GATE ECE 2007 | State Space Analysis Question 21 | Control Systems

GATE ECE 2007

MCQ (Single Correct Answer)

-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 system matrix a is

$$\left[ {\matrix{ 0 & 1 \cr { - 1} & 1 \cr } } \right]$$

$$\left[ {\matrix{ 1 & 1 \cr { - 1} & { - 2} \cr } } \right]$$

$$\left[ {\matrix{ 2 & 1 \cr { - 1} & { - 1} \cr } } \right]$$

$$\left[ {\matrix{ 0 & 1 \cr { - 2} & { - 3} \cr } } \right]$$

GATE ECE 2007

MCQ (Single Correct Answer)

-0.6

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

$$\left[ { - 1,\left[ {\matrix{ 1 \cr { - 1} \cr } } \right]} \right]and\left[ { - 2,\left[ {\matrix{ 1 \cr { - 2} \cr } } \right]} \right]$$

$$\left[ { - 2,\left[ {\matrix{ 1 \cr { - 1} \cr } } \right]} \right]and\left[ { - 1,\left[ {\matrix{ 1 \cr { - 2} \cr } } \right]} \right]$$

$$\left[ { - 1,\left[ {\matrix{ 1 \cr { - 1} \cr } } \right]} \right]and\left[ {2,\left[ {\matrix{ 1 \cr { - 2} \cr } } \right]} \right]$$

$$\left[ {1,\left[ {\matrix{ 1 \cr { - 1} \cr } } \right]} \right]and\left[ { - 2,\left[ {\matrix{ 1 \cr { - 2} \cr } } \right]} \right]$$

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

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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