Find the dynamics of the circuit.

R1I1(s) + LsI1 (s) + LsI2 (s) = V(s)$Ls{{I}_{2}}\left( s \right)+~{{R}_{2}}{{I}_{2}}\left( s \right)+\frac{1}{cs{{I}_{2}}\left( s \right)}+Ls{{I}_{1}}\left( s \right)=~0~$

R1I1 (s) + LsI1(s) – LsI2(s) = V(s)LsI2(s) + R2I2 (S) + Cs I2 (s) + LsI1(s) = 0

R1I1(s) + LsI1 (s) + LsI2(s) = V(s)$L{{s}_{2}}\left( s \right)+~{{R}_{2}}{{I}_{2}}\left( s \right)+\frac{1}{Cs}{{I}_{2}}\left( s \right)-Ls{{I}_{1}}\left( s \right)=0$

R1I1 (s) + LsI1(s) – LsI2(S) = V(s)$Ls{{I}_{2}}\left( s \right)+{{R}_{2}}{{I}_{2}}\left( s \right)+\frac{1}{Cs}{{I}_{2}}\left( s \right)-Ls{{I}_{1}}\left( s \right)=0$

For the given circuit, which one of the following is the correct state equation?

$\frac{d}{{dt}}\left[ {\begin{array}{*{20}{c}} v\\ i \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} { - 4}&{ - 4}\\ { - 2}&4 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} v\\ i \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} 4&4\\ 4&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{i_1}}\\ {{i_2}} \end{array}} \right]$

$\frac{d}{{dt}}\left[ {\begin{array}{*{20}{c}} v\\ i \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} { - 4}&4\\ { - 2}&{ - 4} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} v\\ i \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} 0&4\\ 4&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{i_1}}\\ {{i_2}} \end{array}} \right]$

$\frac{d}{{dt}}\left[ {\begin{array}{*{20}{c}} v\\ i \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} { - 4}&{ - 4}\\ { - 2}&4 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} v\\ i \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} 4&4\\ 4&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{i_1}}\\ {{i_2}} \end{array}} \right]$

$\frac{d}{{dt}}\left[ {\begin{array}{*{20}{c}} v\\ i \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 4&{ - 4}\\ { - 2}&{ - 4} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} v\\ i \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} 0&4\\ 4&4 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{i_1}}\\ {{i_2}} \end{array}} \right]$

$\frac{d}{{dt}}\left[ {\begin{array}{*{20}{c}} v\\ i \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} { - 4}&{ - 4}\\ { - 2}&{ - 4} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} v\\ i \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} 4&0\\ 0&4 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{i_1}}\\ {{i_2}} \end{array}} \right]$

A network is described by the state model as${{\dot{x}}_{1}}=2{{x}_{1}}-{{x}_{2}}+3u$${{\dot{x}}_{2}}=-4{{x}_{2}}-u$$y=3{{x}_{1}}-2{{x}_{2}}$The transfer function H(s) $\left( { = \frac{{Y\left( s \right)}}{{U\left( s \right)}}} \right)$ is

$\frac{{11s - 35}}{{\left( {s - 2} \right)\left( {s + 4} \right)}}$

$\frac{{11s + 35}}{{\left( {s - 2} \right)\left( {s + 4} \right)}}$

$\frac{{11s - 35}}{{\left( {s - 2} \right)\left( {s + 4} \right)}}$

$\frac{{11s + 38}}{{\left( {s - 2} \right)\left( {s + 4} \right)}}$

$\frac{{11s - 38}}{{\left( {s - 2} \right)\left( {s + 4} \right)}}$

Consider the linear system $\dot x = \left[ {\begin{array}{*{20}{c}}{ - 1}&0\\0&{ - 2}\end{array}} \right]x$, with initial condition $x\left( 0 \right) = \left[ {\begin{array}{*{20}{c}}1\\1\end{array}} \right]$. The solution x(t) for this system is

$x\left( t \right) = \left[ {\begin{array}{*{20}{c}}{{e^{ - t}}}&{t{e^{ - 2t}}}\\0&{{e^{ - 2t}}}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}1\\1\end{array}} \right]$

$x\left( t \right) = \left[ {\begin{array}{*{20}{c}}{{e^{ - t}}}&0\\0&{{e^{2t}}}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}1\\1\end{array}} \right]$

$x\left( t \right) = \left[ {\begin{array}{*{20}{c}}{{e^{ - t}}}&{ - {t^2}{e^{ - 2t}}}\\0&{{e^{ - 2t}}}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}1\\1\end{array}} \right]$

$x\left( t \right) = \left[ {\begin{array}{*{20}{c}}{{e^{ - t}}}&0\\0&{{e^{ - 2t}}}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}1\\1\end{array}} \right]$

From the figure, obtain state equation

$\left[ {\dot X} \right] = \left[ {\begin{array}{*{20}{c}} 0&{ - 3}\\ { - 2}&4 \end{array}} \right]\left[ X \right] + \left[ {\begin{array}{*{20}{c}} 1\\ 0 \end{array}} \right]u$

$\left[ {\dot X} \right] = \left[ {\begin{array}{*{20}{c}} 0&{ - 3}\\ { - 2}&4 \end{array}} \right]\left[ X \right] + \left[ {\begin{array}{*{20}{c}} 0\\ 1 \end{array}} \right]u$

$\left[ {\dot X} \right] = \left[ {\begin{array}{*{20}{c}} -2&{ 4}\\ { 0}&-3 \end{array}} \right]\left[ X \right] + \left[ {\begin{array}{*{20}{c}} 0\\ 1 \end{array}} \right]u$

