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EE370/01 Examination No. 1 Fall 2008/09

(50 minutes)

Kuwait University

Electrical Engineering Department

Name :

Student I. D. : .

Signature : .

Problem No. Grade

1 25

2 25

3 25

4 25

Total 100

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Problem 1 (25 points): Drive the expression for)(

)()(

sR

sUsM in terms of )(sGi for the following

system:

Problem 2 (25 points): Can you approximate the step response)10()(

)5.10()(

2

sbsass

ssC as

the step response of 2nd order system with5

2damping ratio and 2 seconds settling time? If yes,

find the values of a and b? Justify your answer.

Problem 3 (25 points): Use the Routh Table to find the range of K that keeps the following

feedback system stable.

Problem 4 (25 points): (a) Find the steady state error, )( te , in terms of a and b for the

following system:

(b) Find the values of a and b that yield zero steady state error.

+

+

_ _

C(s)R(s)

+

_

++

_

++

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EE370/01 Examination No. 1 Fall 2006/07

Kuwait University

Electrical Engineering Department

Name :

Student I. D. : .

Signature : .

Problem No. Grade

1 20

2 20

3 20

4 20

5 20

Total 100

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Problem 1 (20 points): Can you use the partial fraction expansion2

43

2

2

2

1

)2(211

s

k

s

k

s

sk

s

k

to find the inverse Laplace transform of22

2

)2)(1(

3

ss

sif:

a) 01

k b) 0

3 k

c) 32 kk

Explain your answer.

324

044

124

0

)24()44()24()(

)1()2)(1()2()2(3

431

321

4321

32

431321

2

4321

3

32

2

4

2

3

2

2

2

1

2

kkk

kkk

kkkk

kk

kkkskkkskkkkskk

sksskssksks

a) No, you cannot solve the equationsb) No, you cannot solve the equationsc) Yes, you can solve the equations.

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Problem 2 (20 points):

a) (10 points): Drive the expression of the output )(sC in terms of( ), ( ), ( ), ( ), ( )M s H s G s R s and D s for the following system:

b)

(10 points): What conditions on ( )M s and )(sG you will impose if you want to make thesteady state output )(tc equals zero when 0)( sR and )(sD is a step function.

LHPtheinGMofrootstheallplus

MdMG

HG

s

d

GM

HGsCb

DGM

HGR

GM

GCHDMCRHDG

HDGXCa

sss

1

)0(0)0()0(1

)0())0(1(

1

)1(lim)

1

)1(

1

)(

)

+

+

+

+

-

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Problem 3 (20 points): Consider the following closed-loop system:

Find the values of1

k and2

k to get 0.5 settling time.

ionapproximatorderndensuretok

kkw

T

kw

kksks

ks

ss

ksk

ss

ks

sT

n

s

n

20

145.02

84

22

)10()2(

1021

102)(

1

2

2

2

212

2

1

2

1

2

2

1

C(s)R(s)

+

_

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Problem 4 (20 points): Use Routh-Hurwitz criterion to find the values of0

c ,1

c ,2

c that will cause

the polynomial01

2

2

34)( cscscsssM to have two real roots in the LHP, one real root in

the RHP, and one root on the j-axis.

4s 1 2c 0c

3s 1 1c 2

s 12 cc 0c

1s

12

01121

cc

ccccc

0s 0c

Make rows 2 equal zero:12

cc and 00 c

Form the polynomial from scsrows 133

:

Differentiate the polynomial:1

23 cs

You have 3 symmetrical roots, select 01

c (one sign change): one root on the RHP and its

symmetrical root on the LHP plus one root on jw-axis. Also, one root on the LHP since you have

4 roots in total.

Then 012 cc and 00 c

4s 1 2c 0c

3s 1 1c 2

s 3 1c 1

s 1

11

3

2

3

3c

cc

0s 1c

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Problem 5 (20 points): A unity negative feedback system has the open loop transfer function

2

1)(

2

21

ss

skksG . Determine the gains

1k and

2k that minimize the steady state error

due to the step, ramp, or parabolic input.

)(2

)2(lim)(

2

11

lim)()(1

lim)(12

23

3

0

2

21

00 sRkskss

sssR

ss

skk

ssR

sG

ste sss

Then 05.0 12 kk for stability3

s 1 2k 2

s 2 1k 1s 12 5.0 kk 0

s 1k

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EE370/01 Examination no. 1 Fall 2005/06

Kuwait University

Electrical Engineering Department

Name :

Student I.D. :.

