Control Engineering Basics
A unity-feedback system has the loop transfer function K/(s(s + 1)(s + 2)). Which of the following is the range of K for which the closed-loop system is stable?
- A0 < K < 2
- B0 < K < 3
- C0 < K < 6
Show answer and explanation
Answer: C. 0 < K < 6
The characteristic equation is s(s + 1)(s + 2) + K = s³ + 3s² + 2s + K = 0. By the Routh–Hurwitz stability criterion, a cubic s³ + a₂s² + a₁s + a₀ = 0 is stable when all coefficients are positive and a₂a₁ > a₀, so 3 × 2 > K, that is, 0 < K < 6. At K = 6 the poles move to s = ±j√2, giving a sustained oscillation at an angular frequency of about 1.41 rad/s (a period of about 4.4 s). For larger K, the oscillation keeps growing.
- A. This looks only at the coefficient 2 of s. The stability condition is 3 × 2 > K.
- B. This looks only at the coefficient 3 of s².
- C. Correct. This is the range in which the first column of the Routh array is all positive.