$\left[ {\dot X} \right] = \left[ {\begin{array}{*{20}{c}} 0&{ - 3}\\ { - 2}&4 \end{array}} \right]\left[ X \right] + \left[ {\begin{array}{*{20}{c}} 1\\ 0 \end{array}} \right]u$

$\left[ {\dot X} \right] = \left[ {\begin{array}{*{20}{c}} -2&{ 4}\\ { 0}&-3 \end{array}} \right]\left[ X \right] + \left[ {\begin{array}{*{20}{c}} 1\\ 0 \end{array}} \right]u$

For the system governed by the set of equations:$\begin{array}{l}\frac{{d{x_1}}}{{dt}} = 2{x_1} + {x_2} + u\\d{x_2}/dt = -2{x_1} + u\\y = 3{x_1}\end{array}$the transfer function $Y\left( s \right)/U\left( s \right)$ is given by

$\frac{{s + 1}}{{{s^2}-2s + 1}}$

$\frac{{3\left( {s + 1} \right)}}{{{s^2}-2s + 2}}$

$\frac{{s + 1}}{{{s^2}-2s + 1}}$

$\frac{{3\left( {2s + 1} \right)}}{{{s^2}\;-2s + 1}}$

$\frac{{3\left( {2s + 1} \right)}}{{{s^2}\;-2s + 2}}$

Let the state-space representation of an LTI system be ẋ(t) = A x(t) + B u(t), y(t) = C x(t) + d u(t) where A, B, C are matrices, d is a scalar, u(t) is the input to the system, and y(t) is its output. Let B = [0 0 1]T and d = 0. Which one of the following options for A and C will ensure that the transfer function of this LTI system is $H\left( s \right) = \frac{1}{{{s^3} + 3{s^2} + 2s + 1}} = ?$

$A = \left[ {\begin{array}{*{20}{c}} 0&1&0\\ 0&0&1\\ { - 3}&{ - 2}&{ - 1} \end{array}} \right]and\;C = \;\left[ {\begin{array}{*{20}{c}} 1&0&0 \end{array}} \right]$

$A = \left[ {\begin{array}{*{20}{c}} 0&1&0\\ 0&0&1\\ { - 1}&{ - 2}&{ - 3} \end{array}} \right]and\;C = \;\left[ {\begin{array}{*{20}{c}} 1&0&0 \end{array}} \right]$

$A = \left[ {\begin{array}{*{20}{c}} 0&1&0\\ 0&0&1\\ { - 3}&{ - 2}&{ - 1} \end{array}} \right]and\;C = \;\left[ {\begin{array}{*{20}{c}} 1&0&0 \end{array}} \right]$

$A = \left[ {\begin{array}{*{20}{c}} 0&1&0\\ 0&0&1\\ { - 1}&{ - 2}&{ - 3} \end{array}} \right]and\;C = \;\left[ {\begin{array}{*{20}{c}} 0&0&1 \end{array}} \right]$

$A = \left[ {\begin{array}{*{20}{c}} 0&1&0\\ 0&0&1\\ { - 3}&{ - 2}&{ - 1} \end{array}} \right]and\;C = \;\left[ {\begin{array}{*{20}{c}} 0&0&1 \end{array}} \right]$

Consider a system governed by the following equations$\frac{{d{x_1}\left( t \right)}}{{dt}} = {x_2}\left( t \right) - {x_1}\left( t \right)$$\frac{{d{x_2}\left( t \right)}}{{dt}} = {x_1}\left( t \right) - {x_2}\left( t \right)$The initial conditions are such that ${x_1}\left( 0 \right) < {x_2}\left( 0 \right) < \;\infty .$ Let ${x_{1f}} = \mathop {\lim }\limits_{t \to \infty } {x_1}\left( t \right)$ and ${x_{2f}} = \mathop {\lim }\limits_{t \to \infty } {x_2}\left( t \right)$. Which one of the following is true?

${x_{1f}} = {x_{2f}} = \infty$

${x_{1f}} < {x_{2f}} < \infty$

${x_{2f}} < {x_{1f}} < \infty$

${x_{1f}} = \;{x_{2f}} < \infty$

${x_{1f}} = {x_{2f}} = \infty$

Consider a state-variable model of a system$\left[ {\begin{array}{*{20}{c}} {{{\dot x}_1}}\\ {{{\dot x}_2}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 0&1\\ { - \alpha }&{ - 2\beta } \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{x_1}}\\ {{x_2}} \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} 0\\ \alpha \end{array}} \right]r$$y = \left[ {\begin{array}{*{20}{c}} 1&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{x_1}}\\ {{x_2}} \end{array}} \right]$where y is the output, and r is the input. The damping ratio ξ and the undamped natural frequency ωn (rad/sec) of the system are given by

${\rm{\xi }} = \sqrt \alpha ;{\omega _n} = \frac{\beta }{{\sqrt \alpha }}$

${\rm{\xi }} = \frac{{\rm{\beta }}}{{\sqrt \alpha }};{\omega _n} = \sqrt \alpha $

${\rm{\xi }} = \sqrt \alpha ;{\omega _n} = \frac{\beta }{{\sqrt \alpha }}$

${\rm{\xi }} = \frac{{\sqrt \alpha }}{\beta };{\omega _n} = \sqrt \beta$

${\rm{\xi }} = \sqrt \beta ;{\omega _n} = \sqrt \alpha$

A second-order linear time-invariant system is described by the following state equations$\frac{{\rm{d}}}{{{\rm{dt}}}}{{\rm{x}}_1}\left( {\rm{t}} \right) + 2{{\rm{x}}_1}\left( {\rm{t}} \right) = 3{\rm{u}}\left( {\rm{t}} \right)$$ \frac{{\rm{d}}}{{{\rm{dt}}}}{{\rm{x}}_2}\left( {\rm{t}} \right) + {{\rm{x}}_2}\left( {\rm{t}} \right) = {\rm{u}}\left( {\rm{t}} \right) $where x1(t) and x2(t) are the two-state variables and u(t) denotes the input. If the output c(t) = x1(t), then the system is:

observable but not controllable

controllable but not observable

observable but not controllable

both controllable and observable

neither controllable nor observable

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 output. When would the SISO system derived from this diagram be controllable and observable?

