Questions & Answers of Control Systems

The phase cross-over frequency of the transfer function $G\left(S\right)=\frac{100}{{\left(S+1\right)}^3}$ in rad/s is

•  (A)  $\sqrt{3}$
•  (B)  $\frac{1}{\sqrt{3}}$
•  (C)  $3$
•  (D)  $3\sqrt{3}$

Consider the following asymptotic Bode magnitude plot (ω is in rad/s).

•  (A)  $\frac{2s}{\left(1+0.5s\right){\left(1+0.25s\right)}^2}$
•  (B)  $\frac{4\left(1+0.5s\right)}{s\left(1+0.25s\right)}$
•  (C)  $\frac{2s}{\left(1+2s\right)\left(1+4s\right)}$
•  (D)  $\frac{4s}{\left(1+2s\right){\left(1+4s\right)}^2}$

•  (A)  once in clockwise direction
•  (B)  twice in clockwise direction
•  (C)  once in anticlockwise direction
•  (D)  twice in anticlockwise direction

Given the following polynomial equation

$s^3+5.5 s^2+8.5 s+3=0$

the number of roots of the polynomial, which have real parts strictly less than −1, is ________.

The open loop transfer function of a unity feedback control system is given by

$G\left(s\right)=\frac{K\left(s+1\right)}{s\left(1+Ts\right)\left(1+2s\right)}, K>0,t>0.$

The closed loop system will be stable if,

•  (A)  $0<T<\frac{4\left(K+1\right)}{K-1}$
•  (B)  $0<K<\frac{4\left(T+2\right)}{T-2}$
•  (C)  $0<K<\frac{T+2}{T-2}$
•  (D)  $0<T<\frac{8\left(K+1\right)}{K-1}$

•  (A)  $e^{{\lambda }_{1}t}\alpha$
•  (B)  $e^{{\lambda }_{2}t}\beta$
•  (C)  $e^{{\lambda }_{2}t}\alpha$
•  (D)  $e^{{\lambda }_{1}t}\alpha +e^{{\lambda }_{2}t}\beta$

•  (A)  $\frac{1.02}{s^2+5s+100}$
•  (B)  $\frac{102}{s^2+10s+100}$
•  (C)  $\frac{100}{s^2+10s+100}$
•  (D)  $\frac{102}{s^2+5s+100}$

•  (A)  1
•  (B)  2
•  (C)  5
•  (D)  9

A Bode magnitude plot for the transfer function G(s) of a plant is shown in the figure. Which one of the following transfer functions best describes the plant?

•  (A)  $\frac{1000\left(s+10\right)}{s+1000}$
•  (B)  $\frac{10\left(s+10\right)}{s(s+1000)}$
•  (C)  $\frac{s+1000}{10s(s+10)}$
•  (D)  $\frac{s+1000}{10(s+10)}$

For the signal-flow graph shown in the figure, which one of the following expressions is equal to the transfer function ${\overline{)\frac{Y\left(s\right)}{X_2\left(s\right)}}}_{X_1\left(s\right)=0}$?

•  (A)  $\frac{G_1}{1+G_2\left(1+G_1\right)}$
•  (B)  $\frac{G_2}{1+G_1\left(1+G_2\right)}$
•  (C)  $\frac{G_1}{1+G_1{G}_2}$
•  (D)  $\frac{G_2}{1+G_1{G}_2}$

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 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.

The transfer function of a second order real system with a perfectly flat magnitude response of unity has a pole at (2-j3). List all the poles and zeroes.

•  (A)  Poles at $\left(2\pm j3\right)$, no zeroes.
•  (B)  Poles at $\left(\pm 2-j3\right)$, one zero at origin.
•  (C)  Poles at $\left(2-j3\right), \left(-2+j3\right)$, zeroes at $\left(-2-j3\right),\left(2+j3\right)$.
•  (D)  Poles at $\left(2\pm j3\right)$,zeroes at $\left(-2\pm j3\right)$.

Find the transfer function $\frac{Y\left(s\right)}{X\left(s\right)}$ of the system given below.

