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A certain system has transfer function G(s) = (s + 8)/(s² + αs - 4), where α is a parameter. Consider the standard negative unity feedback configuration as shown below. [block diagram of standard negative unity feedback with G(s)] Which of the following statements is true?

1. The closed loop system is never stable for any value of α.
2. For some positive values of α, the closed loop system is stable, but not for all positive values.
3. For all positive values of α, the closed loop system is stable.
4. The closed loop system is stable for all values of α, both positive and negative.

Correct answer: For all positive values of α, the closed loop system is stable.

Solution

Unity feedback gives characteristic equation s^2+as-4 + (s+8) = s^2+(a+1)s+4 = 0. For a second-order polynomial all coefficients positive guarantees stability: a+1>0 and 4>0, so the system is stable for every positive a. The correct option is 'stable for all positive values of a'.

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