When ${u_1}$ is the only input and ${y_2}$ is the only output

When ${u_1}$ is the only input and ${y_1}$ is the only output

When ${u_2}$ is the only input and ${y_1}$ is the only output

When ${u_1}$ is the only input and ${y_2}$ is the only output

When ${u_2}$ is the only input and ${y_2}$ is the only output

Consider the state – space model$\left[ {\begin{array}{*{20}{c}} {{{\dot x}_1}}\\ {{{\dot x}_2}}\\ {{{\dot x}_3}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} { - 1}&1&0\\ 0&{ - 1}&0\\ 0&0&{ - 2} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{x_1}}\\ {{x_2}}\\ {{x_3}} \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} 0\\ 4\\ 0 \end{array}} \right]u$$y = \left[ {\begin{array}{*{20}{c}} 1&1&1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{x_1}}\\ {{x_2}}\\ {{x_3}} \end{array}} \right]$System is

Uncontrollable and Unobservable

A system is represented by $3\dfrac{dy}{dt}+2y=u$, what is the transfer function to the system?

$\dfrac{Y\left( s\right)}{U\left( s\right)}=\dfrac{s+1}{3s+1}$

$\dfrac{Y\left( s\right)}{U\left( s\right)}=\dfrac{1}{s^2+2s+3}$

$\dfrac{Y\left( s\right)}{U\left( s\right)}=\dfrac{1}{3s+2}$

$\dfrac{Y\left( s\right)}{U\left( s\right)}=\dfrac{s+1}{3s+1}$

$\dfrac{Y\left( s\right)}{U\left( s\right)}=\dfrac{1}{2s+3}$

A state-space representation of a system is given by:$A=\left[ \begin{matrix} 0 & 1 \\ -2 & 0 \\ \end{matrix} \right],~y=\left[ 1~-1 \right],~x\left( 0 \right)=\left[ \begin{matrix} 1 \\ 1 \\ \end{matrix} \right]$The time response of the system will be ______.

$\frac{1}{\sqrt{2}}\sin \sqrt{2}t$

$-\frac{1}{\sqrt{2}}\sin \sqrt{2}t$

$\frac{3}{\sqrt{2}}\sin \sqrt{2}t$

$\frac{1}{\sqrt{2}}\sin \sqrt{2}t$

Consider a system described by the state model$\dot{X}=\left[ \begin{matrix} 2 & 1 \\ -1 & 2 \\ \end{matrix} \right] X + \left[ \begin{matrix} 1 \\ 1 \\ \end{matrix} \right] U$Y = [1 1] XThe system is

neither controllable nor observable

controllable but not observable

both controllable and observable

neither controllable nor observable

What is the condition to call a number ‚âà√≠¬¨‚Ñ¢ is an Eigen value of F and a nonzero vector U is the associated Eigen vector?#

(F+ ‚âà√≠¬¨‚Ñ¢I)U=0

(F- ‚âà√≠¬¨‚Ñ¢I)U=0

Which of the following gives the complete definition of the state of a system at time n0?

Amount of information at n0 determines output signal for n‚Äö√Ñ√∂‚àö¬¢‚Äö√Ñ¬¢n0

Input signal x(n) for n‚Äö√Ñ√∂‚àö¬¢‚Äö√Ñ¬¢n0 determines output signal for n‚Äö√Ñ√∂‚àö¬¢‚Äö√Ñ¬¢n0

Input signal x(n) for n‚Äö√Ñ√∂‚àö¬¢‚Äö√Ñ¬¢0 determines output signal for n‚Äö√Ñ√∂‚àö¬¢‚Äö√Ñ¬¢n0

Amount of information at n0+input signal x(n) for n‚Äö√Ñ√∂‚àö¬¢‚Äö√Ñ¬¢n0 determines output signal for n‚Äö√Ñ√∂‚àö¬¢‚Äö√Ñ¬¢n0

47 + Mcqs in State Space System Analysis Structures in Digital Signal Processing Page 1 McqOptions

1.	The parallel form realization is also known as normal form representation.
A.	True
B.	False
Answer» B. False

2.	The determinant \|F-λI\|=0 yields the characteristic polynomial of the matrix F.
A.	True
B.	False
Answer» B. False

3.	What is the condition to call a number λ is an Eigen value of F and a nonzero vector U is the associated Eigen vector?
A.	(F+λI)U=0
B.	(F-λI)U=0
C.	F-λI=0
D.	None of the mentioned
Answer» C. F-λI=0

4.	A closed form solution of the state space equations is easily obtained when the system matrix F is?
A.	Transpose
B.	Symmetric
C.	Identity
D.	Diagonal
Answer» E.

5.	If we interchange the rows and columns of the matrix F, then the system is called as ______________
A.	Identity system
B.	Diagonal system
C.	Transposed system
D.	None of the mentioned
Answer» D. None of the mentioned

6.	From the definition of state of a system, the system consists of only one component called memory less component.
A.	True
B.	False
Answer» C.

