# WBUT Question Papers EE

## Control Systems B Tech 6th Sem June- 2008

Time : 3 Hours )

GROUP – A (Multiple Choice Type Questions)

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

0 A set of state variables for a system is

a) not unique in general  b) always unique

c) never unique           d) may be unique.

ii) State variable approach converts an nth order system into

a)   n second order integro-differential equation

b)              two differential equations

c)               n first order differential equations

d)              a lower order system.

ill) If both eigenvalues of a second order system are real, equal & negative of each other; the origin in the phase portraits is termed as

v) Lyapunov’s stabllty criterion can be used for determination o

a)        linear system b) non-linear system

c)     both (a) & (c)    d) none of these.

vl) In the figure below :

a)              A has unstable limit cycle & B has stable Hmlt cycle

b)              A has stable limit cycle & B has unstable limit cycle

c)              both A & B have unstable limit cycle

d)              none of these.

vii) Lyapunov function is

a)       energy function      b)

c) state function                 d)

viil) Describing function Is based on

a)      Harmonic linearization      b)

c)      Degree of non-linearity      d)

 ‘ – 0-5 0 ‘ 0 ‘ If A = . B = . 0 – 2 . . 1 .

a)              system is controllable

c)               system Is undefined , then

b)                  system is uncontrollable

d)                  none of these.

 1 •X ’ 1 o’ ■ ■ ■ 0 ‘ ss’ + 1 X . 1 1 . – ^2- . 1 .
 CS/B. T0dtfEK)/SM-6/EE-6O3/O6                                      5

 x) A linear system is described by the state equations

 C = X2 where R&.C are the input & output respectively. The transfer function is 1 1
 GROUP-B ( Short Answer Type Questions) Answer any three of the following.
 3x 5= 15

1. 2.      Obtain the state variable model of the system whose transfer function is given by

y(S) S2 + 3S + 3 X/(S) = S3 + 2S2 + 3S+1

 ’ 0 1 ’ ■ 0 ‘ X[ fc+ 1 ) = X(fc) + .0 – 1 . . 1 .
 3.      Determine the unit step response in closed form for the system given by

 – 1 1 0

= 0-lJT;&C = I101]. Determine the

0 -4 2

0                            0 – 10 similarity transformation matrix P to transform the system. X = AX + BU & output Y = CX to the controllable canonical form.

5. Find the describing function of on-off non-linearity.

1. Consider the network shown in the figure. Obtain the state variable formulation.

 v

GROUP -C

l .

( Long Answer Type Questions )

Answer any three of the following.

1. a) Define phase plane, phase trajectory & phase portrait.

b)               Define singular points. Give the detail classification of singular points.

c)               For the linear system shown, sketch the phase trajectories using isocline method.T

——>—1 (<>- £(S)- —

d)               Show that the following quadratic form is positive definite.

v [ x) = 8x] + xl + 4×3 + 2x1 x2 – 4Xj x3-2x2x3.                                           3 + 4 + 5 + 3

a)               Solve the following difference equation using Z-transform method. x(Jc + 2) + 3x(k + l) + 2x(/c) = 0, x(0)=0, x( 1) = 1.

b)               Obtain the discrete time state equation for the continuous time system given in problem, 8 (a) assuming sampling time. T = 0-1 sec.

c)                In continuous time, a system Is given by the transfer function,

G ( s ) = —-—. Find the Z-transfer function, G(z).                     6 + 6 + 3

9. a) Determine the amplitude and frequency of the limit cycle of the non-llnearlty

shown In the figureState Lyapunov’s direct method of investigating stability of non-linear system.

b)               Determine the stability of the system shown In the figureA linear system is described by the state equation X ■x.

Investigate the stability of this system using Lyapunov’s theorem.

 Consider the system defined by X = AX + BU, 0 1 0 0 where A = 0 0 1 , B = 0 – – 1 – 5 – 6 – – 1 –

By using feedback control u = – kx, it Is desired to have closed loop poles at s = – 1 ±J, s = – 10. Determine the state feedback gain matrix k. 4 + 5 + 6

1. Write short notes on any three of the following :

a)               Regulator problems. ,

b)               Harmonic linearization.

c)                Computer control.

d)               Non-conservative systems.

e)                Feedback controller.

END