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Problem 3: Using Routh-Hurwitz Criterion,

a. Determine the number of roots in the right half-plane, in the left half-plane, and on the jw - axis,
b. Determine the stability of the system of the following characteristic equations.
I Q(s) = + 65+ 4 s2 + 12 s +1
II. Q(s) = s5 +5 s - 10 s2 - 4s + 10
III. Q(s) = 5 + 25 + 13 st +16 +56 s2 + 32 s + 80

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To determine the stability of the systems using the Routh-Hurwitz criterion, we need to analyze the characteristic equations provided. Let's go through each equation and find the number of roots in the right half-plane (RHP), left half-plane (LHP), and on the jω-axis.

a) Characteristic Equation I:

Q(s) = 65s^3 + 4s^2 + 12s + 1

To apply the Routh-Hurwitz criterion, we need to construct the Routh array:

s^3 | 65 12

s^2 | 4 1

s^1 | -∞

s^0 | 1

The Routh array indicates that we have 1 sign change in the first column, which means we have 1 root in the RHP.

b) Characteristic Equation II:

Q(s) = s^5 + 5s - 10s^2 - 4s + 10

Constructing the Routh array:

s^5 | 1 -10

s^4 | -4 10

s^3 | 5

s^2 | 10

s^1 | -∞

s^0 | 10

The Routh array shows that we have 2 sign changes in the first column, indicating 2 roots in the LHP.

c) Characteristic Equation III:

Q(s) = 5s^4 + 25s^3 + 13st + 16 + 56s^2 + 32s + 80

Constructing the Routh array:

s^4 | 5 13 56

s^3 | 25 16 0

s^2 | 71 80

s^1 | -∞

s^0 | 80

The Routh array reveals that we have 1 sign change in the first column, indicating 1 root in the RHP.

Summary:

a) Characteristic Equation I: 1 root in the RHP, 0 roots in the LHP, 0 roots on the jω-axis.

b) Characteristic Equation II: 0 roots in the RHP, 2 roots in the LHP, 0 roots on the jω-axis.

c) Characteristic Equation III: 1 root in the RHP, 0 roots in the LHP, 0 roots on the jω-axis.

Based on the Routh-Hurwitz criterion, a system is considered stable if all roots have negative real parts (i.e., no roots in the RHP). Therefore:

a) System I is unstable due to 1 root in the RHP.

b) System II is stable as all roots are in the LHP.

c) System III is unstable due to 1 root in the RHP.

User Jeff Leonard
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