1
GATE ECE 2007
MCQ (Single Correct Answer)
+2
-0.6
Consider a linear system whose state space Representation is $$\mathop x\limits^ \bullet \left( t \right) = AX\left( t \right).$$
If the initial state vector of the system is $$x\left( 0 \right) = \left[ {\matrix{ 1 \cr { - 2} \cr } } \right],$$
then the system response is $$x\left( t \right) = \left[ {\matrix{ {{e^{ - 2t}}} \cr { - 2{e^{ - 2t}}} \cr } } \right].$$
If the initial state vector of the system changes to $$x\left( 0 \right) = \left[ {\matrix{ 1 \cr { - 1} \cr } } \right],$$
then the system response becomes $$x\left( t \right) = \left[ {\matrix{ {{e^{ - t}}} \cr { - {e^{ - t}}} \cr } } \right].$$

The eigen value and eigen vector pairs $$\left( {{\lambda _{i,}}{V_i}} \right)$$ for the system are

A
$$\left[ { - 1,\left[ {\matrix{ 1 \cr { - 1} \cr } } \right]} \right]and\left[ { - 2,\left[ {\matrix{ 1 \cr { - 2} \cr } } \right]} \right]$$
B
$$\left[ { - 2,\left[ {\matrix{ 1 \cr { - 1} \cr } } \right]} \right]and\left[ { - 1,\left[ {\matrix{ 1 \cr { - 2} \cr } } \right]} \right]$$
C
$$\left[ { - 1,\left[ {\matrix{ 1 \cr { - 1} \cr } } \right]} \right]and\left[ {2,\left[ {\matrix{ 1 \cr { - 2} \cr } } \right]} \right]$$
D
$$\left[ {1,\left[ {\matrix{ 1 \cr { - 1} \cr } } \right]} \right]and\left[ { - 2,\left[ {\matrix{ 1 \cr { - 2} \cr } } \right]} \right]$$
2
GATE ECE 2006
MCQ (Single Correct Answer)
+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
$$\left[ {\matrix{ {\cos t} & { - \sin t} \cr {\cos t} & {\sin t} \cr } } \right]$$
3
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.
4
GATE ECE 2004
MCQ (Single Correct Answer)
+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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