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270 Chapter 4 Feedback Control System Characteristics
Closed-Loop Disturbance Step Response
Steady-state error
0.004 0.008 0.012 0.016 0.020
Time (s)
(a)
%Speed Tachometer Example
%
Ra=1; Km=10; J=2; b=0.5; Kb=0.1; Ka=54; Kt=1;
num1=[1]; den1=[J,b]; sys1=tf{num1,den1);
num2=[Ka*Kt]; den2=[1]; sys2=tf(num2,den2);
num3=[Kb]; den3=[1]; sys3=tf(num3,den3);
num4=[Km/Ra]; den4=[1]; sys4=tf(num4,den4);
sysa=parallel(sys2,sys3);
Block diagram reduction
sysb=series(sysa,sys4);
sys_c=feedback(sys1 ,sysb); Change sign of transfer function since the
%
disturbance has negative sign in the diagram.
sys_c=-sys_c <
%
Compute response to
[yc,T]=step(sys_c); M
step disturbance.
plot(T,yc)
tille('Closed-Loop Disturbance Step Response')
FIGURE 4.30 xlabel(Time (s)'), ylabelC\omega_c (rad/s)'), grid
Analysis of the %
closed-loop speed yc(length(T)) *4 Steady-state error —• last value of output yc.
control system.
(a) Response.
(b) m-file script. (b)
We have achieved a remarkable improvement in disturbance rejection. It is clear
that the addition of the negative feedback loop reduced the effect of the disturbance
on the output. This demonstrates the disturbance rejection property of closed-loop
feedback systems. •
EXAMPLE 4.6 English Channel boring machines
The block diagram description of the English Channel boring machines is shown
in Figure 4.17. The transfer function of the output due to the two inputs is
(Equation (4.57))
K + Us 1
Y(s) = 2 R(s) + 2 Us).
s + 12s + K 5 + lZv + K