homework 5 answer automatic control system
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Automatic control SystemTRANSCRIPT
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Homework 5 Answer
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Homework 5 Answer
Problem 1
Using the Routh criterion, determine the stability of the closed-loop system that
has the following characteristic equations. Determine the number of roots of
each equation that are in the right-half s-plane or on the j-axis.
1. 3 220 9 100 0s S s
2. 4 3 23 10 5 2 0s s S s
Answer
1.
3
2
1
0
1 9
20 100
4 0
100
s
s
s
s
The first column of the Routh's tabulation are of the same sign, So the roots of
the equation are all in the left half of the s-plane.
The system is stable.
2. 4
3
2
1
0
3 5 2
10 1 0
4.7 2
3.2553 0
2
s
s
s
s
s
Because there are two sign changes in the first column of the tabulation, the
equation has two roots in the right half of the s-plane.
The system is not stable.
Problem 2
The forward transfer functions of the unity feedback system are as follows:
1. ( )( 1)(0.2 1)
KG s
s s s
2. ( 1)
( )( 1)(0.2 1)
K sG s
s s s
Determine the critical value of K for stability
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Homework 5 Answer
2
Answer
The Routh criterion is best suited to determine the critical value of K for
stability.
1. The characteristic equation
3 20.2 0.8 0S S s K
3
2
1
0
0.2 1
0.8
14
s
s K
Ks
s K
In order to the first column of the Routh's tabulation are of the same sign.
1 04
0
K
K
so there are no solution.
2. The characteristic equation
3 20.2 0.8 ( 1) 0S S K s K
3
2
1
0
0.2 1
0.8
31
4
s K
s K
s K
s K
31 0
4
0
K
K
so 4
3K The system is stable.