Project ForgeA block diagram shows a proportional controller Kc in a cascaded feedback loop with process 1/(τps+1) and measurement lag 1/(τms+1), plus a disturbance input U=0. A unit step input is introduced in the set point. The value of Kc to provide a critically damped response for U=0, τp=8, and τm=1 is
- A3.34
- B2.58
- 1.53
- D1.12
Explanation
With U=0, C/R reduces to a standard closed loop with characteristic polynomial (τps+1)(τms+1)+Kc=0, i.e. τpτm·s²+(τp+τm)s+(1+Kc)=0, i.e. 8s²+9s+(1+Kc)=0. In standard form τ²s²+2ζτs+1 (dividing by 1+Kc): τ²=8/(1+Kc), 2ζτ=9/(1+Kc). Setting ζ=1 (critically damped) and eliminating τ: [4.5/(1+Kc)]²=8/(1+Kc) ⟹ 20.25/(1+Kc)=8 ⟹ 1+Kc=2.531 ⟹ Kc≈1.53.
Source: GATE CH 2012 (2012)
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