Análise de Margem e Estabilidade

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Problem 10.04PP
Consider the closed-loop system shown in Fig.
(a) What is the phase margin if /C = 70,000?
(b) What is the gain margin if /C = 70, 000?
(c) What value of K will yield a phase margin of ~ 70*?
(d) What value of K will yield a phase margin of ~ 0*?
(e) Sketch the root locus with respect to K jo r the system, and determine what value of K causes
(e) Sketch the root locus with respect to K for the system, and determine what value of K causes 
the system to be on the verge of instability.
(f) If the disturbance w is a constant and K = ^Q, 000, what is the maximum allowable value for w 
if yC“ ) is to remain less than 0.1? (Assume r= 0.)
(g) Suppose the specifications require you to allow larger values of w than the value you 
obtained in part (f) but with the same error constraint [|y(°°)| 0.1]. Discuss what steps you could 
take to alleviate the problem.
Figure Control system
W(s)
m • ns)
Step-by-step solution
step 1 of 18
(a)
Step 2 of 18
Refer Figure 10.88 in the textbook.
Step 3 of 18
Derive the transfer function from figure 10.88.
g ( j + i )
G (* )J > W = - ( 1)
 )
Compare Equations (2), and (3).
Number of zeros present in the system is 1 
One pole at break frequency,
Second pole at break frequency, for s in Equation (2).
(3)
G ( » = r (4)
' ' [ y)\
- 2 0 I o g M
= 42.92db
Calculate gain at s .
= | G ( » |
= 2 0 lo g ( l+ l)
= 6.020db
Calculate gain (A) at c t fs o .
- |s lo p e fix )m a )^ ,to « }^ x lo g ^ + A ^ ^
» 0 + 6.020db 
s6.020db
Step 6 of 18
i)at .
to«>.,xlog— 1+A ^
+6.020db
1
1>
at a>=
®caj
+ A .
a -
7.959db
n f l» r t to )
deg
tan"'
(o.oiygA:»
-201ogA: = -7
_7_
'20
a: -2 .238
Therefore, the value of K that yields 7q* phase margin is |Af = 2.23^-
Step 12 of 18
(e)
Write the matlab program and draw the root locus of the transfer function. 
rlocus([1 1],[1,115,1550,5000,0])
Step 13 of 18
Figure 2
Step 14 of 18
Consider the formula and calculate the range of K.
\ + K J > {s )G {s ) = 0 ...... (5)
Substitute equation (2) in equation (5).
i + - r - o
s{s+S){s + 10)(i +100)
Simplify the equation.
5^1154^+I550S*+(5000+a:)4 + a: - o
step 15 of 18
Apply Routh-Array criteria to the equation.
5 3
5 2
5 1
5 0
1 1,550
115 5,000+A"
173,250-A"
115
- a 2 - 155,025 K +866,250,000
115
Figure 2
Step 16 of 18
From figure 2, the system moves to stable condition at Al > 0 ^nd 
-155,02SA:+866,250,000,
- > 0
173,250-A:
Solve -AT* -155,025AT+866,250,000 >0and find the value of K. 
a: >160.424.72
Therefore, the range of K for the system to be stable is |0'(4) G (4
(6)
B '( i ) l+ A J 3 ( s ) .G ( i )
Take the values from figure 10.88 in the textbook. 
1
 occurs when lOOc
step 18 of 18
(g)
Consider the equation y 100c
Assign AT = 10.000and^