•  (A)  $\frac{G_1}{1-H{G}_1}+\frac{G_2}{1-H{G}_2}$
•  (B)  $\frac{G_1}{1+H{G}_1}+\frac{G_2}{1+H{G}_2}$
•  (C)  $\frac{G_1+G_{\mathit{2}}}{1+H\left({\mathit{G}}_{\mathit{1}}\mathit{+}{\mathit{G}}_{\mathit{2}}\right)}$
•  (D)  $\frac{G_1+G_{\mathit{2}}}{1-H\left({\mathit{G}}_{\mathit{1}}\mathit{+}{\mathit{G}}_{\mathit{2}}\right)}$

The open loop poles of a third order unity feedback system are at 0,-1,-2. Let the frequency corresponding to the point where the root locus of the system transits to unstable region be K. Now suppose we introduce a zero in the open loop transfer function at -3, while keeping all the earlier open loop poles intact. Which one of the following is TRUE about the point where the root locus of the modified system transits to unstable region?

•  (A)  It corresponds to a frequency greater than K
•  (B)  It corresponds to a frequency less than K
•  (C)  It corresponds to a frequency K
•  (D)  Root locus of modified system never transits to unstable region

An open loop control system results in a response of $e^{-2t}\left(\sin 5t+\cos 5t\right)$ for a unit impulse input. The DC gain of the control system is _________.

Nyquist plot of two functions  $G_1$(s) and $G_2$(s) are shown in figure.

Nyquist plot of the product of $G_1$ (s) and $G_2$(s) is

•  (A) 
•  (B) 
•  (C) 
•  (D) 

The unit step response of a system with the transfer function $G\left(s\right)=\frac{1-2s}{1+s}$ is given by which one of the following waveforms?

•  (A) 
•  (B) 
•  (C) 
•  (D) 

An open loop transfer function G(s) of a system is
$G\left(s\right)=\frac{K}{s\left(s+1\right)\left(s+2\right)}$
For a unity feedback system, the breakaway point of the root loci on the real axis occurs at,

•  (A)  -0.42
•  (B)  -1.58
•  (C)  -0.42 and -1.58
•  (D)  none of the above

For the system governed by the set of equations:

the transfer function Y(s)/U(s) is given by

•  (A)  $3\left(s+1\right)/\left(s^2-2s+2\right)$
•  (B)  $3\left(2s+1\right)/\left(s^2-2s+1\right)$
•  (C)  $\left(s+1\right)/\left(s^2-2s+1\right)$
•  (D)  $3\left(2s+1\right)/\left(s^2-2s+2\right)$

In the formation of Routh-Hurwitz array for a polynomial, all the elements of a row have zero values. This premature termination of the array indicates the presence of

•  (A) only one root at the origin
•  (B) imaginary roots
•  (C) only positive real roots
•  (D) only negative real roots

The root locus of a unity feedback system is shown in the figure

The closed loop transfer function of the system is

•  (A) $\frac{C\left(s\right)}{R\left(s\right)}=\frac{K}{\left(s+1\right)\left(s+2\right)}$
•  (B) $\frac{C\left(s\right)}{R\left(s\right)}=\frac{-K}{\left(s+1\right)\left(s+2\right)+K}$
•  (C) $\frac{C\left(s\right)}{R\left(s\right)}=\frac{K}{\left(s+1\right)\left(s+2\right)-K}$
•  (D) $\frac{C\left(s\right)}{R\left(s\right)}=\frac{K}{\left(s+1\right)\left(s+2\right)+K}$

For the given system, it is desired that the system be stable. The minimum value of $\alpha$ for this condition is _______________.

The Bode magnitude plot of the transfer function $G\left(S\right)=\frac{K\left(1+0.5s\right)\left(1+as\right)}{s\left(1+{\displaystyle \frac{s}{8}}\right)\left(1+bs\right)\left(1+{\displaystyle \frac{s}{36}}\right)}$ is shown below: Note that -6 dB/octave = -20 dB/decade. The value of $\frac{a}{bk}$ is_____________.

The closed-loop transfer function of a system is $\mathrm{T}\left(\mathrm{S}\right)=\frac{4}{\left({\mathrm{s}}^2+0.4\mathrm{s}+4\right)}$. The steady state error due to unit step input is __________.

