## Control Systems - Jun 2015

### Electronics & Communication (Semester 4)

TOTAL MARKS: 100

TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.

(2) Attempt any **four** from the remaining questions.

(3) Assume data wherever required.

(4) Figures to the right indicate full marks.
**1 (a)** With the help of neat block diagram, define open loop and closed loop control system.(4 marks)
**1 (b)** For a mechanical system shown in Fig. Q1(b) obtain force voltage analogous electrical network.
(8 marks)
**1 (c)** Draw the electrical network based on torque-current analogy and give all the performance equation for the Fig Q1(c).
(8 marks)
**2 (a)** Define the following terms related to signal flow graph with a neat schematic:

i) Forward path

ii) Feedback loop

iii) Self loop

iv) Source node.(6 marks)
**2 (b)** Obtain the transfer function for the block diagram, shown in Fig. Q2(b). Using:

i) Block diagram reducing technique

ii) Mason's gain formula.
(8 marks)
**2 (c)** For the signal flow graph shown in Fig. Q2(c), find the overall transfer function by:

i) Block diagram reduction technique.

ii) Verify the result by Mason's gain formula.
(8 marks)
**3 (a)** Define and derive the expression for: i) Rise time

ii) Peak overshoot of an under-damped second order control system subjected to step input.(6 marks)
**3 (b)** For a unit feedback control system with $$ G(s) = \dfrac {10 (s+2)} {s^2 (s+1)} $$ Find: i) The static error coefficients ii) Steady state error when the input is $$ R(s)= \dfrac {3}{8} - \dfrac {2}{s^2} + \dfrac {1}{3s^3} .$$(6 marks)
**3 (c)** A system is given by differential equation $$ \dfrac {d^2y}{dt^2} + 4\dfrac {dy}{dt} + 8y = 8x, $$ where y=output and x=input. Determine: i) Peak overshoot ii) Settling time iii) Peak time for unit step input.(8 marks)
**4 (a)** Explain Routh-Hurwitz criterion for determining the stability of the system and mention its limitations.(6 marks)
**4 (b)** For a system s^{4}+22s^{2}+10s^{2}+s+k=0, find k_{mar} and ω at k_{mar}.(6 marks)
**4 (c)** Determine the value of 'k' and 'b' so that the system whose open loop transfer function is: $$ G(s) = \dfrac {k(s+1)}{s^3+bs^2 + 3s+1} $$ oscillates at a frequency of oscillations of 2 rad/sec.(8 marks)
**5 (a)** For a unity feedback system, the open loop transfer function is given by: $$ G(s) = \dfrac {K} {s(s+2)(s^2+6s+25)} $$ i) Sketch the root locus for 0≤k≤∞ ii) At what value of 'k' the system becomes unstable

ii) At this point of instability, determine the frequency of oscillation of the system.(15 marks)
**5 (b)** Consider the system with $$ G(s)H(s) = \dfrac{k} {s(s+2)(s+4)} $$ find whether s=-0.75 is point on root locus or not angle condition.(5 marks)
**6 (a)** Explain the procedure for investigating the stability using Nyquist criterion.(5 marks)
**6 (b)** For a certain control system: $$ G(s) H(s) = \dfrac {k} {s(s+2)(s+10)} . $$ Sketch the Nyquist plot and hence calculate the range of value of 'k' for stability.(15 marks)
**7 (a)** Sketch the bode plot for the open loop transfer function: $$ G(s)H(s)= \dfrac {k(1+0.2s)(I+0.025s)}{s^3 (1+0.001s)(1+0.005s)} , $$ Find the range of 'k' for closed loop stability(14 marks)
**7 (b)** Explain the following as applied to bode plots:

i) Gain margin ii) Phase margin iii) Gain and phase cross over frequency.(6 marks)
**8 (a)** Define the following terms: i) State ii) State variable iii) State space iv) State transition.(4 marks)
**8 (b)** A system is described by the differential equation, $$ \dfrac {d^3y}{dt^3}+ \dfrac {3d^2y}{dt^2} + \dfrac {17dy}{dt}+ 5y = 10u(t)), $$ where 'y' is the output and 'u' is input to the system. Determine the state space representation of the system.(6 marks)
**8 (c)** Obtain the state equations for the electrical network shown in Fig. Q8(c).
(10 marks)