1
GATE ECE 2006
+2
-0.6
A linear system is described by the following state equation $$\mathop x\limits^ \bullet \left( t \right) = AX\left( t \right) + BU\left( t \right),A = \left[ {\matrix{ 0 & 1 \cr { - 1} & 0 \cr } } \right].$$$The state-transition matrix of the system is A $$\left[ {\matrix{ {\cos t} & {\sin t} \cr { - \sin t} & {\cos t} \cr } } \right]$$ B $$\left[ {\matrix{ { - \cos t} & {\sin t} \cr { - \sin t} & { - \cos t} \cr } } \right]$$ C $$\left[ {\matrix{ { - \cos t} & { - \sin t} \cr { - \sin t} & {\cos t} \cr } } \right]$$ D $$\left[ {\matrix{ {\cos t} & { - \sin t} \cr {\cos t} & {\sin t} \cr } } \right]$$ 2 GATE ECE 2004 MCQ (Single Correct Answer) +2 -0.6 The state variable equations of a system are: $${\mathop {{x_1} = - 3{x_1} - x}\limits^ \bullet _2} + u$$$ $${\mathop x\limits^ \bullet _2} = 2{x_1}$$$$$y = {x_1} + u.$$$
The system is
A
controllable but not observable.
B
observable but not controllable.
C
neither controllable nor observable.
D
controllable and observable.
3
GATE ECE 2004
+2
-0.6
If A = $$\left[ {\matrix{ { - 2} & 2 \cr 1 & { - 3} \cr } } \right],$$ then sin At is
A
$${1 \over 3}\left[ {\matrix{ {\sin \left( { - 4t} \right) + 2\sin \left( { - t} \right)} & { - 2\sin \left( { - 4t} \right) + 2\sin \left( { - t} \right)} \cr { - \sin \left( { - 4t} \right) + \sin \left( { - t} \right)} & {2\sin \left( { - 4t} \right) + \sin \left( { - t} \right)} \cr } } \right]$$
B
$$\left[ {\matrix{ {\sin \left( { - 2t} \right)} & {\sin \left( {2t} \right)} \cr {\sin \left( t \right)} & {\sin \left( { - 3t} \right)} \cr } } \right]$$
C
$${1 \over 3}\left[ {\matrix{ {\sin \left( {4t} \right) + 2\sin \left( t \right)} & {2\sin \left( { - 4t} \right) - 2\sin \left( { - t} \right)} \cr { - \sin \left( { - 4t} \right) + \sin \left( t \right)} & {2\sin \left( {4t} \right) + \sin \left( t \right)} \cr } } \right]$$
D
$${1 \over 3}\left[ {\matrix{ {\cos \left( { - t} \right) + 2\cos \left( t \right)} & {2\cos \left( { - 4t} \right) + 2\sin \left( { - t} \right)} \cr { - \cos \left( { - 4t} \right) + \sin \left( { - t} \right)} & { - 2\cos \left( { - 4t} \right) + \cos \left( { - t} \right)} \cr } } \right]$$
4
GATE ECE 2004
+2
-0.6
Given A $$= \left[ {\matrix{ 1 & 0 \cr 0 & 1 \cr } } \right],$$ the state transition matrix eAt is given by
A
$$\left[ {\matrix{ 0 & {{e^{ - t}}} \cr {{0^{ - t}}} & 0 \cr } } \right]$$
B
$$\left[ {\matrix{ {{e^t}} & 0 \cr 0 & {{e^t}} \cr } } \right]$$
C
$$\left[ {\matrix{ {{e^{ - t}}} & 0 \cr 0 & {{e^{ - t}}} \cr } } \right]$$
D
$$\left[ {\matrix{ 0 & {{e^t}} \cr {{e^t}} & 0 \cr } } \right]$$
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