State Space Analysis · Control Systems · GATE ECE
Marks 1
1
Consider the state space realization
$$$\left[ {\matrix{
{\mathop x\limits^ \bullet } & {\left( t \right)} \cr
{\mathop x\limits^ \bullet } & {\left( t \right)} \cr
} } \right] = \left[ {\matrix{
0 & 0 \cr
0 & { - 9} \cr
} } \right]\left[ {\matrix{
{{x_1}} & {\left( t \right)} \cr
{{x_2}} & {\left( t \right)} \cr
} } \right] + \left[ {\matrix{
0 \cr
{45} \cr
} } \right]u\left( t \right),$$$ with the initial condition $$\left[ {\matrix{
{{x_1}} & {\left( 0 \right)} \cr
{{x_2}} & {\left( 0 \right)} \cr
} } \right] = \left[ {\matrix{
0 \cr
0 \cr
} } \right];$$
Where u(t) denotes the unit step function.
The value of $$\mathop {Lt}\limits_{x \to \infty } \left| {\sqrt {{x_1}^2\left( t \right) + {x_2}^2\left( t \right)} } \right|$$ is ______
Where u(t) denotes the unit step function.
The value of $$\mathop {Lt}\limits_{x \to \infty } \left| {\sqrt {{x_1}^2\left( t \right) + {x_2}^2\left( t \right)} } \right|$$ is ______
GATE ECE 2017 Set 2
2
Consider the system $${{dx} \over {dt}} = Ax + Bu$$ with $${\rm A} = \left[ {\matrix{
1 & 0 \cr
0 & 1 \cr
} } \right]\,\,\,and\,\,\,{\rm B} = \left[ {\matrix{
p \cr
q \cr
} } \right],$$
where p and q are arbitrary real numbers. Which of the following statesments about the controllability of the system is true?
GATE ECE 2009
3
A linear system is equivalently represented by two sets of state equations. $$\mathop x\limits^ \bullet = \,\,{\rm A}X\,\, + BU$$ and $$\mathop W\limits^ \bullet = \,\,CW\,\, + DU.$$ The eigen values of the representations are also computed as $$\left[ \lambda \right]\,\,\,\,and\,\,\,\,\left[ \mu \right].$$ Which one of the following statements is true?
GATE ECE 2005
4
The transfer function Y(s)/U(s) of a system described by the state equations
$$\mathop x\limits^ \bullet $$(t) = -2x(t)+2u(t)
y(t) = 0.5x(t) is
$$\mathop x\limits^ \bullet $$(t) = -2x(t)+2u(t)
y(t) = 0.5x(t) is
GATE ECE 2002
5
The system mode described by the state equations
$$$X = \left( {\matrix{
0 & 1 \cr
2 & { - 3} \cr
} } \right)x + \left( {\matrix{
0 \cr
1 \cr
} } \right)u,y = \left[ {\matrix{
1 & 1 \cr
} } \right]x$$$
GATE ECE 1999
Marks 2
1
Consider a system $S$ represented in state space as
$$\frac{dx}{dt} = \begin{bmatrix} 0 & -2 \\ 1 & -3 \end{bmatrix}x + \begin{bmatrix} 1 \\ 0 \end{bmatrix}r , \quad y = \begin{bmatrix} 2 & -5 \end{bmatrix}x.$$
Which of the state space representations given below has/have the same transfer function as that of $S$?
GATE ECE 2024
2
The state equation and the output equation of a control system are given below:
$$\mathop x\limits^. = \left[ {\matrix{ { - 4} & { - 1.5} \cr 4 & 0 \cr } } \right]x + \left[ {\matrix{ 2 \cr 0 \cr } } \right]u,$$
$$y = \left[ {\matrix{ {1.5} & {0.625} \cr } } \right]x.$$
The transfer function representation of the system is
$$\mathop x\limits^. = \left[ {\matrix{ { - 4} & { - 1.5} \cr 4 & 0 \cr } } \right]x + \left[ {\matrix{ 2 \cr 0 \cr } } \right]u,$$
$$y = \left[ {\matrix{ {1.5} & {0.625} \cr } } \right]x.$$
The transfer function representation of the system is
GATE ECE 2018
3
A second order LTI system is described by the following state equation.
