GATE ECE 2006 | State Space Analysis Question 24 | Control Systems

GATE ECE 2006

MCQ (Single Correct Answer)

-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

$$\left[ {\matrix{ {\cos t} & {\sin t} \cr { - \sin t} & {\cos t} \cr } } \right]$$

$$\left[ {\matrix{ { - \cos t} & {\sin t} \cr { - \sin t} & { - \cos t} \cr } } \right]$$

$$\left[ {\matrix{ { - \cos t} & { - \sin t} \cr { - \sin t} & {\cos t} \cr } } \right]$$

$$\left[ {\matrix{ {\cos t} & { - \sin t} \cr {\cos t} & {\sin t} \cr } } \right]$$

GATE ECE 2004

MCQ (Single Correct Answer)

-0.6

If A = $$\left[ {\matrix{ { - 2} & 2 \cr 1 & { - 3} \cr } } \right],$$ then sin At is

$${1 \over 3}\left[ {\matrix{ {\sin \left( { - 4t} \right) + 2\sin \left( { - t} \right)} & { - 2\sin \left( { - 4t} \right) + 2\sin \left( { - t} \right)} \cr { - \sin \left( { - 4t} \right) + \sin \left( { - t} \right)} & {2\sin \left( { - 4t} \right) + \sin \left( { - t} \right)} \cr } } \right]$$

$$\left[ {\matrix{ {\sin \left( { - 2t} \right)} & {\sin \left( {2t} \right)} \cr {\sin \left( t \right)} & {\sin \left( { - 3t} \right)} \cr } } \right]$$

$${1 \over 3}\left[ {\matrix{ {\sin \left( {4t} \right) + 2\sin \left( t \right)} & {2\sin \left( { - 4t} \right) - 2\sin \left( { - t} \right)} \cr { - \sin \left( { - 4t} \right) + \sin \left( t \right)} & {2\sin \left( {4t} \right) + \sin \left( t \right)} \cr } } \right]$$

$${1 \over 3}\left[ {\matrix{ {\cos \left( { - t} \right) + 2\cos \left( t \right)} & {2\cos \left( { - 4t} \right) + 2\sin \left( { - t} \right)} \cr { - \cos \left( { - 4t} \right) + \sin \left( { - t} \right)} & { - 2\cos \left( { - 4t} \right) + \cos \left( { - t} \right)} \cr } } \right]$$

GATE ECE 2004

MCQ (Single Correct Answer)

-0.6

Given A $$ = \left[ {\matrix{ 1 & 0 \cr 0 & 1 \cr } } \right],$$ the state transition matrix e^At is given by

$$\left[ {\matrix{ 0 & {{e^{ - t}}} \cr {{0^{ - t}}} & 0 \cr } } \right]$$

$$\left[ {\matrix{ {{e^t}} & 0 \cr 0 & {{e^t}} \cr } } \right]$$

$$\left[ {\matrix{ {{e^{ - t}}} & 0 \cr 0 & {{e^{ - t}}} \cr } } \right]$$

$$\left[ {\matrix{ 0 & {{e^t}} \cr {{e^t}} & 0 \cr } } \right]$$

GATE ECE 2004

MCQ (Single Correct Answer)

-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

controllable but not observable.

observable but not controllable.

neither controllable nor observable.

controllable and observable.

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