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Problem 4.33PP
For the system in Fig., compute the following steady-state errors: 
(a) to a unit-step reference input;
(b) to a unit-ramp reference input;
(c) to a unit-step disturbance input;
(d) for a unit-ramp disturbance input.
(e) Verify your answers to (a) and (d) using Matlab. Note that a ramp response can be generated
(e) Verify your answers to (a) and (d) using Matlab. Note that a ramp response can be generated 
as a step response of a system modified by an added integrator at the reference input.
Figure DC Motor speed-control block diagram
Step-by-step solution
Step 1 of 16
Refer to Figure 4.44 in the text book for the block diagram of a DC motor speed-control. 
The controller transfer function is,
D ,(5 ) = * , + i
Consider the differential equation of the DC motor.
+ 60;> = 600v, - l.SOOw 
Apply Laplace transform on both sides. 
»r(i)+601 '(s ) = 600F .(i)- l,500»'(j)
Consider the armature voltage value in PI control.
Apply Laplace transform on both sides.
Step 2 of 16
Calculate the transfer function of the PI controller.
( i + 6 0 ) r ( i ) - 6 0 0 ^ - * , ,£ ( j ) - ^ £ ( j ) j - l , 5 0 0 » '( s ) 
{s + 6 0 )Y (s ) = 6 0 0 ^ ^ * , j £ ( j ) j-1 ,5 0 0 » '(* )
( j+ 6 0 ) r ( j ) + 6 0 0 ^ ^ * ,+ ^ j £ ( j ) j = - l ,5 0 0 » '( j) 
^s + 60 + 6 0 0 t ,+ 6 0 0 ^ j £ ( s ) = -l,5 0 0 » '(» )
-1,500
^i+60 + 600*, + 6 0 0 ^ j
-1.500s
( j “+60s + 600*,j + 600*,)
Consider the value of R as zero and write the value of error detector.
£ (» )— l'( s )
1,500s
»'(s) s '+ 60 (l + 101:,)s+6001:,
Step 3 of 16
The roots of characteristics equation is - ^ + 6 0 J y -6 0 -6 0 J ■ 
Write the characteristic equation from the roots. 
(s+60+60y){s+60 -60y) = s’ + 120s+7200 
Write the general characteristic equation.
5*+2Co^ + o),* « 0
Calculate natural frequency from equations (7) and (8).
« ),=V 7i200
^ )“ 
k,^12
Step 5 of 16
Calculate proportional constant k ^ .
60(1+1 Ot, ) = 2(0.707)(60V2)
60
lO/t,
2(0.707)(60^) ^
60
* , = 0.1
Step 6 of 16
(a)
To determine the transfer function with reference input, take disturbance input, » '(s ) 
as zero.
Determine the transfer function from the output to the reference input.
1+600
6 0 0 ( V + it ; )
s(s + 60) ♦ 600(^^ + k,)
Step 7 of 16
Determine the steady state error.
£(s ) = « (s ) - l '(s )
I y (» )
£(s) £(s)
I 600(*,s + * ,)
s(s+60) + 600(*,s+*,) 
s(s + 60)
s (s + 60)+600( t ,s + * ,)
£ (s ) =
j (j -f60)
s(s + 60) + 600(k^ + k,)
g(s)
Step 8 of 16
The unit-step reference input is, 
I£ (s ) = l
Determine the steady-state emor to unit-step reference input.
= lin i5 £ (^ )
s l im j
+ 60) 1
5 (j + 6 0 )+ 6 0 0 ( it^ + /t,) 5
Thus, the steady state error to unit step reference input is .
Step 9 of 16
(b)
The unit-ramp reference input is.
Determine the steady-state emor to unit-step reference input.
g | im j£ ( f )
g(j+60) 1
s l im ;
g (g + 6 0 )+ 6 0 0 ( it^ + * ,)4 ^
= lim -
(j+ 60 )
• 5 (g + 60) + 6 0 0 ( + Jfc,) 
60 
600A;,
I
" lo t ,
The value of k, is 12.
1
10(12)
1
*120
Thus, the steady state error to unit ramp reference input is
Step 10 of 16
(c)
The transfer function from the output to the disturbance input is,
£ ( i ) -1,500»
B '( i) « '+ 6 0 (l + 10*,)s+600*,
Substitute 12 for k, and 0.1 for k^-
£ ( £ ) = . -l,50te
^ (s ) j ’ +60(l + 10x0.1)f + 600(12) 
-l,500i
j ’ + 120j + 7200
The error function is.
E (s) = ~i-------------------fY(s)
g’ +120g + 7200
The unit-step disturbance input is.
Step 11 of 16
Determine the steady-state emor to unit-step reference input. 
e„^ \m sW {s)
-hSOOs 1slimg-
g* + 120g + 7200g
= 0
Thus, the steady state error to unit step disturbance input is .
Step 12 of 16
(d)
The unit-ramp disturbance input is.
Determine the steady-state emor to unit-step reference input. 
e„ = \m sE{s)
-USOQg 1
s lim g - ra g* + 120g + 7 2 0 0 g *
.. -1,500 ^
.................« - » i ' + 120f + 7200 
1500
= -0.208
Thus, the steady state error to unit ramp disturbance input is |-0.2081-
Step 13 of 16
(e)
The error function is.
E (s) =
g(s + 60)
s ( s + 6 O ) + 6 O 0 { k p S + k ,
j ( g + 60)
■ , ’ + (60 + 6 0 0 t , ) j + 600*,
Write the MATLAB code to verify the steady state emor for unit step reference input.
ki=12;
kp=0.1;
n=[1 60 0];
d=[1 60+600*kp 600*ki];
sys=tf(n,d);
step(sys)
Step 14 of 16
Get the MATLAB output for the step response.
Step 15 of 16
The transfer function to disturbance input is.
y ( j) -l,50to 
» '( i) +120s+ 7200
Note that the ramp response is obtained by added an integrator tenn to the transfer function.
y (s) -1.500s 
B '( s ) " s ( s ’ + 120$ + 7200)
Write the MATLAB code to verify the steady state emor for unit ramp disturbance input.
n=[-1500 0];
d=[1 120 7200 0];
sys=tf(n,d);
step(sys)
Step 16 of 16
Get the MATLAB output for the ramp response.
Hence, the result is verified.