$$$\eqalign{
& {d \over {dt}}{x_1}\left( t \right) - {x_2}\left( t \right) = 0 \cr
& {d \over {dt}}{x_2}\left( t \right) + 2{x_1}\left( t \right) + 3{x_2}\left( t \right) = r\left( t \right) \cr} $$$
When x1(t) and x2(t) are the two state variables and r(t) denotes the input. The output c(t)=X1(t). The systyem is
GATE ECE 2017 Set 2
4
A second-order linear time-invariant system is described by the following state equations
$$$\eqalign{& {d \over {dt}}{x_1}\left( t \right) + 2{x_1}\left( t \right) = 3u\left( t \right) \cr
& {d \over {dt}}{x_2}\left( t \right) + {x_2}\left( t \right) = u\left( t \right) \cr} $$$
Where x1(t), then the system is
GATE ECE 2016 Set 3
5
A network is described by the state model as
$$$\eqalign{
& {\mathop x\limits^ \bullet _1} = 2{x_1} - {x_2} + 3u, \cr
& \mathop {{x_2}}\limits^ \bullet = - 4{x_2} - u, \cr
& y = 3{x_1} - 2{x_2} \cr} $$$
the transfer function H(s)$$\left[ { = {{Y\left( s \right)} \over {U\left( s \right)}}} \right]is$$
GATE ECE 2015 Set 3
6
The state variable representation of a system is given as
$$$\eqalign{
& \mathop x\limits^ \bullet = \left[ {\matrix{
0 & 1 \cr
0 & { - 1} \cr
} } \right]x;x\left( 0 \right) = \left[ {\matrix{
1 \cr
0 \cr
} } \right] \cr
& y = \left[ {\matrix{
0 & 1 \cr
} } \right]x \cr} $$$
The response y(t) is
GATE ECE 2015 Set 2
7
Consider the state space model of a system, as given below
The system is
GATE ECE 2014 Set 1
8
The state transition matrix $$\phi \left( t \right)$$ of a system $$$\left[ {\matrix{
{\mathop {{x_1}}\limits^ \bullet } \cr
{\mathop {{x_2}}\limits^ \bullet } \cr
} } \right] = \left[ {\matrix{
0 & 1 \cr
0 & 0 \cr
} } \right]\left[ {\matrix{
{{x_1}} \cr
{{x_2}} \cr
} } \right] is$$$
GATE ECE 2014 Set 4
9
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
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
GATE ECE 2014 Set 3
10
Consider the state space system expressed by the signal flow diagram shown in the figure.
The corresponding system is
GATE ECE 2014 Set 2
11
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
GATE ECE 2014 Set 2
12
The state diagram of a system is shown below. A system is shown below. A system is described by the state variable equations
The state transition matrix eAt of the system shown in the figure above is
GATE ECE 2013
13
The state diagram of a system is shown below. A system is shown below. A system is described by the state variable equations
The state-variable equations of the system shown in the figure above are
GATE ECE 2013
14
The state variable description of an LTI system is given by
where y is the output and u is input. The system is controllable for
GATE ECE 2012
15
The block diagram of a system with one input u and two outputs y1 and y2 is given below
A state space model of the above system in terms of the state vector $$\underline x $$ and the output vector $$\underline y = {\left[ {\matrix{ {{y_1}} & {{y_2}} \cr } } \right]^\tau }$$ is
GATE ECE 2011
16
The signal flow graph of a system is shown below.
The state variable representation of the system can be
GATE ECE 2010
17
The signal flow graph of a system is shown below.
The transfer function of the system is
GATE ECE 2010
18
A signal flow graph of a system is given below.
