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Problem 4.24PP
(a) Compute the transfer function from R(s) to E(s) and determine the steady-state error {ess) for 
a unit-step reference input signal, and a unit-ramp reference input signal.
Consider the system shown in Fig.
(b) Determine the locations of the closed-loop poles of the system..
(c) Select the system parameters {k, kP, kl) such that the closed-loop system has damping 
coefficient ^ = 0.707 and ojn= What percent overshoot would you expect in y(t) for unit-step 
reference input?
fdXJFinrlthft,trackinn prmr « ( * ) (1)
1 + G(5)Dc(»)
Substitute the plant gain of the system and controller gain from Figure 4.39 in the textbook for 
G {.) in equation (1).
1E {s)= -
1 +
1+
(‘-4 )
s ^ + k jM + k ,k
R(s)
J?(i) s* + k ^ + k , k
■(2)
Hence, the transfer function of the system from J?(;)to £ ( j ) is £ W = .
R{s) + k ^ + k ,k
Step 2 of 10 ^
Calculate the = from equation (2).
-2 1
»-»» s ^ + k fk s + k jk s
s . « H = o (3)
Hence, the steady-state error for the unit step input is signal is |e_ ̂ (oo) = 0 |.
Step 3 of 10 ^
Calculate the from equation (2).
= lim-
*-*® s^+kpks+kfk
= 0 ...... (4)
Hence, the steady-state error for the unit ramp input is signal is |e ̂ (oo) = 0 |.
Step 4 of 10
(b)
Write the denominator part of the equation (2).
s ^ + k ^ + k , k » 0 ...... (5)
Find the roots of the above equation.
■■ 2
Hence, the roots of the closed loop poles from the transfer function from J?(5)to is
- k ^ ± y l { k ^ y - 4 ( k , k )
Step 5 of 10
(c)
Write the characteristic equation of the second order system.
5’ -f 2Co^ - K i>/ » 0 ...... (6)
Equate the equations (5) and (6).
s *+ k fk s + k fk ^ s ^ + 2 fy a ^ + to * ...... (7)
Substitute 0.707 for ^ and 1 for .)
Write the general formula for percent of peak overshoot.
W , = e”* ' '^ x l0 0 (14)
Substitute 0.707 for ^ and 1 for h -
- ( * * , ) ’ » 0
4tt, » ( * * , ) ’ (20)
^ > 0 ...... (21)
2
Step 9 of 10 ^
Hence, In PI controller the relation between parameters ifc^and ifej,must be satisfy the
4 jM : ,» ( J M :^ fa n d ^ > 0 condition.
Step 10 of 10
(f)
Determine the transient behavior of the tracking emor for unit ramp input from equation (19).
. fj4tt,-(**,)’
8in J s O 
2
M - ( * * , ) ’ .' _ r — n
V.
From equation (22), the overshoot time is finite and rather small for practical purposes. 
Hence, the relation must satisfy the condition |4jbfĉ