Question

For the system of Figure P1: 1. a) Find the range of K for stability. b) Find the value of K for marginal stability. c) Find the actual location of the closed-loop poles when the system is marginally stable. Cis) s(s+i)(s+2)6+5)

Answer 1

Characteristic equation = 1 + G (s) h (s) = 0 1 + k/s (s + 1) (s + 2) (s + 5) times 1 = 0 s (s + 1) (s + 2) (s + 5) + k = 0 (s^2 + s) (s + 2) (s + 5) + k = 0 (s^2 + s) (s^2 + 7s + 10) + k = 0 s^4 + 7s^3 + 10s^2 + s^3 + 7s^2 + 10s + k = 0 s^4 + 8s^3 + 17s^2 + 10s + k = 0 a) By RH Criteria s^4 1 17 k s^3 8 10 s^2 17 times 5 - 10/8 15.75 k s^1 (15.75 times 10) - 8k/15.75 s^0 k For stability k greaterthan 0, and (15.75 times 10) - 8k/15.75 greaterthan 0157.5 - 8k greaterthan 0 8k < 157.5 k < 157.5/ k < 19.6 range of k for stability 0 < k < 19.6 \r\n b) value of k for marginal stability is

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