The state transition matrix for the system

$\left[\begin{array}{c}\stackrel{.}{x_1}\\ \stackrel{.}{x_2}\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+\left[\begin{array}{c}1\\ 1\end{array}\right]u$

is

•  (A) $\left[\begin{array}{cc}{e}^t& 0\\ e^t& e^t\end{array}\right]$
•  (B) $\left[\begin{array}{cc}{e}^t& 0\\ t^2{e}^t& e^t\end{array}\right]$
•  (C) $\left[\begin{array}{cc}{e}^t& 0\\ t{e}^t& e^t\end{array}\right]$
•  (D) $\left[\begin{array}{cc}{e}^t& t{e}^t\\ 0& e^t\end{array}\right]$

A system with the open loop transfer function

$G\left(S\right)=\frac{K}{s\left(s+2\right)\left(s^2+2s+2\right)}$

is connected in a negative feedback configuration with a feedback gain of unity. For the closed loop system to be marginally stable, the value of K is ______

For the transfer function

$G\left(S\right)=\frac{5\left(s+4\right)}{s\left(s+0.25\right)\left(s^2+4s+25\right)}$

The values of the constant gain term and the highest corner frequency of the Bode plot respectively are

•  (A) 3.2, 5.0
•  (B) 16.0, 4.0
•  (C) 3.2, 4.0
•  (D) 16.0, 5.0

The second order dynamic system

$\frac{dx}{dt}=PX+Qu$

$y=RX$

has the matrices P, Q and R as follows:

$P=\left[\begin{array}{cc}-1& 1\\ 0& -3\end{array}\right] Q=\left[\begin{array}{c}0\\ 1\end{array}\right] R=\left[\begin{array}{cc}0& 1\end{array}\right]$

The system has the following controllability and observability properties:

•  (A) Controllable and observable
•  (B) Not controllable but observable
•  (C) Controllable but not observable
•  (D) Not controllable and not observable

The signal flow graph of a system is shown below. U(s) is the input and C(s) is the output.

Assuming, h₁=b₁ and h₀=b₀-b₁a₁, the input-output transfer function, $G\left(S\right)=\frac{C\left(s\right)}{U\left(s\right)}$ of the system is given by

•  (A) $G\left(S\right)=\frac{b_{0}s+b_1}{s^2+a_{0}s+a_1}$
•  (B) $G\left(S\right)=\frac{a_{1}s+a_0}{s^2+b_{1}s+b_0}$
•  (C) $G\left(S\right)=\frac{b_{1}s+b_0}{s^2+a_{1}s+a_0}$
•  (D) $G\left(S\right)=\frac{a_{0}s+a_1}{s^2+b_{0}s+b_1}$

A single-input single-output feedback system has forward transfer function $G\left(s\right)$ and feedback transfer function $H\left(s\right)$. It is given that $\left|G\left(s\right)H\left(s\right)\right|<1.$ Which of the following is true about the stability of the system?

•  (A) The system is always stable
•  (B) The system is stable if all zeros of $G\left(s\right)$$H\left(s\right)$ are in left half of the s-plane
•  (C) The system is stable if all poles of $G\left(s\right)$$H\left(s\right)$ are in left half of the s-plane
•  (D) It is not possible to say whether or not the system is stable from the information given

The block diagram of a system is shown in the figure

If the desired transfer function of the system is

$\frac{C\left(s\right)}{R\left(s\right)}=\frac{s}{s^2+s+1}$

then G(s) is

•  (A) 1
•  (B) s
•  (C) 1/s
•  (D) $\frac{-s}{s^3+s^2-s-2}$

Consider the system described by following state space equations

$\left[\frac{{\displaystyle \stackrel{.}{x_1}}}{\stackrel{.}{x_2}}\right]=\left[\begin{array}{cc}0& 1\\ -1& -1\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+\left[\begin{array}{c}0\\ 1\end{array}\right]u; y=\left[\begin{array}{cc}1& 0\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]$

If u is unit step input, then the steady state error of the system is

•  (A) 0
•  (B) 1/2
•  (C) 2/3
•  (D) 1

The magnitude Bode plot of a network is shown in the figure

The maximum phase angle $\phi_m$ and the corresponding gain G_m respectively, are

•  (A) −30°and 1.73dB
•  (B) −30°and 4.77dB
•  (C) +30°and 4.77dB
•  (D) +30°and 1.73dB

Assuming zero initial condition, the response y(t) of the system given below to a unit step input u(t) is

•  (A) $u\left(t\right)$
•  (B) $tu\left(t\right)$
•  (C) $\frac{t^2}{2}u\left(t\right)$
•  (D) $e^{-t}u\left(t\right)$

The Bode plot of a transfer function G(s) is shown in the figure below.

