What is the stability condition for a system with the transfer function G(s) = 1

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Question: What is the stability condition for a system with the transfer function G(s) = 1/(s^2 + 4s + 5)?

Options:

Correct Answer: All poles in the left half-plane

Solution:

The system is stable if all poles of the transfer function are in the left half-plane.

What is the stability condition for a system with the transfer function G(s) = 1/(s^2 + 4s + 5)?

What is the stability condition for a system with the transfer function G(s) = 1/(s^2 + 4s + 5)?

Step 1: Identify the transfer function G(s) = 1/(s^2 + 4s + 5).
Step 2: Find the denominator of the transfer function, which is s^2 + 4s + 5.
Step 3: To determine stability, we need to find the poles of the system. The poles are the values of s that make the denominator equal to zero.
Step 4: Set the denominator equal to zero: s^2 + 4s + 5 = 0.
Step 5: Use the quadratic formula to solve for s: s = (-b ± √(b^2 - 4ac)) / (2a), where a = 1, b = 4, and c = 5.
Step 6: Calculate the discriminant: b^2 - 4ac = 4^2 - 4*1*5 = 16 - 20 = -4.
Step 7: Since the discriminant is negative, the roots (poles) will be complex numbers.
Step 8: Calculate the poles using the quadratic formula: s = (-4 ± √(-4)) / 2 = (-4 ± 2i) / 2 = -2 ± i.
Step 9: The poles are -2 + i and -2 - i. Both poles have a real part of -2.
Step 10: Since both poles have negative real parts, they are in the left half-plane.
Step 11: Conclude that the system is stable because all poles are in the left half-plane.

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