Consider a unity negative feedback control system with forward path gain $G(s) = \frac{K}{(s + 1)(s + 2)(s + 3)}$ as shown.

The impulse response of the closed-loop system decays faster than $e^{-t}$ if ________.

A closed loop system is shown in the figure where $$k>0$$ and $$\alpha > 0$$. The steady state error due to a ramp input $$(R(s) = \alpha /{s^2})$$ is given by

Two linear time-invariant systems with transfer functions

$${G_1}(s) = {{10} \over {{s^2} + s + 1}}$$ and $${G_2}(s) = {{10} \over {{s^2} + s\sqrt {10} + 10}}$$

have unit step responses y₁(t) and y₂(t), respectively. Which of the following statements is/are true?

The block diagram of a closed-loop control system is shown in the figure. R(s), Y(s), and D(s) are the Laplace transforms of the time-domain signals r(t), y(t), and d(t), respectively. Let the error signal be defined as e(t) = r(t) $$-$$ y(t). Assuming the reference input r(t) = 0 for all t, the steady-state error e($$\infty$$), due to a unit step disturbance d(t), is __________ (rounded off to two decimal places).