PYQs Based test 15: Control Systems

Consider a system governed by the following equations

$\frac{d{x}_1\left(t\right)}{dt}=x_2\left(t\right)-x_1\left(t\right)$

$\frac{d{x}_2\left(t\right)}{dt}=x_1\left(t\right)-x_2\left(t\right)$

The initial conditions are such that $x_1\left(0\right)<x_2\left(0\right)<\infty .$ Let $x_{1f}=\underset{t\to \infty }{\lim }{x}_1\left(t\right)$ and $x_{2f}=\underset{t\to \infty }{\lim }{x}_2\left(t\right)$. Which one of the following is true?

The number of roots of the polynomial, s⁷ + s⁶ + 7s⁵ + 14s⁴ + 31s³ + 73s² + 25s + 200, in the open left half of the complex plane is

The unit step response of a system with the transfer function $G\left(s\right)=\frac{1-2s}{1+s}$ is given by which one of the following waveforms?

In the signal flow diagram given in the figure, $u_{1}and{u}_2$ are possible inputs whereas $y_{1}and{y}_2$ are possible output. When would the SISO system derived from this diagram be controllable and observable?

For the closed-loop system shown, the transfer function $\frac{E(s)}{R(s)}$ is