1
GATE ECE 2014 Set 4
+2
-0.6
The state transition matrix $$\phi \left( t \right)$$ of a system $$\left[ {\matrix{ {\mathop {{x_1}}\limits^ \bullet } \cr {\mathop {{x_2}}\limits^ \bullet } \cr } } \right] = \left[ {\matrix{ 0 & 1 \cr 0 & 0 \cr } } \right]\left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr } } \right] is$$$A $$\left[ {\matrix{ t & 1 \cr 1 & 0 \cr } } \right]$$ B $$\left[ {\matrix{ 1 & 0 \cr t & 1 \cr } } \right]$$ C $$\left[ {\matrix{ 0 & 1 \cr 1 & t \cr } } \right]$$ D $$\left[ {\matrix{ 1 & t \cr 0 & 1 \cr } } \right]$$ 2 GATE ECE 2014 Set 3 MCQ (Single Correct Answer) +2 -0.6 The state equation of a second-order linear system is given by $$\mathop x\limits^ \bullet \left( t \right) = Ax\left( t \right),x\left( 0 \right) = {x_0}.$$ For $${x_0} = \left[ {\matrix{ 1 \cr { - 1} \cr } } \right],x\left( t \right) = \left[ {\matrix{ {{e^{ - t}}} \cr { - {e^{ - t}}} \cr } } \right]$$ and for $${x_0} = \left[ {\matrix{ 0 \cr 1 \cr } } \right],x\left( t \right) = \left[ {\matrix{ {{e^{ - t}}} & { - {e^{ - 2t}}} \cr { - {e^{ - t}}} & { + 2{e^{ - 2t}}} \cr } } \right]$$ when $${x_0} = \left[ {\matrix{ 3 \cr 5 \cr } } \right],x\left( t \right)$$ is A $$\left[ {\matrix{ { - 8{e^{ - t}}} & { + 11{e^{ - 2t}}} \cr {8{e^{ - t}}} & { - 22{e^{ - 2t}}} \cr } } \right]$$ B $$\left[ {\matrix{ {11{e^{ - t}}} & { - 8{e^{ - 2t}}} \cr { - 11{e^{ - t}}} & { + 16{e^{ - 2t}}} \cr } } \right]$$ C $$\left[ {\matrix{ {3{e^{ - t}}} & { - 5{e^{ - 2t}}} \cr { - 3{e^{ - t}}} & { + 10{e^{ - 2t}}} \cr } } \right]$$ D $$\left[ {\matrix{ {5{e^{ - t}}} & { - 3{e^{ - 2t}}} \cr { - 5{e^{ - t}}} & { + 6{e^{ - 2t}}} \cr } } \right]$$ 3 GATE ECE 2014 Set 2 MCQ (Single Correct Answer) +2 -0.6 An unforced linear time invariant (LTI) system is represented by $$\left[ {\matrix{ {\mathop {{x_1}}\limits^ \bullet } \cr {\mathop {{x_2}}\limits^ \bullet } \cr } } \right] = \left[ {\matrix{ { - 1} & 0 \cr 0 & { - 2} \cr } } \right]\left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr } } \right].$$$

If the initial conditions are x1(0)= 1 and x2(0)=-1, the solution of the state equation is

A
$${x_1}\left( t \right) = - 1,{x_2}\left( t \right) = 2$$
B
$${x_1}\left( t \right) = - {e^{ - t}},{x_2}\left( t \right) = 2{e^{ - t}}$$
C
$${x_1}\left( t \right) = {e^{ - t}},{x_2}\left( t \right) = - {e^{ - 2t}}$$
D
$${x_1}\left( t \right) = {e^{ - t}},{x_2}\left( t \right) = - 2{e^{ - t}}$$
4
GATE ECE 2014 Set 2
+2
-0.6
Consider the state space system expressed by the signal flow diagram shown in the figure.

The corresponding system is

A
always controllable
B
always observable
C
always stable
D
always unstable
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