7.	Find the dynamics of the circuit.
A.	R1I1(s) + LsI1 (s) + LsI2 (s) = V(s)\(Ls{{I}_{2}}\left( s \right)+~{{R}_{2}}{{I}_{2}}\left( s \right)+\frac{1}{cs{{I}_{2}}\left( s \right)}+Ls{{I}_{1}}\left( s \right)=~0~\)
B.	R1I1 (s) + LsI1(s) – LsI2(s) = V(s)LsI2(s) + R2I2 (S) + Cs I2 (s) + LsI1(s) = 0
C.	R1I1(s) + LsI1 (s) + LsI2(s) = V(s)\(L{{s}_{2}}\left( s \right)+~{{R}_{2}}{{I}_{2}}\left( s \right)+\frac{1}{Cs}{{I}_{2}}\left( s \right)-Ls{{I}_{1}}\left( s \right)=0\)
D.	R1I1 (s) + LsI1(s) – LsI2(S) = V(s)\(Ls{{I}_{2}}\left( s \right)+{{R}_{2}}{{I}_{2}}\left( s \right)+\frac{1}{Cs}{{I}_{2}}\left( s \right)-Ls{{I}_{1}}\left( s \right)=0\)
Answer» E.

8.	Dynamics of system is described by \(\frac{{m{d^2}y}}{{dt}} + \frac{{bdy}}{{dt}} + ky = u\) where m, b, k are constants, y, u are time varying quantities. How many state variables are required for this system to be represented in state-space form?
A.	1
B.	3
C.	4
D.	2
Answer» E.

9.	For the given circuit, which one of the following is the correct state equation?
A.	\(\frac{d}{{dt}}\left[ {\begin{array}{{20}{c}} v\\ i \end{array}} \right] = \left[ {\begin{array}{{20}{c}} { - 4}&4\\ { - 2}&{ - 4} \end{array}} \right]\left[ {\begin{array}{{20}{c}} v\\ i \end{array}} \right] + \left[ {\begin{array}{{20}{c}} 0&4\\ 4&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{i_1}}\\ {{i_2}} \end{array}} \right]\)
B.	\(\frac{d}{{dt}}\left[ {\begin{array}{{20}{c}} v\\ i \end{array}} \right] = \left[ {\begin{array}{{20}{c}} { - 4}&{ - 4}\\ { - 2}&4 \end{array}} \right]\left[ {\begin{array}{{20}{c}} v\\ i \end{array}} \right] + \left[ {\begin{array}{{20}{c}} 4&4\\ 4&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{i_1}}\\ {{i_2}} \end{array}} \right]\)
C.	\(\frac{d}{{dt}}\left[ {\begin{array}{{20}{c}} v\\ i \end{array}} \right] = \left[ {\begin{array}{{20}{c}} 4&{ - 4}\\ { - 2}&{ - 4} \end{array}} \right]\left[ {\begin{array}{{20}{c}} v\\ i \end{array}} \right] + \left[ {\begin{array}{{20}{c}} 0&4\\ 4&4 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{i_1}}\\ {{i_2}} \end{array}} \right]\)
D.	\(\frac{d}{{dt}}\left[ {\begin{array}{{20}{c}} v\\ i \end{array}} \right] = \left[ {\begin{array}{{20}{c}} { - 4}&{ - 4}\\ { - 2}&{ - 4} \end{array}} \right]\left[ {\begin{array}{{20}{c}} v\\ i \end{array}} \right] + \left[ {\begin{array}{{20}{c}} 4&0\\ 0&4 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{i_1}}\\ {{i_2}} \end{array}} \right]\)
Answer» B. \(\frac{d}{{dt}}\left[ {\begin{array}{{20}{c}} v\\ i \end{array}} \right] = \left[ {\begin{array}{{20}{c}} { - 4}&{ - 4}\\ { - 2}&4 \end{array}} \right]\left[ {\begin{array}{{20}{c}} v\\ i \end{array}} \right] + \left[ {\begin{array}{{20}{c}} 4&4\\ 4&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{i_1}}\\ {{i_2}} \end{array}} \right]\)

10.	A network is described by the state model as\({{\dot{x}}_{1}}=2{{x}_{1}}-{{x}_{2}}+3u\)\({{\dot{x}}_{2}}=-4{{x}_{2}}-u\)\(y=3{{x}_{1}}-2{{x}_{2}}\)The transfer function H(s) \(\left( { = \frac{{Y\left( s \right)}}{{U\left( s \right)}}} \right)\) is
A.	\(\frac{{11s + 35}}{{\left( {s - 2} \right)\left( {s + 4} \right)}}\)
B.	\(\frac{{11s - 35}}{{\left( {s - 2} \right)\left( {s + 4} \right)}}\)
C.	\(\frac{{11s + 38}}{{\left( {s - 2} \right)\left( {s + 4} \right)}}\)
D.	\(\frac{{11s - 38}}{{\left( {s - 2} \right)\left( {s + 4} \right)}}\)
Answer» B. \(\frac{{11s - 35}}{{\left( {s - 2} \right)\left( {s + 4} \right)}}\)

11.	linear time-invariant single-input single-output system has a state space model given by \(\frac{{dx}}{{dt}} = Fx + Gu;y = Hx\)Where,\(F = \left[ {\begin{array}{{20}{c}} 0&1\\ { - 4}&{ - 2} \end{array}} \right];\;G = \left[ {\begin{array}{{20}{c}} 0\\ 1 \end{array}} \right];\;H = \left[ {\begin{array}{*{20}{c}} 1&0 \end{array}} \right]\)Here, x is the sate vector, u is the input and y is the output.The damping ratio of the system is
A.	0.25
B.	0.5
C.	1
D.	2
Answer» C. 1