The gain $(20log\left|G\left(s\right)\right|)$ is 32 dB and –8 dB at 1 rad/s and 10 rad/s respectively. The phase is negative for all ω. Then G(s) is

•  (A) $\frac{39.8}{s}$
•  (B) $\frac{39.8}{s^2}$
•  (C) $\frac{32}{s}$
•  (D) $\frac{32}{s^2}$

The open-loop transfer function of a dc motor is given as $\frac{\omega \left(s\right)}{V_a\left(s\right)}=\frac{10}{1+10s}$ When connected in feedback as shown below, the approximate value of K_a that will reduce the time constant of the closed loop system by one hundred times as compared to that of the open-loop system is

•  (A) 1
•  (B) 5
•  (C) 10
•  (D) 100

The signal flow graph for a system is given below. The transfer function $\frac{Y\left(s\right)}{U\left(s\right)}$ for this system is

•  (A) $\frac{s+1}{5s^2+6s+2}$
•  (B) $\frac{s+1}{s^2+6s+2}$
•  (C) $\frac{s+1}{s^2+4s+2}$
•  (D) $\frac{1}{5s^2+6s+2}$

The state variable formulation of a system is given as

$\left[\begin{array}{c}\stackrel{.}{x_1}\\ \stackrel{.}{x_2}\end{array}\right]=\left[\begin{array}{cc}-2& 0\\ 0& -1\end{array}\right] \left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+\left[\begin{array}{c}1\\ 1\end{array}\right]u, x_1\left(0\right)=0, x_2\left(0\right)=0 and y=\left[\begin{array}{cc}1& 0\end{array}\right] \left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]$

The system is

•  (A) controllable but not observable
•  (B) not controllable but observable
•  (C) both controllable and observable
•  (D) both not controllable and not observable

The state variable formulation of a system is given as

$\left[\begin{array}{c}\stackrel{.}{x_1}\\ \stackrel{.}{x_2}\end{array}\right]=\left[\begin{array}{cc}-2& 0\\ 0& -1\end{array}\right] \left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+\left[\begin{array}{c}1\\ 1\end{array}\right]u, x_1\left(0\right)=0, x_2\left(0\right)=0 and y=\left[\begin{array}{cc}1& 0\end{array}\right] \left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]$

The response y(t) to a unit step input is

•  (A) $\frac{1}{2}-\frac{1}{2}{e}^{-2t}$
•  (B) $1-\frac{1}{2}{e}^{-2t}-\frac{1}{2}{e}^{-t}$
•  (C) $e^{-2t}-e^{-t}$
•  (D) $1-e^{-t}$

A system with transfer function

G(s)=$\frac{\left(s^2+9\right)\left(s+2\right)}{\left(s+1\right)\left(s+3\right)\left(s+4\right)}$

is excited by $\sin \left(\omega t\right)$. The steady-state output of the system is zero at

•  (A) $\omega$=1 rad/s
•  (B) $\omega$=2 rad/s
•  (C) $\omega$=3 rad/s
•  (D) $\omega$=4 rad/s

The state variable description of an LTI system is given by

$\left(\begin{array}{c}\stackrel{.}{x_1}\\ \stackrel{.}{x_2}\\ \stackrel{.}{x_3}\end{array}\right)= \left(\begin{array}{ccc}0& a_1& 0\\ 0& 0& a_2\\ a_3& 0& 0\end{array}\right) \left(\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right)+\left(\begin{array}{c}0\\ 0\\ 1\end{array}\right)u$

$y=\left(\begin{array}{ccc}1& 0& 0\end{array}\right)\left(\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right)$

where y is the output and u is the input. The system is controllable for

•  (A) $a_1\ne 0, a_2=0, a_3\ne 0$
•  (B) $a_1=0, a_2\ne 0, a_3\ne 0$
•  (C) $a_1=0, a_2\ne 0, a_3=0$
•  (D) $a_1\ne 0, a_2\ne 0, a_3=0$

The feedback system shown below oscillates at 2 rad/s when

•  (A) K = 2 and a = 0.75
•  (B) K = 3 and a = 0.75
•  (C) K = 4 and a = 0.5
•  (D) K = 2 and a = 0.5

The transfer function of a compensator is given as

$G_c\left(s\right)=\frac{s+a}{s+b}$.

 $G_c\left(s\right)$ is a lead compensator if

•  (A) a =1, b = 2
•  (B) a = 3, b = 2
•  (C) a = –3, b = –1
•  (D) a = 3, b = 1

The transfer function of a compensator is given as

$G_c\left(s\right)=\frac{s+a}{s+b}$.

