GATE EE
Control Systems
State Variable Analysis
Previous Years Questions

## Marks 1

The state transition matrix for the system $$\left[ {\matrix{ {\mathop {{x_1}}\limits^ \bullet } \cr {\mathop {{x_2}}\limits^ \bullet } \cr... For a system with the transfer function$$H\left( s \right) = {{3\left( {s - 2} \right)} \over {{s^3} + 4{s^2} - 2s + 1}},\,\,$$the matrix$$A$$in t... A second order system starts with an initial condition of$$\left( {\matrix{ 2 \cr 3 \cr } } \right)$$without any external input. The st... The state transition matrix for the system$$\mathop X\limits^ \bullet = AX\,\,$$with initial state$$X(0)$$is Given the homogeneous state-space equation$$\mathop X\limits^ \bullet = \left[ {\matrix{ { - 3} & 1 \cr 0 & { - 2} \cr } } \ri...
A system is described by the state equation $$\mathop X\limits^ \bullet = AX + BU$$ , The output is given by $$Y=CX$$ Where $$A = \left( {\matrix{ ... The matrix of any state space equations for the transfer function$$C(s)/R(s)$$of the system, shown below in. Figure is ... Consider a second order system whose state space representation is of the form$$\mathop X\limits^ \bullet = AX + BU.$$If$$\,{x_1}\,\,\left( t \ri...
The transfer function for the state variable representation $$\mathop X\limits^ \bullet = AX + BU,\,\,Y = CX + DU,$$ is given by

## Marks 2

Consider the system described by the following state space representation \eqalign{ & \left[ {\matrix{ {\mathop {{x_1}}\limits^ \bullet \... The transfer function of the systemY\left( s \right)/U\left( s \right)$$, whose state-space equations are given below is:$$\eqalign{ & \lef...
Consider the following state - space representation of a linear time-invariant system. $$\mathop x\limits^ \bullet \left( t \right) = \left[ {\matrix... For the system governed by the set of equations:$$\eqalign{ & d{x_1}/dt = 2{x_1} + {x_2} + u \cr & d{x_2}/dt = - 2{x_1} + u \cr ... In the signal flow diagram given in the figure, $${u_1}$$ and $${u_2}$$ are possible inputs whereas $${y_1}$$ and $${y_2}$$ are possible outputs. When... Consider the system described by the following state space equations \eqalign{ & \left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr } } \r... The second order dynamic system{{dX} \over {dt}} = PX + Qu,\,\,\,y = RX$$has the matrices$$P,Q,$$and$$R$$as follows:$$P = \left[ {\matrix{ ... The state variable formulation of a system is given as $$\left[ {\matrix{ {\mathop {{x_1}}\limits^ \bullet } \cr {\mathop {{x_2}}\limits^ \b... The state variable formulation of a system is given as$$\left[ {\matrix{ {\mathop {{x_1}}\limits^ \bullet } \cr {\mathop {{x_2}}\limits^ \b... The state variable description of an $$LTI$$ system is given by $$\left( {\matrix{ {\mathop {{x_1}}\limits^ \bullet } \cr {\mathop {{x_2}}\... The system$$\mathop X\limits^ \bullet = AX + BU$$with$$A = \left[ {\matrix{ { - 1} & 2 \cr 0 & 2 \cr } } \right],B = \l... A system is described by the following state and output equations $${{d{x_1}\left( t \right)} \over {dt}} = - 3{x_1}\left( t \right) + {x_2}\left( t... A system is described by the following state and output equations$${{d{x_1}\left( t \right)} \over {dt}} = - 3{x_1}\left( t \right) + {x_2}\left( t...
The state space equation of a system is described by $$\mathop X\limits^ \bullet = AX + BU,\,\,Y = Cx$$ where $$X$$ is state vector, $$U$$ is input...
The state space equation of a system is described by $$\mathop X\limits^ \bullet = AX + BU,\,\,Y = Cx$$ where $$X$$ is state vector, $$U$$ is input...
A state variable system $$\mathop X\limits^ \bullet \left( t \right) = \left( {\matrix{ 0 & 1 \cr 0 & { - 3} \cr } } \right)X\le... A state variable system$$\mathop X\limits^ \bullet \left( t \right) = \left( {\matrix{ 0 & 1 \cr 0 & { - 3} \cr } } \right)X\le...
The state variable description of a linear autonomous system is, $$\mathop X\limits^ \bullet = AX,\,\,$$ where $$X$$ is the two dimensional state v...
The following equation defines a separately exited $$dc$$ motor in the form of a differential equation $${{{d^2}\omega } \over {d{t^2}}} + {{B\,d\omeg... For the system$$X = \left[ {\matrix{ 2 & 3 \cr 0 & 5 \cr } } \right]X + \left[ {\matrix{ 1 \cr 0 \cr } } \right]u,$$... For the system$$\mathop X\limits^ \bullet = \left[ {\matrix{ 2 & 0 \cr 0 & 4 \cr } } \right]X + \left[ {\matrix{ 1 \cr ...

## Marks 5

Obtain a state variable representation of the system governed by the differential equation: $${{{d^2}y} \over {d{t^2}}} + {{dy} \over {dt}} - 2y = u\l... Consider the state equation$$\mathop X\limits^ \bullet \left( t \right) = Ax\left( t \right)$$Given :$${e^{AT}} = \left[ {\matrix{ {{e^{ - t}}...
The state-space representation of a system is given by $$\left[ {\matrix{ {\mathop {{X_1}}\limits^ \bullet } \cr {\mathop {{X_2}}\limits^ \bu... Determine the transfer function of the system having the following state variable representation:$$\eqalign{ & X = \left[ {\matrix{ 0 & ...
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