Optimum State Space Control of BLDC Motor

(Simulation and Experiment)

Auralius Manurung

RIS-Lab, GSNU

December, 2009

Outline

• Model Identification

• Control Design

• Simulation Results

• Experiment Results

• Conclusions

• References

Model Identification

• Well defined model is very important for optimum control design.

• Input data (applied voltage) and output data (rotational speed) were recorded during experiment.

• Numerical process to identify the system was conducted by using MATLAB.

Model Identification (cont’d)Parameter estimation was conducted by using MATLAB.

Model Identification (cont’d)• The BLDC motor is a nonlinear system. Coloumb friction is considered as

its main cause.

• Precise model is difficult to get.

Model Identification (cont’d)

• From model identification, the transfer function of the BLDC motor is:

• The state space of the system are:

• Where x1(t) is the angular velocity of the motor shaft, x2(t) is the armature current of the motor, and u(t) is voltage applied to the motor driver.

1167010340

1923902 ++ ss

[ ]97.460

0

32

0128

17.9110340

=

=

−−=

C

B

A

Control Design

• State feedback controller with LQR design to minimize the quadratic cost function:

• Optimal feedback gain:

• Solution of P is solved using algebraic Ricatti equation:

∫∞

+=0

))()())()(( tRututQxtxJ TT

PBRK T1−=

01 =+−+ − QPBPBRPAPA TT

Control Design (cont’d)

• Feedback control signal:

• Feed-forward term is used to compensate disturbance and model uncertainties:

• To reduce steady state error, integral term is introduced:

rVKxu ω+−=

11 ))(( −−−−= BBKACV

∫ −=t

ar dt0

)( ωωε

Control Design (cont’d)

Speed Regulator Structure

+

-

Simulation Result• clc;• close all;•• A = [ -10340 -91.17 ; 128 0] ;• B = [ 32 ; 0];• C = [0 46.97];•• %Weighting matrices• Q = [ 1 0 ; 0 0.1 ];• R = 1;•• %State feedback controller gains that minizimizes the• % cost functional with the above given weighting matrices• K = lqr(A,B,Q,R)• 'Multiplied with xR'• B*K•• %Regulator closed loop system• Ac = A-B*K;• eig(Ac)• Bc = [ 0; 1 ];• Cc = eye(2);•• Dc = [ 0 ];• sys = ss(Ac, Bc, Cc, Dc);•• V = inv(C*(inv(A-B*K)*B))• %Step response• figure• step(sys)

Simulation Result (cont’d)

• Using weighting matrices:

• We can get:

– Kω = 0.0564

– Ki = 0.5599

– Kff = 0.0610

Q = [ 1 0 ; 0 0.1 ];R = 1;

Experiment Results

• LabVIEW 8.5.

• NI PXI-7358 controller.

• Maxon servo amplifier DES 70/10.

• Maxon EC45 BLDC motor with choke inductor.

NI-PXI Maxon DES 70/10 Maxon EC

Experiment Results (cont’d)

With Only Integrator as Controller

Time delay

Experiment Results (cont’d)With Integrator and LQR as Controller

%Weighting matricesQ = [ 1 0 ; 0 0.1 ];R = 1;

Time delay

Experiment Results (cont’d)

%Weighting matricesQ = [ 1 0 ; 0 0.1 ];R = 1;

With Integrator, LQR, and Feed Forward as Controller

Experiment Results (cont’d)Disturbance Test

With Feed-forward Term Without Feed-forward Term

Time delay

Kff = 0.0610 Kff = 0

Conclusions

• We have designed a control system for a BLDC motor based on system identification and LQR.

• LQR control provides an optimal state feedback control that minimizes the quadratic error and control effort. In our case, it makes the transient respond time of our system become better.

• In general DC motor, the effect of Coloumb friction can be reduced by introducing feed-forward control to the system, increasing its disturbance rejection capability and reduce delay time.

• This control method has been implemented to real system and considerably well suited for BLDC motor.

References

• M. Ruderman, J. Krettek, F. Hoffmann, and T. Bertram, Optimal State Space Control of DC Motor, Proceedings of the 17th World Congress IFAC, Korea, 2008.

• P. Chevrel, L. Sicot, and S. Siala, Switched LQ controllers for DC motor speed and current control: a comparison with cascade control, Power Electronics Specialists Conference, Italy, 1996.

• Aimin Liu, Yaru Liang, Shang Gao and Jun Gao, Modified linear quadratic optimal control method and application in linear brushless direct current motor, . International Conference on Electrical Machines and Systems, 2007.