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.