NATIONAL INSTITUTE OF TECHNOLOGY, ROURKELA-8
B.Tech. End Semester Examination, 2023-24
SUBJECT: Principle of Control Systems Engineering SUBJECT CODE: EE3301
Duration of Examination: 3 hours FULL MARKS: 50
Figures at the right hand margin indicate marks. All parts of a question should be
answered at one place. Assume any missing information appropriately.
Question 1 is mandatory and answer any 4 from the rest
1
Q1. a) Consider a unity feedback system with plant G ( s ) = 2 and cascade
s + s +1
2
controller D ( s ) = K P + . Find the range of K P for which the system is stable.
s
b) Closed loop transfer function of a unity feedback system is given
Y ( s) 1
by = . Find the velocity error coefficient ( K v ) of the system.
R( s ) s + 1
c) Root locus of a feedback system as gain K is varied is shown in figure given
below (Fig. 1). Determine the range of K for which system response to step input
is non oscillatory. Justify your answer.
j
Im
−2 −1
K = 0.4 =0 −1 = Re
K =6
K
G ( j ) − plane
→
Fig. 1 Fig. 2
d) Fig. 2 given above shows the Nyquist plot of a unity feedback system having open
loop transfer function G(s) with one pole in the right half of s-plane. Comments
about the close loop stability using Nyquist Stability Criterion with proper
justification.
K1
e) A compensator is of the form D( s) = K 2 + ; K 0, K1 0, 1 0 .State is it a
1s + 1 2
lead or lag compensator with proper justifications. [2*5=10]
Q2. a) Write the dynamic equation and find the transfer function for the circuit given
below [3+2=5]
Rf
C
vin
− vo
R2 R1
+
1
s +1+
b) A system transfer function is given by G ( s) = . Compute the unit step
( s + 1)( s + 2)
response. Analyse your result for −1,1 . [4+1=5]
Q3. (a) For the system shown in Fig. 3(a), (i) sketch the root locus and find the (ii)
asymptotes and (iii) break away points. [3*1.5=4.5]
(b) Find the (i) range of K for stability and (ii) value of K to yield a 0.7 damping ratio
for the dominant second-order pair. [2]
(c) To improve stability, we desire the root locus to cross the jω-axis at j5.5 and the
open-loop function is cascaded with a zero as shown in Fig. 3(b). Then find the
value of α and sketch the new root locus. What improvement in transient response do
you notice? [1+1.5+1=3.5]
Fig. 3(a) Fig. 3(b)
Q4. (a) Fig. 4 shows a block diagram of a space-vehicle control system. Determine the
gain K such that the phase margin is 50°.What is the gain margin in this case? Draw
the Bode plot (gain and phase diagrams) of the closed loop system. [2*2.5+2=7]
Fig 4
C ( s) n 2
b) For the standard second-order system = 2 , show that the
R( s) s + 2n s + n 2
b is given by b = n (1 − 2 + )
1/2
bandwidth 4 4 + 4 2 + 2 . Plot a curve of
b n versus . [2+1=3]
Q5. (a)Draw a Nyquist locus for − + and the unity-feedback control system with the
open-loop transfer function
K (1 − s)
G( s) =
( s + 1)
Using the Nyquist stability criterion, determine the stability of the closed-loop system.
[1.5+2.5=4]
2
(b) A system with the open-loop transfer function
K
G ( s) =
s (T1s + 1)
2
is inherently unstable. This system can be stabilized by adding derivative control,
say, Gp (s) = (T2 s + 1) . Find the closed loop transfer function for both these cases.
Sketch the polar plots for the open-loop transfer function with and without
derivative control. [1+2.5+2.5=6]
Q6. A unity feedback system with open-loop transfer function is given as
1
G (s) =
s2
Design a lead compensator such that the acceleration error constant K a 20 and
phase margin (PM) is greater than 400 . [10]
