Conference
on Decision & Control, pp. 1177-1 182, December 1998
[7] Allan R. Hambley, Electrical Engineering: Principles and Application, Prentice Hall, New Jersey 1997.
[8] Chee-Mun Ong, Dynamic Simulation of Electric Machinery, Prentice Hall, New Jersey, 1998.

    Authors Profile:




                                   VK.Chakravarthy Naik. received his B-Tech degree from Sri Venkateswara
                                   University, A.P, India in 2009 and presently doing his M-Tech in G.Pulla Reddy
                                   Engineering College, Kurnool, Andhra Pradesh, INDIA.




                                   G.Venkata Subba Reddy. received his M-Tech degree from Nagarjuna
                                   University, Andhra Pradesh, India. And he is presently working as Assistant
                                   Professor in G. Pulla Reddy Engineering College, Kurnool, Andhra Pradesh,
                                   INDIA.




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