GATE 2017 EC (EC2) – Question 57

Control Systems · Transient and Steady-State Analysis of LTI Systems · 2 marks · Multiple choice

A second-order LTI system satisfies $\dot x_1(t)-x_2(t)=0$ and $\dot x_2(t)+2x_1(t)+3x_2(t)=r(t)$. The variables x1 and x2 are states, r(t) is the input, and the output is c(t)=x1(t). The system is

undamped (oscillatory)
underdamped
critically damped
overdamped

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Show answer and explanation

Correct answer: (D) overdamped

Explanation

The first equation gives $x_2=\dot x_1$. Substitute it into the second to obtain $\ddot x_1+3\dot x_1+2x_1=r$.

The characteristic polynomial is $s^2+3s+2=(s+1)(s+2)$, with two distinct negative real roots. Equivalently, $\omega_n=\sqrt2$ and $\zeta=3/(2\sqrt2)>1$.

The response is **overdamped, D**, rather than oscillatory or critically damped.

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