Respuesta :
The corresponding gain is 7.81×10-³
With compensation G(s) = k / (s+3) (s+1)
a) closed loop characteristic equation is
1 + G(s) = 0
(s+3) (s+1) + k = 0
s² + 4s + k + 3 = 0 ------ (1)
standard form of characteristic equation is
s² + 2ξ ωn + ωn² = 0 ------ (2)
Given ξ = 1/√2
comparing equation (1) and (2)
ωn² = k+3 , ωn = √(k+3)
2 ξ ωn = 4
2 × 1/√2 × √ (k+3) = 4
√(k+3) = 2√2
k + 3 = 8, k = 5
b) closed loop tranfer function
T(s) = G(s) / 1+G(s)
= 5/ s²+4s+8
magnitude |G(s)| = 5/√ [ (64-ω²)² + (4ω)²]
For maximum gain, denominator --> maximum
d/dω [ (64-ω²)² + 16ω²] = 0
2(64-ω²)² (-2ω) + 32ω = 0
4ω ( -4096 - ω⁴+128ω²+8 ) = 0
ω⁴ + 128ω² - 4088 = 0
ω = 26.6 rad/s
Gn = 5/√ [ (64-ω²)² + (4ω)²]
Gain = 7.81×10-³
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