1
GATE EE 2014 Set 3
+2
-0.6
Consider the system described by the following state space equations \eqalign{ & \left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr } } \right] = \left[ {\matrix{ 0 & 1 \cr { - 1} & { - 1} \cr } } \right]\left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr } } \right] + \left[ {\matrix{ 0 \cr 1 \cr } } \right]u; \cr & y = \left[ {\matrix{ 1 & 0 \cr } } \right]\left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr } } \right] \cr}\$

If $$u$$ unit step input, then the steady state error of the system is

A
$$0$$
B
$$1/2$$
C
$$2/3$$
D
$$1$$
2
GATE EE 2014 Set 2
+2
-0.6
The second order dynamic system $${{dX} \over {dt}} = PX + Qu,\,\,\,y = RX$$ has the matrices $$P,Q,$$ and $$R$$ as follows: $$P = \left[ {\matrix{ { - 1} & 1 \cr 0 & { - 3} \cr } } \right]\,\,Q = \left[ {\matrix{ 0 \cr 1 \cr } } \right]$$
$$R = \left[ {\matrix{ 0 & 1 \cr } } \right]$$ The system has the following controllability and observability properties:
A
Controllable and observable
B
Not controllable but observable
C
Controllable but not observable
D
Not controllable and not observable
3
GATE EE 2013
+2
-0.6
The state variable formulation of a system is given as
$$\left[ {\matrix{ {\mathop {{x_1}}\limits^ \bullet } \cr {\mathop {{x_2}}\limits^ \bullet } \cr } } \right] = \left[ {\matrix{ { - 2} & 0 \cr 0 & { - 1} \cr } } \right]\left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr } } \right] + \left[ {\matrix{ 1 \cr 1 \cr } } \right]u,\,\,{x_1}\left( 0 \right) = 0,$$
$${x_2}\left( 0 \right) = 0$$ and $$y = \left[ {\matrix{ 1 & 0 \cr } } \right]\left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr } } \right]$$

The system is

A
controllable but not observable
B
not controllable but obserable
C
both controllable and observable
D
both not controllable and not Observable
4
GATE EE 2013
+2
-0.6
The state variable formulation of a system is given as
$$\left[ {\matrix{ {\mathop {{x_1}}\limits^ \bullet } \cr {\mathop {{x_2}}\limits^ \bullet } \cr } } \right] = \left[ {\matrix{ { - 2} & 0 \cr 0 & { - 1} \cr } } \right]\left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr } } \right] + \left[ {\matrix{ 1 \cr 1 \cr } } \right]u,\,\,{x_1}\left( 0 \right) = 0,$$
$${x_2}\left( 0 \right) = 0$$ and $$y = \left[ {\matrix{ 1 & 0 \cr } } \right]\left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr } } \right]$$

The response $$y(t)$$ to a unit step input is

A
$${1 \over 2} - {1 \over 2}{e^{ - 2t}}$$
B
$$1 - {1 \over 2}{e^{ - 2t}} - {1 \over 2}{e^{ - t}}$$
C
$${e^{ - 2t}} - {e^{ - t}}$$
D
$$1 - {e^{ - t}}$$
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