16.	A system has the state variable representation \(\dot X = \begin{bmatrix} 0 & 1 \\\ 0 & -1 \end{bmatrix} X + \begin{bmatrix} 0 \\\ 1 \end{bmatrix} u : y = \begin{bmatrix} 1 & 1 \end{bmatrix}X\). Its transfer function is
A.	1
B.	\(\frac 1 s\)
C.	\(\frac {1}{s + 1}\)
D.	\(\frac {1}{s^2}\)
Answer» C. \(\frac {1}{s + 1}\)

20.	Number of state variables of discrete-time system, described by\(y[n] - \frac{3}{4}y[n - 1] + \frac{1}{8}y[n - 2] = x[n]\) is
A.	2
B.	3
C.	4
D.	1
Answer» B. 3

21.	All the constant – N loci in G-plane intersect the real axis in points
A.	-1 and origin
B.	-0.5 and +0.5
C.	-1 and +1
D.	Origin and +1
Answer» B. -0.5 and +0.5

25.	Bounded-input bounded-output stability implies asymptotic stability for1. Completely controllable system2. Completely observable system3. Uncontrollable system4. Unobservable systemWhich of the above statements are correct?
A.	1 and 4 only
B.	1 and 2 only
C.	2 and 3 only
D.	3 and 4 only
Answer» C. 2 and 3 only

12.	Consider the linear system \(\dot x = \left[ {\begin{array}{{20}{c}}{ - 1}&0\\0&{ - 2}\end{array}} \right]x\), with initial condition \(x\left( 0 \right) = \left[ {\begin{array}{{20}{c}}1\\1\end{array}} \right]\). The solution x(t) for this system is
A.	\(x\left( t \right) = \left[ {\begin{array}{{20}{c}}{{e^{ - t}}}&{t{e^{ - 2t}}}\\0&{{e^{ - 2t}}}\end{array}} \right]\left[ {\begin{array}{{20}{c}}1\\1\end{array}} \right]\)
B.	\(x\left( t \right) = \left[ {\begin{array}{{20}{c}}{{e^{ - t}}}&0\\0&{{e^{2t}}}\end{array}} \right]\left[ {\begin{array}{{20}{c}}1\\1\end{array}} \right]\)
C.	\(x\left( t \right) = \left[ {\begin{array}{{20}{c}}{{e^{ - t}}}&{ - {t^2}{e^{ - 2t}}}\\0&{{e^{ - 2t}}}\end{array}} \right]\left[ {\begin{array}{{20}{c}}1\\1\end{array}} \right]\)
D.	\(x\left( t \right) = \left[ {\begin{array}{{20}{c}}{{e^{ - t}}}&0\\0&{{e^{ - 2t}}}\end{array}} \right]\left[ {\begin{array}{{20}{c}}1\\1\end{array}} \right]\)
Answer» E.

13.	From the figure, obtain state equation
A.	\(\left[ {\dot X} \right] = \left[ {\begin{array}{{20}{c}} 0&{ - 3}\\ { - 2}&4 \end{array}} \right]\left[ X \right] + \left[ {\begin{array}{{20}{c}} 0\\ 1 \end{array}} \right]u\)
B.	\(\left[ {\dot X} \right] = \left[ {\begin{array}{{20}{c}} -2&{ 4}\\ { 0}&-3 \end{array}} \right]\left[ X \right] + \left[ {\begin{array}{{20}{c}} 0\\ 1 \end{array}} \right]u\)
C.	\(\left[ {\dot X} \right] = \left[ {\begin{array}{{20}{c}} 0&{ - 3}\\ { - 2}&4 \end{array}} \right]\left[ X \right] + \left[ {\begin{array}{{20}{c}} 1\\ 0 \end{array}} \right]u\)
D.	\(\left[ {\dot X} \right] = \left[ {\begin{array}{{20}{c}} -2&{ 4}\\ { 0}&-3 \end{array}} \right]\left[ X \right] + \left[ {\begin{array}{{20}{c}} 1\\ 0 \end{array}} \right]u\)
Answer» C. \(\left[ {\dot X} \right] = \left[ {\begin{array}{{20}{c}} 0&{ - 3}\\ { - 2}&4 \end{array}} \right]\left[ X \right] + \left[ {\begin{array}{{20}{c}} 1\\ 0 \end{array}} \right]u\)

14.	For the system governed by the set of equations:\(\begin{array}{l}\frac{{d{x_1}}}{{dt}} = 2{x_1} + {x_2} + u\\d{x_2}/dt = -2{x_1} + u\\y = 3{x_1}\end{array}\)the transfer function \(Y\left( s \right)/U\left( s \right)\) is given by
A.	\(\frac{{3\left( {s + 1} \right)}}{{{s^2}-2s + 2}}\)
B.	\(\frac{{s + 1}}{{{s^2}-2s + 1}}\)
C.	\(\frac{{3\left( {2s + 1} \right)}}{{{s^2}\;-2s + 1}}\)
D.	\(\frac{{3\left( {2s + 1} \right)}}{{{s^2}\;-2s + 2}}\)
Answer» B. \(\frac{{s + 1}}{{{s^2}-2s + 1}}\)

15.	A second-order LTI system is described by the following state equations.\(\frac{d}{{dt}}{x_1}\left( t \right) - {x_2}\left( t \right) = 0\)\(\frac{d}{{dt}}{x_2}\left( t \right) + 2{x_1}\left( t \right) + 3{x_2}\left( t \right) = r\left( t \right)\) where x1(t) and x2(t) are the two state variables and r(t) denotes the input. The output c(t) = x1(t). The system is
A.	undamped (oscillatory)
B.	underdamped
C.	critically damped
D.	overdamped
Answer» E.

