Solutions Manual for Automatic Control Systems (10th Edition) by Farid Golnaraghi and Benjamin C. Kuo

All Chapters Covered




SOLUTION MANUAL

,
,Automatic Control Systems, 10th Edition Chapter 2 Solutions Golnaraghi, Kuo


Chapter 2

2‐1 (a) Poles: s = 0, 0, 1, 10; (b) Poles: s = 2, 2;

Zeros: s = 2, , , . Zeros: s = 0.

The pole and zero at s = 1 cancel each other.




(c) Poles: s = 0, 1 + j, 1 j; (d) Poles: s = 0, 1, 2, .
Zeros: s = 2.




2-2) a) ‫ܩ‬ሺ‫ݏ‬ሻ ൌ ሺ௦ାଵሻ
௦ሺ௦ାଶሻሺ௦ାଷሻమ
௦
b) ‫ܩ‬ሺ‫ݏ‬ሻ ൌ మ
ሺ௦ାଵሻሺ௦ାସሻ

c) ‫ܩ‬ሺ‫ݏ‬ሻ ൌ ௦మ ିଵ
௦మሺ௦ାଷሻሺ௦ାଵሻమ




2-3)

MATLAB code:




2‐1

,Automatic Control Systems, 10th Edition Chapter 2 Solutions Golnaraghi, Kuo


clear all; s = tf('s')



'Generated transfer function:'

Ga=10*(s+2)/(s^2*(s+1)*(s+10)) 'Poles:'

pole(Ga)

'Zeros:'

zero(Ga)



'Generated transfer function:'

Gb=10*s*(s+1)/((s+2)*(s^2+3*s+2)) 'Poles:';

pole(Gb)

'Zeros:'

zero(Gb)



'Generated transfer function:' Gc=10*(s+2)/(s*(s^2+2*s+2))

'Poles:';

pole(Gc)

'Zeros:'

zero(Gc)



'Generated transfer function:' Gd=pade(exp(-

2*s),1)/(10*s*(s+1)*(s+2)) 'Poles:';

pole(Gd)

'Zeros:'

zero(Gd)




2‐2

,Automatic Control Systems, 10th Edition Chapter 2 Solutions Golnaraghi, Kuo




Poles and zeros of the above functions:

(a)

Poles: 0 0 ‐10 ‐1

Zeros: ‐2

(b)

Poles: ‐2.0000 ‐2.0000 ‐1.0000

Zeros: 0 ‐1

(c)

Poles:

0

‐1.0000 + 1.0000i

‐1.0000 ‐ 1.0000i

Zeros: ‐2

Generated transfer function:

(d) using first order Pade approximation for exponential term Poles:

0

‐2.0000

‐1.0000 + 0.0000i

‐1.0000 ‐ 0.0000i



Zeros:

1




2‐3

,Automatic Control Systems, 10th Edition Chapter 2 Solutions Golnaraghi, Kuo


2-4) Mathematical representation:

In all cases substitute ‫ ݏ‬ൌ ݆߱ and simplify. The use MATLAB to verify.

10 22 1 2
2
102 2
R ;
2 2
( 1)( 2 100)

10( j 2)
2
( j 1)( j 10) tan 1 2 2
2


10( j 2) ( j 1)( j 10)
1 2
 2 2 2
2
( j 1)( j 10) ( j 1)( j 10)

10( j 2)( j 1)( j 10)
a) 2 2
( 1)( 2 100) tan 1 12
2 1
j 2 j 1 j 10 12
R 2 2 
2 1 2 102 2

R(e j 1 e j 2 e j 3 ) tan 1 102 2
3
10
102  2
1 2 3




2 2
10 1 9
R ;
( 2 1)2 ( 2 9)

10
( j 1) ( j 3) 2
tan 1 12
1 1
10 ( j 1)( j 1)( j 3)
 12
( j 1)( j 1)( j 3) ( j 1)( j 1)( j 3) 
10( j 1)( j 1)( j 3) 12
b) tan 1
( 2 1) 2 ( 2 9) 2 1
R
j 1 j 1 j 3 12
1 2 1 2 9 2 
1
R(e j 1 e j 2 e j 3 ) 3 tan 9 2
3
9 2
1 2 3




2‐4

,Automatic Control Systems, 10th Edition Chapter 2 Solutions Golnaraghi, Kuo


10
j ( j2 2 2 )
10 j (2 2
j2 )
( j2 2 2 ) (2 2
j2 )
10( 2 (2 2 ) j)
c)
(4 2 (2 2 )2 ) 2
2 (2 ) j
R
4 2 (2 2 )2
R(e j )

10 4 2 (2 2 2
)
R 10
;
(4 2
(2 2 2
)) 4 2 (2 2 )2
2 2
tan 1 4 2 (2 2 )2
2
4  (2 2 )2
2


1
R ;
2 2 2
e 2j
10 2  1
10 j ( j 1)( j 2) 2  2 2

j( j 1)(2 j 2) 22j tan 1 2
d)  10 ( 1)( 2) e 2 2
1 2


j 2 j 1 2j j /2 
R e
22 2 1 2
tan 1
12
R(e j 1 e j 2 e j 3 )
2
1
12
1 2 3


MATLAB code:

clear all; s = tf('s')



'Generated transfer function:'

Ga=10*(s+2)/(s^2*(s+1)*(s+10)) figure(1)




2‐5

,Automatic Control Systems, 10th Edition Chapter 2 Solutions Golnaraghi, Kuo


Nyquist(Ga)



'Generated transfer function:'

Gb=10*s*(s+1)/((s+2)*(s^2+3*s+2)) figure(2)

Nyquist(Gb)



'Generated transfer function:' Gc=10*(s+2)/(s*(s^2+2*s+2))

figure(3)

Nyquist(Gc)




'Generated transfer function:' Gd=pade(exp(-

2*s),1)/(10*s*(s+1)*(s+2)) figure(4)

Nyquist(Gd)



Nyquist plots (polar plots):
Part(a)




2‐6

,Automatic Control Systems, 10th Edition Chapter 2 Solutions Golnaraghi, Kuo

Nyquist Diagram
15



10



5


Imaginary Axis
0



-5



-10



-15
-300 -250 -200 -150 -100 -50 0
Real Axis




Part(b)
Nyquist Diagram
1.5



1



0.5
Imaginary Axis




0



-0.5



-1



-1.5
-1 -0.5 0 0.5 1 1.5 2 2.5
Real Axis




Part(c)




2‐7

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