The phase of the above lead compensator is maximum at

•  (A) $\sqrt{2}$ rad/s
•  (B) $\sqrt{3}$ rad/s
•  (C) $\sqrt{6}$ rad/s
•  (D) 1/$\sqrt{3}$ rad/s

The frequency response of a linear system G(jω) is provided in the tubular form below

The gain margin and phase margin of the system are

•  (A) 6 dB and 30^oc
•  (B) 6 dB and −30^oc
•  (C) −6 dB and 30^oc
•  (D) −6 dB and −30^oc

The steady state error of a unity feedback linear system for a unit step input is 0.1. The steady state error of the same system, for a pulse input  r (t) having a magnitude of 10 and a duration of one second, as shown in the figure is 

•  (A) 0
•  (B) 0.1
•  (C) 1
•  (D) 10

An open loop system represented by the transfer function $G\left(s\right)=\frac{\left(s-1\right)}{\left(s+2\right)\left(s+3\right)}$ is

•  (A) Stable and of the minimum phase type
•  (B) table and of the non - minimum phase type
•  (C) Unstable and of the minimum phase type
•  (D) Unstable and of non-minimum phase type

The open loop transfer function $G\left(s\right)$ of a unity feedback control system is given as

$G\left(s\right)=\frac{k\left(s+{\displaystyle \frac{2}{3}}\right)}{s^2\left(s+2\right)}$

From the root locus, it can be inferred that when k tends to positive infinity,

•  (A) Three roots with nearly equal real parts exist on the left half of the s-plane
•  (B) One real root is found on the right half of the s-plane
•  (C) The root loci cross the $j\omega$ axis for a finite value of k ; k $\ne$ 0
•  (D) Three real roots are found on the right half of the s-plane

The response $h\left(t\right)$ of a linear time invariant system to an impulse $\delta \left(t\right)$, under initially relaxed condition is $h\left(t\right)=e^{-t}+e^{-2t}$. The response of this system for a unit step input $u\left(t\right)$ is

•  (A) $u\left(t\right)+e^{-t}+e^{-2t}$
•  (B) $\left(e^{-t}+e^{-2t}\right)u\left(t\right)$
•  (C) $\left(1.5-e^{-t}-0.5e^{-2t}\right)u\left(t\right)$
•  (D) $e^{-t}\delta \left(t\right)+e^{-2t}u\left(t\right)$

A two loop position control system is shown below

The gain K of the Tacho-generator influences mainly the

•  (A) Peak overshoot
•  (B) Natural frequency of oscillation
•  (C) Phase shift of the closed loop transfer function at very low frequencies (ω → 0)
•  (D) Phase shift of the closed loop transfer function at very high frequencies (ω → ∞)

As shown in the figure, a negative feedback system has an amplifier of gain 100 with ±10% tolerance in the forward path, and an attenuator of value 9/100 in the feedback path. The overall system gain is approximately

•  (A) 10±1%
•  (B) 10±2%
•  (C) 10±5%
•  (D) 10±10%

For the system $\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$\left(s+1\right)$}\right.$ the approximate time taken for a step response to reach 98% of its final value is

•  (A) 1s
•  (B) 2s
•  (C) 4s
•  (D) 8s

The frequency response of G(s) = 1 / [s (s + 1) (s + 2) ] plotted in the complex G(jω) plane (for o < ω < ∞) is

•  (A) 
•  (B) 
•  (C) 
•  (D) 

The system   x   =  Ax  + Bu with   $A=\left[\begin{array}{cc}-1& 2\\ 0& 2\end{array}\right],B=\left[\begin{array}{c}0\\ 1\end{array}\right]$ is

•  (A) stable and controllable
•  (B) stable but uncontrollable
•  (C) unstable but controllable
•  (D) unstable and uncontrollable

The characteristic equation of a closed-loop system is s(s+1)(s+3)+k(s+2)=0, k>0. Which of the following statements is true?