17.	Consider the state space system expressed by the signal flow diagram shown in the figure.The corresponding system is
A.	always controllable
B.	always observable
C.	always stable
D.	always unstable
Answer» B. always observable

18.	Let the state-space representation of an LTI system be ẋ(t) = A x(t) + B u(t), y(t) = C x(t) + d u(t) where A, B, C are matrices, d is a scalar, u(t) is the input to the system, and y(t) is its output. Let B = [0 0 1]T and d = 0. Which one of the following options for A and C will ensure that the transfer function of this LTI system is \(H\left( s \right) = \frac{1}{{{s^3} + 3{s^2} + 2s + 1}} = ?\)
A.	\(A = \left[ {\begin{array}{{20}{c}} 0&1&0\\ 0&0&1\\ { - 1}&{ - 2}&{ - 3} \end{array}} \right]and\;C = \;\left[ {\begin{array}{{20}{c}} 1&0&0 \end{array}} \right]\)
B.	\(A = \left[ {\begin{array}{{20}{c}} 0&1&0\\ 0&0&1\\ { - 3}&{ - 2}&{ - 1} \end{array}} \right]and\;C = \;\left[ {\begin{array}{{20}{c}} 1&0&0 \end{array}} \right]\)
C.	\(A = \left[ {\begin{array}{{20}{c}} 0&1&0\\ 0&0&1\\ { - 1}&{ - 2}&{ - 3} \end{array}} \right]and\;C = \;\left[ {\begin{array}{{20}{c}} 0&0&1 \end{array}} \right]\)
D.	\(A = \left[ {\begin{array}{{20}{c}} 0&1&0\\ 0&0&1\\ { - 3}&{ - 2}&{ - 1} \end{array}} \right]and\;C = \;\left[ {\begin{array}{{20}{c}} 0&0&1 \end{array}} \right]\)
Answer» B. \(A = \left[ {\begin{array}{{20}{c}} 0&1&0\\ 0&0&1\\ { - 3}&{ - 2}&{ - 1} \end{array}} \right]and\;C = \;\left[ {\begin{array}{{20}{c}} 1&0&0 \end{array}} \right]\)

19.	Consider the following standard state-space description of a linear time-invariant single input single output system:\(\dot x = Ax + Bu,y = Cx + Du\)Which one of the following statements about the transfer function should be true if D ≠ 0?
A.	The system is stable
B.	The system is strictly proper
C.	The system is low pass
D.	The system is of type zero
Answer» B. The system is strictly proper

22.	Consider a system governed by the following equations\(\frac{{d{x_1}\left( t \right)}}{{dt}} = {x_2}\left( t \right) - {x_1}\left( t \right)\)\(\frac{{d{x_2}\left( t \right)}}{{dt}} = {x_1}\left( t \right) - {x_2}\left( t \right)\)The initial conditions are such that \({x_1}\left( 0 \right) < {x_2}\left( 0 \right) < \;\infty .\) Let \({x_{1f}} = \mathop {\lim }\limits_{t \to \infty } {x_1}\left( t \right)\) and \({x_{2f}} = \mathop {\lim }\limits_{t \to \infty } {x_2}\left( t \right)\). Which one of the following is true?
A.	\({x_{1f}} < {x_{2f}} < \infty\)
B.	\({x_{2f}} < {x_{1f}} < \infty\)
C.	\({x_{1f}} = \;{x_{2f}} < \infty\)
D.	\({x_{1f}} = {x_{2f}} = \infty\)
Answer» D. \({x_{1f}} = {x_{2f}} = \infty\)

23.	Consider the following properties attributed to state model of a system:1. State model is unique.2. Transfer function for the system is unique.3. State model can be derived from transfer function of the system.Which of the above statements are correct?
A.	1, 2 and 3
B.	1 and 2 only
C.	2 and 3 only
D.	1 and 3 only
Answer» D. 1 and 3 only

24.	A second-order system represented by state variables has \(A = \left[ {\begin{array}{*{20}{c}} -2&-4\\ 1&0 \end{array}} \right]\)The values of natural frequency and damping factor are respectively
A.	2 and 0.5
B.	2 and 1
C.	1 and 2
D.	0.5 and 2
Answer» B. 2 and 1

Explore topic-wise MCQs in Digital Signal Processing.

The parallel form realization is also known as normal form representation.

The determinant |F-λI|=0 yields the characteristic polynomial of the matrix F.

What is the condition to call a number λ is an Eigen value of F and a nonzero vector U is the associated Eigen vector?

A closed form solution of the state space equations is easily obtained when the system matrix F is?

If we interchange the rows and columns of the matrix F, then the system is called as ______________

From the definition of state of a system, the system consists of only one component called memory less component.

Find the dynamics of the circuit.

Dynamics of system is described by \(\frac{{m{d^2}y}}{{dt}} + \frac{{bdy}}{{dt}} + ky = u\) where m, b, k are constants, y, u are time varying quantities. How many state variables are required for this system to be represented in state-space form?

For the given circuit, which one of the following is the correct state equation?

