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1

### GATE EE 2009

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
$${{2s - 5} \over {{s^2} + 5s + 6}}$$
2

The state space equation of a system is described by $$\mathop X\limits^ \bullet = AX + BU,\,\,Y = Cx$$ where $$X$$ is state vector, $$U$$ is input, $$Y$$ is output and $$A = \left( {\matrix{ 0 & 1 \cr 0 & { - 2} \cr } } \right)\,\,B = \left( {\matrix{ 0 \cr 1 \cr } } \right)\,\,C = \left[ {\matrix{ 1 & 0 \cr } } \right]$$$A unity feedback is provided to the above system $$G(s)$$ to make it a closed loop system as shown in figure. For a unit step input $$r(t),$$ the steady state error in the input will be A $$0$$ B $$1$$ C $$2$$ D $$\infty$$ 3 ### GATE EE 2008 MCQ (Single Correct Answer) The state space equation of a system is described by $$\mathop X\limits^ \bullet = AX + BU,\,\,Y = Cx$$ where $$X$$ is state vector, $$U$$ is input, $$Y$$ is output and $$A = \left( {\matrix{ 0 & 1 \cr 0 & { - 2} \cr } } \right)\,\,B = \left( {\matrix{ 0 \cr 1 \cr } } \right)\,\,C = \left[ {\matrix{ 1 & 0 \cr } } \right]$$$

The transfer function $$G(s)$$ of this system will be

A
$${s \over {\left( {s + 2} \right)}}$$
B
$${{s + 1} \over {s\left( {s - 2} \right)}}$$
C
$${s \over {\left( {s - 2} \right)}}$$
D
$${1 \over {s\left( {s + 2} \right)}}$$
4

### GATE EE 2005

A state variable system
$$\mathop X\limits^ \bullet \left( t \right) = \left( {\matrix{ 0 & 1 \cr 0 & { - 3} \cr } } \right)X\left( t \right) + \left( {\matrix{ 1 \cr 0 \cr } } \right)u\left( t \right)$$ with the initial condition $$X\left( 0 \right) = {\left[ { - 1\,\,3} \right]^T}$$ and the unit step input $$u(t)$$ has

The state transition equation

A
$$X\left( t \right) = \left( {\matrix{ {t - {e^{ - t}}} \cr {{e^{ - t}}} \cr } } \right)$$
B
$$X\left( t \right) = \left( {\matrix{ {t - {e^{ - t}}} \cr {3{e^{ - 3t}}} \cr } } \right)$$
C
$$X\left( t \right) = \left( {\matrix{ {t - {e^{ - 3t}}} \cr {3{e^{ - 3t}}} \cr } } \right)$$
D
$$X\left( t \right) = \left( {\matrix{ {t - {e^{ - 3t}}} \cr {{e^{ - t}}} \cr } } \right)$$
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