1. a. Find out the settling time for a second-order closed loop transfer function

2

9

4 9T s

s s

(calculate the settling time for 5% band as well as for 2% band).

1. b. A Linear time-invariant system initially at rest, when subjected to a unit-step input, gives a response y(t) = te-t , t > 0. Find out the transfer function for the system.

1. c. Find the phase angle of the system

2

3 2

4

2 5 10n

s s

s s s s

at ω=0 rad/sec and at ω=∞

rad/sec.

1. d. The closed loop transfer function of a control system is given by

1

1

C s

R s s

.

Find out the steady state value of the output c(t) when the system is excited by an input r(t)=sint.

1. e. Find out the amount of maximum phase shift that can be obtained from a lead

compensator whose transfer function is given by 1 12

1 6c

sG s

s

1. f. Find the open loop transfer function of a dynamic system (c whose root locus plot (location of poles not given) has been shown in Fig.P1f.

Fig. P1f

1. g. Find out the steady state error for the system shown in Fig. P1g

Fig. P1g

1. h. Find out the range of Gain K for which the system shown in Fig P1h will exhibit a stable response

Fig. P1h

1. i. The steady state error of a unity feedback linear system for a unit step input is 0.1. Calculate the steady state error of the same system, for a pulse input r(t) having a magnitude of 10 and a duration of one second, as shown in the figure P1i.

Fig. P1i

1. j. The second order system is described by the state equation: X AX with

A=0 1

2 3

. Find out the value of damping ratio and natural frequency of

oscillation for the system.

2X10=20

10 

1sec

2. a. Consider the following characteristic equation:

4 3 2 1 0s Ks s s

Determine the range of gain K for stable operation. 6

b. What is the unit of damping ratio? Justify your answer. 3

c. State and explain the Hurwitz stability criterion for an nth order system. 4

d. Show that the sensitivity of the transfer function

with respect to the parameter K is given by

3. a. Show that the angle of departure from any complex pole of the open loop system can be found using the relation (where Фp and Ф carry their usual meanings).

0180 2 1 ........ 0,1, 2,p q q

b. Sketch the root Locus plot (0<k<∞) for the system, whose open loop transfer function

is given by 24 4 13

KG s H s

s s s s

.

Also, find the breakaway point(s) (if any), the asymptotes, and their centroid. Find the angle of departure from complex poles (if any). In addition, find the values of k at each breakaway point.

4. a. Prove that the polar plot of the first order Low-pass filter is a semi-circle.

b. Determine the transfer function of an open loop system whose experimental frequency-response curves are shown in Figures P4b(i) and P4b(ii):

7

5 

15

7 

13 

Fig P4b(i). Bode Magnitude Plot

Fig P4b(ii). Bode Phase Plot (Slope of the phase curve at very high value of ω>10000 is almost equal to -0.02rad/(rad/sec).

5. a. Consider a Lag-Lead network defined by

show that at frequency ω1, where 1

1 2

1

T T the phase angle Gc(jω)=0. 6

b. The forward path transfer function of a unity feedback system is given by

( )

2 30

KG s

s s s

Design a suitable compensator such that the system will satisfy the following specifications. Also justify your choice of the compensator citing suitable reasons:

a) Phase Margin >= 350

b) Gain Margin >= 20db

c) Steady State Error for ramp input <=25/sec 14

6. Find out the generalized solution of Linear time varying state equation given by X A t X B t u 12

b. The maximum overshoot for a unity feedback control system having its forward path transfer function as 1G s K s sT is to be reduced from 60% to 20%. The

system input is a unit step function. Determine the factor by which k should be reduced to achieve the aforesaid reduction. 8

7. State and prove the necessary and sufficient condition of arbitrary pole placement using feedback design. 20

8. a. Check the controllability of the system using Gilbert’s method. Where A and B matrices of the input state equation are as follows:

0 1 0

0 0 1

6 11 6

A

0

0

1

B

6

b. Consider the input-state equation of a linear time invariant deterministic system described by X AX BU , where the matrices A and B are

1 0 0

0 2 0

0 0 3

A

1

1

1

B

Find the feedback gain matrix K=[k1 k2 k3] in a state variable feedback structure such

that the closed loop poles are placed at p1= -2, p2= -3, p3= -4. 14