1
GATE EE 2023
Numerical
+2
-0

Consider the state-space description of an LTI system with matrices

$$A = \left[ {\matrix{ 0 & 1 \cr { - 1} & { - 2} \cr } } \right],B = \left[ {\matrix{ 0 \cr 1 \cr } } \right],C = \left[ {\matrix{ 3 & { - 2} \cr } } \right],D = 1$$

For the input, $$\sin (\omega t),\omega > 0$$, the value of $$\omega$$ for which the steady-state output of the system will be zero, is ___________ (Round off to the nearest integer).

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2
GATE EE 2017 Set 2
Numerical
+2
-0
Consider the system described by the following state space representation
$$\eqalign{ & \left[ {\matrix{ {\mathop {{x_1}}\limits^ \bullet \left( t \right)} \cr {\mathop {{x_2}}\limits^ \bullet \left( t \right)} \cr } } \right] = \left[ {\matrix{ 0 & 1 \cr 0 & { - 2} \cr } } \right]\left[ {\matrix{ {{x_1}\left( t \right)} \cr {{x_2}\left( t \right)} \cr } } \right] + \left[ {\matrix{ 0 \cr 1 \cr } } \right]u\left( t \right) \cr & y\left( t \right) = \left[ {\matrix{ 1 & 0 \cr } } \right]\left[ {\matrix{ {{x_1}\left( t \right)} \cr {{x_2}\left( t \right)} \cr } } \right] \cr} $$

If $$u(t)$$ is a unit step input and $$\left[ {\matrix{ {{x_1}\left( 0 \right)} \cr {{x_2}\left( 0 \right)} \cr } } \right] = \left[ {\matrix{ 1 \cr 0 \cr } } \right],$$ the value of output $$y(t)$$ at $$t=1$$ sec (rounded off to three decimal places) is _____________.

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3
GATE EE 2017 Set 1
MCQ (Single Correct Answer)
+2
-0.6
The transfer function of the system $$Y\left( s \right)/U\left( s \right)$$ , whose state-space equations are given below is:
$$\eqalign{ & \left[ {\matrix{ {\mathop {{x_1}}\limits^ \bullet \left( t \right)} \cr {\mathop {{x_2}}\limits^ \bullet \left( t \right)} \cr } } \right] = \left[ {\matrix{ 1 & 2 \cr 2 & 0 \cr } } \right]\left[ {\matrix{ {{x_1}\left( t \right)} \cr {{x_2}\left( t \right)} \cr } } \right] + \left[ {\matrix{ 1 \cr 2 \cr } } \right]u\left( t \right) \cr & y\left( t \right) = \left[ {\matrix{ 1 & 0 \cr } } \right]\left[ {\matrix{ {{x_1}\left( t \right)} \cr {{x_2}\left( t \right)} \cr } } \right] \cr} $$
A
$${{\left( {s + 2} \right)} \over {\left( {{s^2} - 2s - 2} \right)}}$$
B
$${{\left( {s + 2} \right)} \over {\left( {{s^2} + s - 4} \right)}}$$
C
$${{\left( {s - 4} \right)} \over {\left( {{s^2} + s - 4} \right)}}$$
D
$${{\left( {s + 4} \right)} \over {\left( {{s^2} - s - 4} \right)}}$$
4
GATE EE 2016 Set 1
Numerical
+2
-0
Consider the following state - space representation of a linear time-invariant system.
$$\mathop x\limits^ \bullet \left( t \right) = \left[ {\matrix{ 1 & 0 \cr 0 & 2 \cr } } \right]\,\,x\left( t \right),\,\,y\left( t \right) = {c^T}x\left( t \right),\,c = \left[ {\matrix{ 1 \cr 1 \cr } } \right]$$ and
$$x\left( 0 \right) = \left[ {\matrix{ 1 \cr 1 \cr } } \right]$$

The value of $$y(t)$$ for $$t\,\,\, = \,\,{\log _e}2$$ ___________.

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