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1

GATE EE 2012

MCQ (Single Correct Answer)
The state variable description of an $$LTI$$ system is given by $$$\left( {\matrix{ {\mathop {{x_1}}\limits^ \bullet } \cr {\mathop {{x_2}}\limits^ \bullet } \cr {\mathop {{x_3}}\limits^ \bullet } \cr } } \right) = \left( {\matrix{ 0 & {{a_1}} & 0 \cr 0 & 0 & {{a_2}} \cr {{a_3}} & 0 & 0 \cr } } \right)\left( {\matrix{ {{x_1}} \cr {{x_2}} \cr {{x_3}} \cr } } \right) + \left( {\matrix{ 0 \cr 0 \cr 1 \cr } } \right)u,$$$ $$$y = \left( {\matrix{ 1 & 0 & 0 \cr } } \right)\left( {\matrix{ {{x_1}} \cr {{x_2}} \cr {{x_3}} \cr } } \right)$$$

where $$y$$ is the output and $$u$$ is the input. The system is controllable for

A
$${a_1} \ne 0,\,\,{a_2} = 0,\,\,{a_3} \ne 0$$
B
$${a_1} = 0,\,\,{a_2} \ne 0,\,\,{a_3} \ne 0$$
C
$${a_1} = 0,\,\,{a_2} \ne 0,\,\,{a_3} = 0$$
D
$${a_1} \ne 0,\,\,{a_2} \ne 0,\,\,{a_3} = 0$$
2

GATE EE 2010

MCQ (Single Correct Answer)
The system $$\mathop X\limits^ \bullet = AX + BU$$ with $$A = \left[ {\matrix{ { - 1} & 2 \cr 0 & 2 \cr } } \right],$$ $$B = \left[ {\matrix{ 0 \cr 1 \cr } } \right]$$ is
A
stable and controllable
B
stable but uncontrollable
C
unstable but controllable
D
unstable and uncontrollable
3

GATE EE 2009

MCQ (Single Correct Answer)
A system is described by the following state and output equations $$${{d{x_1}\left( t \right)} \over {dt}} = - 3{x_1}\left( t \right) + {x_2}\left( t \right) + 2u\left( t \right)$$$ $$${{d{x_2}\left( t \right)} \over {dt}} = - 2{x_2}\left( t \right) + u\left( t \right)$$$

$$y\left( t \right) = {x_1}\left( t \right)$$ when $$u(t)$$ is the input and $$y(t)$$ is the output

The state $$-$$ transition matrix of the above system is

A
$$\left( {\matrix{ {{e^{ - 3t}}} & 0 \cr {{e^{ - 2t}} + {e^{ - 3t}}} & {{e^{ - 2t}}} \cr } } \right)$$
B
$$\left( {\matrix{ {{e^{ - 3t}}} & {{e^{ - 2t}} - {e^{ - 3t}}} \cr 0 & {{e^{ - 2t}}} \cr } } \right)$$
C
$$\left( {\matrix{ {{e^{ - 3t}}} & {{e^{ - 2t}} + {e^{ - 3t}}} \cr 0 & {{e^{ - 2t}}} \cr } } \right)$$
D
$$\left( {\matrix{ {{e^{3t}}} & {{e^{ - 2t}} - {e^{ - 3t}}} \cr 0 & {{e^{ - 2t}}} \cr } } \right)$$
4

GATE EE 2009

MCQ (Single Correct Answer)
A system is described by the following state and output equations $$${{d{x_1}\left( t \right)} \over {dt}} = - 3{x_1}\left( t \right) + {x_2}\left( t \right) + 2u\left( t \right)$$$ $$${{d{x_2}\left( t \right)} \over {dt}} = - 2{x_2}\left( t \right) + u\left( t \right)$$$

$$y\left( t \right) = {x_1}\left( t \right)$$ when $$u(t)$$ is the input and $$y(t)$$ is the output

The system transfer function is

A
$${{s + 2} \over {{s^2} + 5s - 6}}$$
B
$${{s + 3} \over {{s^2} + 5s + 6}}$$
C
$${{2s + 5} \over {{s^2} + 5s + 6}}$$
D
$${{2s - 5} \over {{s^2} + 5s + 6}}$$

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