GATE 2017 EE (EE2) – Question 16

Control Systems · Transient and Steady-state analysis of linear time invariant systems · 1 mark · Multiple choice

A unity-feedback system has closed-loop transfer function $C(s)/R(s)=(Ks+b)/(s^2+as+b)$ with a,b,K>0. Its steady-state error to a unit ramp is

$0$
$a/b$
$(a+K)/b$
$(a-K)/b$

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Show answer and explanation

Correct answer: (D) $(a-K)/b$

Explanation

Use the supplied closed-loop transfer T, rather than treating it as the open-loop gain. The error transfer is
$$1-T(s)=\frac{s^2+(a-K)s}{s^2+as+b}.$$
For a unit ramp R=1/s², the final value theorem gives $e_{ss}=\lim_{s\to0}s(1-T)/s^2=(a-K)/b$.

The denominator is stable for positive a,b. Hence **D**; positive parameters do not by themselves require a positive tracking error.

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