The open loop transfer function for a unit feedback system is G(s) = K/s(sT + 1). By what factor the amplitude gain K should be reduced so that the peak overshoot becomes 25% from 75%?
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Percent overshoot fixes the damping ratio through M_p=e^{−πζ/√(1−ζ²)}. For G(s)=K/[s(sT+1)], the closed-loop damping ratio varies inversely with √K. Converting 75% and 25% overshoot to ζ values and squaring their ratio gives the required gain reduction factor, matching 20.
Formula / Key Relation
Mp = exp[−πζ/√(1−ζ²)]
ζ = −ln(Mp)/√[π²+ln²(Mp)]
ζ75% ≈ 0.091
ζ25% ≈ 0.404
Knew/Kold=(ζold/ζnew)²
≈(0.091/0.404)²≈0.05
So K is reduced by ≈20 times.
Detailed Explanation
Concept & ReasoningThe closed-loop characteristic equation is Ts²+s+K=0. In standard form s²+2ζω_ns+ω_n²=0, giving ω_n²=K/T and 2ζω_n=1/T, so ζ=1/(2√(KT)). Therefore K∝1/ζ² for fixed T. Determine ζ from each overshoot and use K_new/K_old=(ζ_old/ζ_new)². The strong drop in overshoot requires a substantial reduction in K.How to Solve It in the ExamOvershoot → ζ first; then use ζ∝1/√K.Important Exam PointUse M_p as decimal (0.75, 0.25), not percentage numbers 75 and 25.Related Revision PathThis question sits in the revision path Control Systems → Transient Response → Second-Order Systems .
Concept & Reasoning
The closed-loop characteristic equation is Ts²+s+K=0. In standard form s²+2ζω_ns+ω_n²=0, giving ω_n²=K/T and 2ζω_n=1/T, so ζ=1/(2√(KT)). Therefore K∝1/ζ² for fixed T. Determine ζ from each overshoot and use K_new/K_old=(ζ_old/ζ_new)². The strong drop in overshoot requires a substantial reduction in K.
How to Solve It in the Exam
Overshoot → ζ first; then use ζ∝1/√K.
Important Exam Point
Use M_p as decimal (0.75, 0.25), not percentage numbers 75 and 25.
Related Revision Path
This question sits in the revision path Control Systems → Transient Response → Second-Order Systems .
Quick Trick
Overshoot → ζ first; then use ζ∝1/√K.
Common Mistake
Assuming overshoot is directly proportional to gain.
Exam Tip
Use M_p as decimal (0.75, 0.25), not percentage numbers 75 and 25.
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