The set of equations that correspond to this signal flow graph is
GATE ECE 2008
19
The state space representation of a separately excited DC servo motor dynamics is given as
$$$\left[ {\matrix{
{{{d\omega } \over {dt}}} \cr
{{{d{i_a}} \over {dt}}} \cr
} } \right] = \left[ {\matrix{
{ - 1} & 1 \cr
{ - 1} & { - 10} \cr
} } \right]\left[ {\matrix{
\omega \cr
{{i_a}} \cr
} } \right] + \left[ {\matrix{
0 \cr
{10} \cr
} } \right]u.$$$
Where 'ω' is the speed of the motor, 'ia' is the armature current and u is the armature voltage. The transfer function $${{\omega \left( s \right)} \over {U\left( s \right)}}$$ of the motor is
GATE ECE 2007
20
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 $$x\left( t \right) = \left[ {\matrix{ {{e^{ - t}}} \cr { - {e^{ - t}}} \cr } } \right].$$
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
GATE ECE 2007
21
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 $$x\left( t \right) = \left[ {\matrix{ {{e^{ - t}}} \cr { - {e^{ - t}}} \cr } } \right].$$
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
GATE ECE 2007
22
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
The state-transition matrix of the system is
GATE ECE 2006
23
If A = $$\left[ {\matrix{
{ - 2} & 2 \cr
1 & { - 3} \cr
} } \right],$$ then sin At is
GATE ECE 2004
24
Given A $$ = \left[ {\matrix{
1 & 0 \cr
0 & 1 \cr
} } \right],$$ the state transition matrix eAt is given by
GATE ECE 2004
25
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
The system is
GATE ECE 2004
26
The zero, input response of a system given by the state space equation
$$$\left[ {{{\mathop {{x_1}}\limits^ \bullet } \over {\mathop {{x_2}}\limits^ \bullet }}} \right] = \left[ {\matrix{
1 & 0 \cr
1 & 1 \cr
} } \right]\left[ {\matrix{
{{x_1}} \cr
{{x_2}} \cr
} } \right]and\left[ {\matrix{
{{x_1}} & {\left( 0 \right)} \cr
{{x_2}} & {\left( 0 \right)} \cr
} } \right] = \left[ {\matrix{
1 \cr
0 \cr
} } \right]is$$$
GATE ECE 2003
27
For the system described by the state
equation $$$\mathop x\limits^ \bullet = \left[ {\matrix{
0 & 1 & 0 \cr
0 & 0 & 1 \cr
{0.5} & 1 & 2 \cr
} } \right]x + \left[ {\matrix{
0 \cr
0 \cr
1 \cr
} } \right]u.$$$
If the control signal u is given by u=(-0.5-3-5)x+v, then the eigen values of the closed loop system will be
GATE ECE 1999
28
A certain linear time invariant system has the state and the output equations given below
$$$\left[ {\matrix{
{\mathop {{x_1}}\limits^ \bullet } \cr
{\mathop {{x_2}}\limits^ \bullet } \cr
} } \right] = \left[ {\matrix{
1 & { - 1} \cr
0 & 1 \cr
} } \right]\left[ {\matrix{
{{x_1}} \cr
{{x_2}} \cr
} } \right] + \left[ {\matrix{
0 \cr
1 \cr
} } \right]u$$$
$$$y = \left[ {\matrix{
1 & 1 \cr
} } \right]\left[ {\matrix{
{{x_1}} \cr
{{x_2}} \cr
} } \right], if$$$
$${x_1}\left( 0 \right) =1 ,{x_2}\left( 0 \right) = - 1,$$ $$u\left( 0 \right) = 0,$$ then $${{dy} \over {dt}}{|_{t = 0}}$$ is
GATE ECE 1997
29
A linear time-invariant system is described by the state variable model
$$$\left[ {\matrix{
{{{\mathop x\limits^ \bullet }_1}} \cr
{{{\mathop x\limits^ \bullet }_2}} \cr
} } \right] = \left[ {\matrix{
{ - 1} & 0 \cr
0 & { - 2} \cr
} } \right]\left[ {\matrix{
{{x_1}} \cr
{{x_2}} \cr
} } \right] + \left[ {\matrix{
0 \cr
1 \cr
} } \right]u.$$$
$$$Y = \left[ {\matrix{
1 & 2 \cr
} } \right]\left[ {\matrix{
{{x_1}} \cr
{{x_2}} \cr
} } \right]$$$
GATE ECE 1992
30
A linear second order single input continuous-time system is described by the
following set of differential equations
$$$\eqalign{
& \mathop {{x_1}}\limits^ \bullet \left( t \right) = - 2{x_1}\left( t \right) + 4{x_2}\left( t \right) \cr
& \mathop {{x_1}}\limits^ \bullet \left( t \right) = 2{x_1}\left( t \right) - {x_2}\left( t \right) + u\left( t \right) \cr} $$$
Where x1(t) and x2(t) are the state variables and u (t) is the control variable. The system is
Where x1(t) and x2(t) are the state variables and u (t) is the control variable. The system is
GATE ECE 1991
Marks 5
1
The block diagram of a linear time invariant system is given in Figure is
(a) Write down the state variable equations for the system in matrix form
assuming the state vector to be $${\left[ {{x_1}\left( t \right)\,\,{x_2}\left( t \right)} \right]^T}$$
(b) Find out the state transition matrix.
