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Problem 4.38PP
Consider the DC motor speed-control system shown in Fig. with proportional control, (a) Add 
feedfonward control to eliminate the steady-state tracking error for a step reference input, (b) Also 
add feedforward control to eliminate the effect of a constant output disturbance signal, w, on the 
output of the system.
Figure Block diagram
Step-by-step solution
step 1 of 7
(a)
Consider the gain of the plant.
59.292C ( .) =
i"+6.978s + 15.123
Consider the gain of the controller.
= 3 ......(2)
Consider the gain of sensor.
H{$) = \ ..... (3)
Determine the DC gain of the plant.
15.123
(1)
G " (0 ) = i59.292 
C -'(0 ) = 0.2551 ..... (4)
Write the output equation for error detector. 
£ ( j ) = j t ( j ) - y ( 5 ) ...... (5)
step 2 of 7
Refer from Figure 4.49 in the textbook and write the transfer function equation with respect to 
zero Wvalue.
(6)y(«) = G(»)[*,£(»)+G-'(0)J?(»)]
Substitute equation (5) in equation (6).
y ( » ) = G W [ * , ( « M - r W ) + G - ' ( o ) J iW ] 
i ' ( 4 ) = G ( 4 ) [ M ( 4 ) - t , i ' ( 4 ) ) + c r - ( o ) « ( * ) ] 
r ( j ) = G (f ) * ,« ( j ) -G ( i ) t , l ' ( j )+ G ( j )G - ' {0)R(s) 
( l + G ( * ) * , ) y ( , ) = « W [ G W * , + G W G - '(0 ) ]
y(») ft ,-fG -'(0 )lG (» )
« ( i ) l+ * ,G ( i)
(7)
Step 3 of 7
Write the MATLAB program for tracking response with feedfonward from equations (1), and (7). 
clc;
s=tf('s');
G=59.292/(s''2+6.978*s+15.123); 
kp=3;
dcgain1=dcgain(G);
T1 =G*{1/dcgain1 +kp)/(1 +kp*G);
t=0:.01:5;
y1=step(T1,t):
plot(t,y1);
xlabel{'Time (sec)'); 
ylabel{'$y(t)$'.'interpreter'.'latex'):
The output for MATLAB program is given below.
Figure 1
Step 4 of 7
The figure 1 clarifies the influence of feedfonward control in eliminating the steady-state tracking 
error.
Hence, the addition of feedfonward control results in zero steady-state tracking error for a step 
reference input and the dc gain of the closed loop system is I unity!.
Step 5 of 7
(b)
Refer from Figure 4.49 in the textbook and write the transfer function equation with Wvalue.
r ( j ) = iy ( i ) + G ( s ) [ * , £ ( j ) + G - ' ( o ) » ' ( j ) ] (8)
Write the output equation for error detector.
f(4 ) = J?(4)-y(5) ..... (9)
Consider the value of J?(4)- 
= 0 ...... (10)
Substitute equation (10) in equation (9).
£ ( 4 ) = - r ( 4 ) ...... (11)
Substitute equation (11) in equation (8). 
y ( , )= » '( * )+ G W [* , ( -y M ) -G - '(o )» '(s ) ] 
y (s ) . (y ( i ) - t ,G ( j ) r (s ) -G - '(o )G (s ) ) r ( j ) 
[ N . t , G W ] y ( s ) = [ i - G - ( o ) G ( * ) ] iy W
r ( 4 [ i - G - ' ( o ) G M ]
W(s)~ [ l+ t ,G ( j ) ]
(12)
Step 6 of 7
Write the MATLAB program for disturbance rejection response with feedfonvard systems from
equations (1), and (12).
s=tf('s');
G=59.292/(s''2+6.978*s+15.123); 
kp=3;
dcgain1=dcgain(G);
t=0:.01:5;
Tw1=(1-1/dcgain1*G)/(1+kp*G);
yw1 =step(Tw1 ,t);
figureO
plot(t,yw1);
xlabel('Time (sec)');
ylabel('$y(t)$'.'interpreter'.'latex');
nicegrid
The output for MATLAB program is given below.
Figure 2
Step 7 of 7
The figure 2 clarifies the influence of feedfonward control in eliminating the steady-state tracking 
error for a step output disturbance.
Hence, the dc gain of the closed loop system is |zero|.