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1

### GATE EE 2010

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
2

### 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 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
$$\left( {\matrix{ {{e^{3t}}} & {{e^{ - 2t}} - {e^{ - 3t}}} \cr 0 & {{e^{ - 2t}}} \cr } } \right)$$
3

### 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
$${{2s - 5} \over {{s^2} + 5s + 6}}$$
4

### GATE EE 2008

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
$$\infty$$
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