A network is described by the state model as\({{\dot{x}}_{1}}=2{{x}_{1}}-{{x}_{2}}+3u\)\({{\dot{x}}_{2}}=-4{{x}_{2}}-u\)\(y=3{{x}_{1}}-2{{x}_{2}}\)The transfer function H(s) \(\left( { = \frac{{Y\left( s \right)}}{{U\left( s \right)}}} \right)\) is

Consider the linear system \(\dot x = \left[ {\begin{array}{*{20}{c}}{ - 1}&0\\0&{ - 2}\end{array}} \right]x\), with initial condition \(x\left( 0 \right) = \left[ {\begin{array}{*{20}{c}}1\\1\end{array}} \right]\). The solution x(t) for this system is

From the figure, obtain state equation

For the system governed by the set of equations:\(\begin{array}{l}\frac{{d{x_1}}}{{dt}} = 2{x_1} + {x_2} + u\\d{x_2}/dt = -2{x_1} + u\\y = 3{x_1}\end{array}\)the transfer function \(Y\left( s \right)/U\left( s \right)\) is given by

A system has the state variable representation \(\dot X = \begin{bmatrix} 0 & 1 \\\ 0 & -1 \end{bmatrix} X + \begin{bmatrix} 0 \\\ 1 \end{bmatrix} u : y = \begin{bmatrix} 1 & 1 \end{bmatrix}X\). Its transfer function is

Consider the state space system expressed by the signal flow diagram shown in the figure.The corresponding system is

Consider the following standard state-space description of a linear time-invariant single input single output system:\(\dot x = Ax + Bu,y = Cx + Du\)Which one of the following statements about the transfer function should be true if D ≠ 0?

Number of state variables of discrete-time system, described by\(y[n] - \frac{3}{4}y[n - 1] + \frac{1}{8}y[n - 2] = x[n]\) is

All the constant – N loci in G-plane intersect the real axis in points

Consider the following properties attributed to state model of a system:1. State model is unique.2. Transfer function for the system is unique.3. State model can be derived from transfer function of the system.Which of the above statements are correct?

A second-order system represented by state variables has \(A = \left[ {\begin{array}{*{20}{c}} -2&-4\\ 1&0 \end{array}} \right]\)The values of natural frequency and damping factor are respectively

Bounded-input bounded-output stability implies asymptotic stability for1. Completely controllable system2. Completely observable system3. Uncontrollable system4. Unobservable systemWhich of the above statements are correct?

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 output. When would the SISO system derived from this diagram be controllable and observable?

In the state variable model of a system, the square matrix ϕ(t) that converts the initial states of the system x(0) to a new state x(t) at a later time t is called:

In the state model of a linear system the output equation is represented as:

A system is represented by \(3\dfrac{dy}{dt}+2y=u\), what is the transfer function to the system?

A state-space representation of a system is given by:\(A=\left[ \begin{matrix} 0 & 1 \\ -2 & 0 \\ \end{matrix} \right],~y=\left[ 1~-1 \right],~x\left( 0 \right)=\left[ \begin{matrix} 1 \\ 1 \\ \end{matrix} \right]\)The time response of the system will be ______.

Consider a system described by the state model\(\dot{X}=\left[ \begin{matrix} 2 & 1 \\ -1 & 2 \\ \end{matrix} \right] X + \left[ \begin{matrix} 1 \\ 1 \\ \end{matrix} \right] U\)Y = [1 1] XThe system is

How many vector equations forms the state model of the linear system?

THE_PARALLEL_FORM_REALIZATION_IS_ALSO_KNOWN_AS_NORMAL_FORM_REPRESENTATION.?$

The determinant |F- ‚âà√≠¬¨‚Ñ¢I|=0 yields the characteristic polynomial of the matrix F?#

What is the condition to call a number ‚âà√≠¬¨‚Ñ¢ is an Eigen value of F and a nonzero vector U is the associated Eigen vector?#

A closed form solution of the state space equations is easily obtained when the system matrix F is:

A single input-single output system and its transpose have identical impulse responses and hence the same input-output relationship.

If we interchange the rows and columns of the matrix F, then the system is called as:

From the definition of state of a system ,the system consists of only one component called memory less component.

Which of the following gives the complete definition of the state of a system at time n0?

State variables provide information about all the internal signals in the system.

The state space or the internal description of the system still involves a relationship between the input and output signals, what are the additional set of variables it also involves?

Consider the linear system \(\dot x = \left[ {\begin{array}{{20}{c}}{ - 1}&0\\0&{ - 2}\end{array}} \right]x\), with initial condition \(x\left( 0 \right) = \left[ {\begin{array}{{20}{c}}1\\1\end{array}} \right]\). The solution x(t) for this system is

26.	Consider a state-variable model of a system\(\left[ {\begin{array}{{20}{c}} {{{\dot x}_1}}\\ {{{\dot x}_2}} \end{array}} \right] = \left[ {\begin{array}{{20}{c}} 0&1\\ { - \alpha }&{ - 2\beta } \end{array}} \right]\left[ {\begin{array}{{20}{c}} {{x_1}}\\ {{x_2}} \end{array}} \right] + \left[ {\begin{array}{{20}{c}} 0\\ \alpha \end{array}} \right]r\)\(y = \left[ {\begin{array}{{20}{c}} 1&0 \end{array}} \right]\left[ {\begin{array}{{20}{c}} {{x_1}}\\ {{x_2}} \end{array}} \right]\)where y is the output, and r is the input. The damping ratio ξ and the undamped natural frequency ωn (rad/sec) of the system are given by
A.	\({\rm{\xi }} = \frac{{\rm{\beta }}}{{\sqrt \alpha }};{\omega _n} = \sqrt \alpha \)
B.	\({\rm{\xi }} = \sqrt \alpha ;{\omega _n} = \frac{\beta }{{\sqrt \alpha }}\)
C.	\({\rm{\xi }} = \frac{{\sqrt \alpha }}{\beta };{\omega _n} = \sqrt \beta\)
D.	\({\rm{\xi }} = \sqrt \beta ;{\omega _n} = \sqrt \alpha\)
Answer» B. \({\rm{\xi }} = \sqrt \alpha ;{\omega _n} = \frac{\beta }{{\sqrt \alpha }}\)

