# Control Systems B Tech 6th Sem

Time Allotted : 3 Hours

Full Marks : 70

. The figures in the margin indicate full marks.

Candidates are required to give their answers in their own words

as far as practicable.

GROUP ~ A

( Multiple Choice Type Questions )

1. Choose the correct alternatives for any ten of the following :

10 x 1 = 10

u

-3J L1 j

y = [ 1 0}x The system is

a)             controllable 8& observable

b)             uncontrollable & observable

c)              controllable & unobservable

d)             uncontrollable & unobservable.

ii)

The state variable description of a linear autonomous system is X = AX when X is a state vector &

0            2

2 °J                                     *

The poles of the system are located at

a)     -2 and + 2    b) -2 j and + 2 j

c)      -2 and – 2    d) + 2 and + 2.

d y                       . *

iii)          The value of a matrix in — = AX

dt ‘

for the system described by the differential equation

 1       0 -1       -2 1       0 -3       -2
 b)
 0 1 -2 -1
 c)
 d)

The transfer function of a ZOH is

 a)
 b)
 d)
 C)

vi)           Stable focus is represented by

vii)         The variable gradient method is used to find

a)           Lyapunov function

b)           describing function

c)            state transition matrix

d)           eigenvectors.

viii)       Nonlinear system can display

a)           only one equilibrium point & limit cycle

b)           multiple equilibrium point & limit cycle

c)            only one equilibrium point

d)           only one limit cycle.

ix)          The curve traced out by all possible point [x1{t),x2{t)]

is called phase trajectory

a)           at t is varied from 0 to a

b)           at t is varied from – a to a

c)            as t is varied from 0 to -a

d)           as t is varied from any value.

x)           The characteristic equation of a system is KG (s) H(s) = -1 . Stability condition for such system is

a)              \G( > )H(p ) | < 1/K

and <G(jv )H( jo> ) = -180°

b)             \G(ja>        ) | <K

and < G (p) ) H (ju> ) = 180°

c)              \G(ju )H(ju> ) | < 1

and < G f jco ) H (ju> ) = -180°

d)             \G(ju ) H (jn ) | > 1/K

and < G (jc o jHfjco ) – -180°.

xi)          The example of positive semi definite function is

a)           (Xj + x2f ,                                  b) x}2 + x2

c)               -x2 -fa + x2)2       d) *1*2 + *22

xii)        The describing function for _____________ Ll*. with inPut

a)          **                                   b) ^

1              nX

, 4                                                           HI 4M

c      —                                    d)

1     TtX                                            71

GROUP -B ( Short Answer Type Questions )

Answer any three of the following.

1. The overall transfer function of a SISO system is given

hv ZM _ s + 4s + j_. Obtain state model of the system. ms) s3 + 5s2 + 4s

1. The state space representation of a system is Xi = -xx + v

.X.2 ~ 2%2 ^

Comment on controllability and observability of the system.

1. Solve the difference equation given below :

y{k + 2) + 3y{k + \) + 2y(k) = 0 for y(-l) = ~}^, y (-2) = % •

\ *

1. Obtain pulse transfer function of the system shown below

‘ 20 with T = 0 ■ 5 s and G„(s) =

s (s + 5)

wi.3—O—

1. Find state-transition matrix for the homogeneous state

equation

X(k + l) = FX(k) , where F =

GROUP -C ( Long Answer Type Questions )

Answer any three of the following.

1. a) For the system represented by
 ‘-7 r 2′ X + -1 u -12 0
 X =

Y = [3 – 4] jc + [ 2 ] it

Compute output response when u(t) = 3e and

X =

Determine the state feedback gain matrix so that the closed loop poles of the following system are located at

– 2 ± J3 • 464 , – 5 . Give a block diagram of the control

d ‘ u

configuration.

 ‘ 0 1 o– ‘ O’ X = 0 0 1 x + 0 -6 -11 -6 -1
 u

7 + 8

Y = [ 1 0 0] x .

1. a) Derive describing function of a relay with saturation & dead zone nonlinearity.

b)         Investigate stability of a system shown below using describing function technique.

1. a) Explain jump resonance of a nonlinear system.

b)             Explain the concept of limit cycle with a suitable example.

c)              For a spring mass system, construct the phase trajectory on X – X plane using isocline method with

v,     %

initial conditions x(0) = -l and x(0) = 0. Comment on the kind’of singularity obtained.

1. a) Discuss the concept of Lyapunov’s first and second

stability analysis.

b)             Investigate stability using Laypunov’s second method for the system represented by

0 1 -1 -1

Describing function is based on

a)           first harmonic approximation

b)           approximation at an operating point

c)            stability of an operating point

d)           finding of Lyapunov function.