•  (A) Its roots are always real
•  (B) It cannot have a breakaway point in the range −1<Re[s]  <0
•  (C) Two of its roots tend to infinity along the asymptotes Re[s] = -1
•  (D) It may have complex roots in the right half plane

The measurement system shown in the figure uses three sub-systems in cascade whose gains are specified as G₁,G₂ and $\frac{1}{G_3}$. The relative small errors associated with each respective subsystem G₁,G₂ and G₃ are ε₁,ε₂ and ε₃. The error associated with the output is:

•  (A) ${\epsilon }_1+{\epsilon }_2+\frac{1}{{\epsilon }_3}$
•  (B) $\frac{{\epsilon }_1·{\epsilon }_2}{{\epsilon }_3}$
•  (C) ${\epsilon }_1+{\epsilon }_2-{\epsilon }_3$
•  (D) ${\epsilon }_1+{\epsilon }_2+{\epsilon }_3$

The polar plot of an open loop stable system is shown below. The closed loop system is

•  (A) Always stable
•  (B) Marginally stable
•  (C) Unstable with one pole on the RH s-plane
•  (D) Unstable with two poles on the RH s-plane

The first two rows of Routh's tabulation of a third order equation are as follows.

$\begin{array}{ccc}{\mathrm{s}}^3& 2& 2\\ {\mathrm{s}}^2& 4& 4\end{array}$

This means there are

•  (A) two roots at s ± j and one root in right half s-plane
•  (B) two roots at s ± j2 and one root in left half s-plane
•  (C) two roots at s ± j2 and one root in right half s-plane
•  (D) two roots at s ± j  and one root in left half s-plane

The asymptotic approximation of the log-magnitude vs frequency plot of a system containing only real poles and zeros is shown. Its transfer function is

•  (A) $\frac{10\left(s+5\right)}{s\left(s+2\right)\left(s+25\right)}$
•  (B) $\frac{100\left(s+5\right)}{s^2\left(s+2\right)\left(s+25\right)}$
•  (C) $\frac{100\left(s+5\right)}{s\left(s+2\right)\left(s+25\right)}$
•  (D) $\frac{80\left(s+5\right)}{s^2\left(s+2\right)\left(s+25\right)}$

The unit-step response of a unity feedback system with open loop transfer function G(s) = K/((s + 1)(s + 2)) is shown in the figure. The value of K is

•  (A) 0.5
•  (B) 2
•  (C) 4
•  (D) 6

The open loop transfer function of a unity feedback system is given by G(s) = (e^-0.1s)/s. The gain margin of this system is

•  (A) 11.95 dB
•  (B) 17.67 dB
•  (C) 21.33 dB
•  (D) 23.9 dB

A system is described by the following state and output equations

$\frac{d{x}_1\left(t\right)}{dt}=-3x_1\left(t\right)+x_2\left(t\right)+2u\left(t\right)$

$\frac{d{x}_2\left(t\right)}{dt}=-2x_2\left(t\right)+u\left(t\right)$

$y\left(t\right)=x_1\left(t\right)$

where u(t) is the input and y(t) is the output

The system transfer function is

•  (A) $\frac{s+2}{s^2+5s-6}$
•  (B) $\frac{s+3}{s^2+5s+6}$
•  (C) $\frac{2s+5}{s^2+5s+6}$
•  (D) $\frac{2s-5}{s^2+5s-6}$

A system is described by the following state and output equations

$\frac{d{x}_1\left(t\right)}{dt}=-3x_1\left(t\right)+x_2\left(t\right)+2u\left(t\right)$

$\frac{d{x}_2\left(t\right)}{dt}=-2x_2\left(t\right)+u\left(t\right)$

$y\left(t\right)=x_1\left(t\right)$

where u(t) is the input and y(t) is the output

The state transition matrix of the above system is

•  (A) $\left[\begin{array}{cc}{e}^{-3t}& 0\\ e^{-2t}+e^{-3t}& e^{-2t}\end{array}\right]$
•  (B) $\left[\begin{array}{cc}{e}^{-3t}& e^{-2t}-e^{-3t}\\ 0& e^{-2t}\end{array}\right]$
•  (C) $\left[\begin{array}{cc}{e}^{-3t}& e^{-2t}+e^{-3t}\\ 0& e^{-2t}\end{array}\right]$
•  (D) $\left[\begin{array}{cc}{e}^{3t}& e^{-2t}-e^{-3t}\\ 0& e^{-2t}\end{array}\right]$

A function y(t) satisfies the following differential equation :

$\frac{dy\left(t\right)}{dt}+y\left(t\right)=\delta \left(t\right)$

where δ(t) is the delta function. Assuming zero initial condition, and denoting the unit step function by u(t), y(t) can be of the form

•  (A) e^t
•  (B) e^-t
•  (C) e^tu(t)
•  (D) e^-tu(t)