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.