A closed-loop system is shown in the figure. The system parameter α is not known. The condition for asymptotic stability of the closed loop system is
- Aα < -0.5
- B-0.5 < α < 0.5
- C0 < α < 0.5
- Dα > 0.5
Solution & Step-by-step Explanation
We know that characteristic equation
1 + G(s) H(s) = 0
s³ + 2αs² + αs + 1 + s + α = 0
s³ + 2αs² + (1 + α)s + (1 + α) = 0
Now apply R.H. criteria
*20c s^3&1&1 + α \\ s^2&2α &1 + α \\ s^1&{{2α ≤ft( 1 + α ) - ≤ft( 1 + α )}2α }&0\\ s^0&α + 1&0
For sufficient condition
2α > 1
∴ α > 0.5
1 + G(s) H(s) = 0
s³ + 2αs² + αs + 1 + s + α = 0
s³ + 2αs² + (1 + α)s + (1 + α) = 0
Now apply R.H. criteria
*20c s^3&1&1 + α \\ s^2&2α &1 + α \\ s^1&{{2α ≤ft( 1 + α ) - ≤ft( 1 + α )}2α }&0\\ s^0&α + 1&0
For sufficient condition
2α > 1
∴ α > 0.5