27.	A system is described by\(\dot x\left( t \right) = \left[ {\begin{array}{{20}{c}}0&1&0\\3&0&2\\{ - 12}&{ - 7}&{ - 6}\end{array}} \right]x\left( t \right) + \left[ {\begin{array}{{20}{c}}1\\0\\2\end{array}} \right]u\left( t \right)\)\(y\left( t \right) = \left[ {\begin{array}{*{20}{c}}1&0&0\end{array}} \right]x\left( t \right)\)Its Eigenvalues are
A.	1, 2, 3
B.	2, 2, 3
C.	12, 6, 7
D.	-1, -2, -3
Answer» E.

29.	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 output. When would the SISO system derived from this diagram be controllable and observable?
A.	When \({u_1}\) is the only input and \({y_1}\) is the only output
B.	When \({u_2}\) is the only input and \({y_1}\) is the only output
C.	When \({u_1}\) is the only input and \({y_2}\) is the only output
D.	When \({u_2}\) is the only input and \({y_2}\) is the only output
Answer» C. When \({u_1}\) is the only input and \({y_2}\) is the only output

30.	Consider the state – space model\(\left[ {\begin{array}{{20}{c}} {{{\dot x}_1}}\\ {{{\dot x}_2}}\\ {{{\dot x}_3}} \end{array}} \right] = \left[ {\begin{array}{{20}{c}} { - 1}&1&0\\ 0&{ - 1}&0\\ 0&0&{ - 2} \end{array}} \right]\left[ {\begin{array}{{20}{c}} {{x_1}}\\ {{x_2}}\\ {{x_3}} \end{array}} \right] + \left[ {\begin{array}{{20}{c}} 0\\ 4\\ 0 \end{array}} \right]u\)\(y = \left[ {\begin{array}{{20}{c}} 1&1&1 \end{array}} \right]\left[ {\begin{array}{{20}{c}} {{x_1}}\\ {{x_2}}\\ {{x_3}} \end{array}} \right]\)System is
A.	Controllable and observable
B.	Uncontrollable and observable
C.	Uncontrollable and Unobservable
D.	Controllable and Unobservable
Answer» C. Uncontrollable and Unobservable

31.	In the state variable model of a system, the square matrix ϕ(t) that converts the initial states of the system x(0) to a new state x(t) at a later time t is called:
A.	controller matrix
B.	process matrix
C.	state transition matrix
D.	plant matrix
Answer» D. plant matrix

32.	In the state model of a linear system the output equation is represented as:
A.	Y(t) = C X(t) + D U(t)
B.	Ẋ(t) = A X(t) + B U̇(t)
C.	Y(t) = A X(t) + B U(t)
D.	Ẋ(t) = C X(t) + D U̇(t)
Answer» B. Ẋ(t) = A X(t) + B U̇(t)

33.	A system is represented by \(3\dfrac{dy}{dt}+2y=u\), what is the transfer function to the system?
A.	\(\dfrac{Y\left( s\right)}{U\left( s\right)}=\dfrac{1}{s^2+2s+3}\)
B.	\(\dfrac{Y\left( s\right)}{U\left( s\right)}=\dfrac{1}{3s+2}\)
C.	\(\dfrac{Y\left( s\right)}{U\left( s\right)}=\dfrac{s+1}{3s+1}\)
D.	\(\dfrac{Y\left( s\right)}{U\left( s\right)}=\dfrac{1}{2s+3}\)
Answer» C. \(\dfrac{Y\left( s\right)}{U\left( s\right)}=\dfrac{s+1}{3s+1}\)

34.	A state-space representation of a system is given by:\(A=\left[ \begin{matrix} 0 & 1 \\ -2 & 0 \\ \end{matrix} \right],~y=\left[ 1~-1 \right],~x\left( 0 \right)=\left[ \begin{matrix} 1 \\ 1 \\ \end{matrix} \right]\)The time response of the system will be ______.
A.	sin √2t
B.	\(-\frac{1}{\sqrt{2}}\sin \sqrt{2}t\)
C.	\(\frac{3}{\sqrt{2}}\sin \sqrt{2}t\)
D.	\(\frac{1}{\sqrt{2}}\sin \sqrt{2}t\)
Answer» D. \(\frac{1}{\sqrt{2}}\sin \sqrt{2}t\)

36.	A discrete system is represented by the difference equation\(\left[ {\begin{array}{{20}{c}}{{X_1}\left( {k + 1} \right)}\\{{X_2}\left( {k + 1} \right)}\end{array}} \right] = \left[ {\begin{array}{{20}{c}}a&{a - 1}\\{a + 1}&a\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{X_1}\left( k \right)}\\{{X_2}\left( k \right)}\end{array}} \right]\)It has initial conditions X1(0) = 1; X2(0) = 0. The pole locations of the system for a = 1, are
A.	1 ± j0
B.	-1 ± j0
C.	±1 + j0
D.	0 ± j1
Answer» B. -1 ± j0

37.	How many vector equations forms the state model of the linear system?
A.	4
B.	2
C.	3
D.	1
Answer» C. 3