(c) Determine y(t), t ≥ 0, when the initial values of the state at time t = 0 are $${x_1}$$(0) = 1, and $${x_2}$$(0) = 1.
(a) Write down the state variable equations for the system in matrix form
assuming the state vector to be $${\left[ {{x_1}\left( t \right)\,\,{x_2}\left( t \right)} \right]^T}$$
(b) Find out the state transition matrix.
(c) Determine y(t), t ≥ 0, when the initial values of the state at time t = 0 are $${x_1}$$(0) = 1, and $${x_2}$$(0) = 1.
GATE ECE 2002
2
A certain linear, time-invariant system has the state and output representation shown below:
$$$\eqalign{
& \left[ {\matrix{
{\mathop {{x_1}}\limits^ \bullet } \cr
{\mathop {{x_2}}\limits^ \bullet } \cr
} } \right] = \left[ {\matrix{
{ - 2} & 1 \cr
0 & { - 3} \cr
} } \right]\left[ {\matrix{
{{x_1}} \cr
{{x_2}} \cr
} } \right] + \left[ {\matrix{
1 \cr
0 \cr
} } \right]u \cr
& y = \left( {\matrix{
1 & 1 \cr
} } \right)\left[ {\matrix{
{{x_1}} \cr
{{x_2}} \cr
} } \right] \cr} $$$
(a) Find the eigen values (natural frequencies) of the system.
(b)If u(t)=$$\delta \left( t \right)$$ and x1(0+)=x2(0+)=0, find x1(t),x2(t) and y(t), for t>0.
(c)When the input is zero, choose initial conditions $${x_1}\left( {{0^ + }} \right)$$ and $${x_2}\left( {{0^ + }} \right)$$ such that $$y\left( t \right) = A{e^{ - 2t}}$$ for t>0
(a) Find the eigen values (natural frequencies) of the system.
(b)If u(t)=$$\delta \left( t \right)$$ and x1(0+)=x2(0+)=0, find x1(t),x2(t) and y(t), for t>0.
(c)When the input is zero, choose initial conditions $${x_1}\left( {{0^ + }} \right)$$ and $${x_2}\left( {{0^ + }} \right)$$ such that $$y\left( t \right) = A{e^{ - 2t}}$$ for t>0
GATE ECE 2000
3
For the circuit shown in the figure, choose state variables as $${x_{1,}}{x_{2,}}{x_3}$$ to be $${i_{L1}}\left( t \right),{v_{c2}}\left( t \right),{i_{L3}}\left( t \right)$$
Wriote the state equations
$$$\left[ {\matrix{ {\mathop {{x_1}}\limits^ \bullet } \cr {\mathop {{x_2}}\limits^ \bullet } \cr {\mathop {{x_3}}\limits^ \bullet } \cr } } \right] = A\left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr {{x_3}} \cr } } \right] + B\left[ {e\left( t \right)} \right]$$$GATE ECE 1997
4
Obtain a state space representation in diagonal form for the following system
$$${{{d^3}y} \over {d{t^3}}} + 6{{{d^2}y} \over {d{t^2}}} + 11{{dy} \over {dt}} + 6y = 6u\left( t \right)$$$
GATE ECE 1996
