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1

### GATE EE 2017 Set 1

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
$${{\left( {s + 4} \right)} \over {\left( {{s^2} - s - 4} \right)}}$$
2

### GATE EE 2017 Set 2

Numerical
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 _____________.

Correct answer is between 1.280 and 1.287
3

### GATE EE 2016 Set 1

Numerical
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 $$t\,\,\, = \,\,{\log _e}2$$ ___________.

Correct answer is between 5.9 and 6.1
4

### GATE EE 2015 Set 2

For the system governed by the set of equations: \eqalign{ & d{x_1}/dt = 2{x_1} + {x_2} + u \cr & d{x_2}/dt = - 2{x_1} + u \cr & \,\,\,\,\,\,y = 3{x_1} \cr}\$
the transfer function $$Y(s)/U(s)$$ is given by
A
$$3\left( {s + 1} \right)/\left( {{s^2} - 2s + 2} \right)$$
B
$$3\left( {2s + 1} \right)/\left( {{s^2} - 2s + 1} \right)$$
C
$$\left( {s + 1} \right)/\left( {{s^2} - 2s + 1} \right)$$
D
$$3\left( {2s + 1} \right)/\left( {{s^2} - 2s + 2} \right)$$
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