What is the stability condition for a system with the characteristic equation s^

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Question: What is the stability condition for a system with the characteristic equation s^2 + 3s + 2 = 0?

Options:

Correct Answer: Stable

Solution:

The roots of the characteristic equation are s = -1 and s = -2, both of which are in the left half of the s-plane, indicating stability.

What is the stability condition for a system with the characteristic equation s^2 + 3s + 2 = 0?

What is the stability condition for a system with the characteristic equation s^2 + 3s + 2 = 0?

Step 1: Identify the characteristic equation, which is given as s^2 + 3s + 2 = 0.
Step 2: Use the quadratic formula to find the roots of the equation. The quadratic formula is s = (-b ± √(b² - 4ac)) / (2a). Here, a = 1, b = 3, and c = 2.
Step 3: Calculate the discriminant (b² - 4ac). For our equation, it is 3² - 4(1)(2) = 9 - 8 = 1.
Step 4: Substitute the values into the quadratic formula. This gives us s = (-3 ± √1) / (2*1).
Step 5: Calculate the two roots. The first root is s = (-3 + 1) / 2 = -1, and the second root is s = (-3 - 1) / 2 = -2.
Step 6: Analyze the roots. Both roots, s = -1 and s = -2, are negative and located in the left half of the s-plane.
Step 7: Conclude that since both roots are in the left half of the s-plane, the system is stable.

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