Problem no. Grade1 20

2 20

3 20

4 20

5 20

Bonus 20

Total 120

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Problem 1 (20 points): Given the following differential equation:

2

2

( ) ( ) ( 0.01)3 2 ( ) ( 0.01).

d y t d y t d u t y t u t

dt dt dt

Use Laplace transform to find ( )y t for a unit step input ( )u t and zero initial conditions.

Problem 2 (20 points): Find the value of 0a that will cause the following system to have somepoles on the j-axis. Use Routh-Hurwitz criterion.

Problem 3 (20 points):

c) (10 points): Drive the expression of the error ( ) ( ) ( )E s R s C s in terms of( ), ( ), ( ), ( ), ( )M s H s G s R s and D s for the following stable system:

d) (10 points): If ( )R s and ( )D s are unit step functions, find the relation between( )H s and ( )M s in order to get zero steady state error.

+

-

+

- + +

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Problem 4 (20 points): Consider the following closed-loop system:

Can you find the values of 1 0k and 2 0k to meet the following specifications:

Closed-loop damping ratio 0.8 . Closed-loop settling time 10sT sec. Steady state error of 0.1 due to a unit step input.

Write the reason if you cannot find 1k and 2k .

Problem 5 (20 points):a) (16 points): Sketch the root locus for the following system.

Show the following on the root locus:

Breakaway and/or break-in points. Angles of departure and/or arrival.

b) (4 points): Can you find 0k to stabilize the closed-loop system.

Bonus Problem (20 points): Consider the following feedback system

where 7 1a . Find the value of 0k that will minimize the steady state error due to a unitstep input for

C(s)R(s)+ +

- -

+

-

+

-

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EE370/01 Examination no. 1 Spring 2006

Kuwait University

Electrical Engineering Department

Name :

Student I.D. :.

Problem no. Grade

1 25

2 25

3 25

4 25

Bonus 20

Total 120

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Problem 1 (25 points): Given the closed-loop transfer function2

( ) 3 4( ) :

( ) 12 12

Y s sT s

R s s s

for the

following system

e) (10 points): Determine the closed-loop poles and zeros. Also, show if there is any dominantpole(s).

f) (15 points): Find the transfer function ( )G s ?

Problem 2 (25 points): Given the following system

a) (10

points): Determine the value of 0k that will minimize the steady state error

due to a unit step input ( )r t .

b) (15 points): Determine the value of 0k that will minimize the sensitivity of the error ( )E s to a change in the parameter a .

Problem 3 (25 points): Given3 2

( ) 6 M s s s ks c , find the range of k in term of c so that all

the roots of ( )M s have a negative real part. Use Routh-Hurwitz criterion.

Problem 4 (25 points): Consider the system:

Determine the values of , ,a b c and k to have a unit step response ( )y t with: steady state value of 2,

maximum amplitude of 3 and settling time of 4 sec.

+

-

2

-

k

-

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Bonus (20 points): Given the Laplace transform of

( ( 1))d f a t

dt

equals G(s), find the

Laplace transform of ( )f t in term of G(s)?

EE370/51 Examination No. 1 Spring 2009

(50 minutes)

Kuwait University

Electrical Engineering Department

Name (Arabic) :

Student I. D. : .

Signature : .

Problem No. Grade

1 25

2 25

3 25

4 25

Total 100

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Problem 1 (25 points): For the following system, find )(sU ?

++

+_ __

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Problem 2 (25 points): Consider the following translational mechanical system. A 2 N force f(t)

is applied for 0t , find the values of vf , M, and K such that the response x(t) is given by the

plot shown below.

0 0.5 20

1

1.1

Response

x(t)

Step Response

Time (sec)

Amplitude

fv

K

f (t)

x (t)

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Problem 3 (25 points): Use the Routh Table to find the range of that makes the followingfeedback system unstable.

Problem 4 (25 points): (a: 15 points) Derive the expression for the error )()()( sCsRsE in term

of )(sH for the following system:

(b: 10 points) Find the value of )0( sH that minimizes the steady state error.

C(s)R(s)+

_

+

_

+

+