GATE 2014 EC (EC4) – Question 54
A causal LTI system with zero initial conditions and impulse response h(t) obeys $y^{\prime\prime}(t)+\alpha y^\prime(t)+\alpha^2y(t)=x(t)$. Define $g(t)=\alpha^2\int_0^th(\tau)d\tau+h^\prime(t)+\alpha h(t)$. If G(s) is its Laplace transform, find the number of poles of G(s).
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Correct answer: 0.99 to 1.01
Explanation
The differential equation gives H(s)=1/(s²+αs+α²). With zero initial conditions and h(0+)=0, G(s)=[α²/s+s+α]H(s)=(s²+αs+α²)H(s)/s=1/s. Thus **G has1pole**, at the origin. Cancelled factors do not remain poles of the simplified transfer function.