## Control Systems - Dec 2013

### 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)** Define a control system. Explain with examples, open loop and closed loop control systems. List the merits and demerits of open loop and closed loop control system.(10 marks)
**1 (b)** For the mechanical system shown in Fig. Q1(b).

i) Draw the mechanical network

ii) Write the differential equations description the system.

iii) Draw the F-V analogous electrical circuit after writing the corresponding electrical equations.
(10 marks)
**2 (a)** For the circuit shown in Fig Q2(a). Draw the block diagram and determine the transfer function $$ \dfrac {V_p(S)}{V_1(S)} $$ using block diagram rules.
(10 marks)
**2 (b)** For the system represented by the following equations, find the transfer function X(s)/U(s) by signal flow graph techniques

$$ x=x_1+\alpha_3 U \\ x_1=-\beta_1x_1+x_2+\alpha_2U \\ x_2=-\beta_2x_1+\alpha_1U $$(10 marks)
**3 (a)** Explain the following time domain specification of a second order systems, with neat sketch i) Peak time ii) Delay time iii) Rise time iv) Maximum over shoot

v) Settling time.(6 marks)
**3 (b)** A system description by $$ \dfrac {d^2 y}{dt^2}+ \dfrac {8dy}{dt}+ 25y(t)=50x(t) $$ Evaluate the response and maximum output for a step of 2.5 units.(8 marks)
**3 (c)** In the block shown in Fig. Q3(c) G(s)=A/S^{2} and H(s)=(ms+n). For A=10, determine the values of m and n for a step input with a time constant 0.1 sec; which give a peak over shoot of 30%.
(6 marks)
**4 (a)** What are the difficulties encountered while assessing Routh-Hurwitz criteria and how do you eliminate these difficulties, explain with examples.(6 marks)
**4 (b)** The open loop transfer function of a feedback control system is given by $$ G(s)H(s)= \dfrac {k}{S(s+4)(s^2+2s+2)} $$

i) Using R-H criterion determine the range of K" for which the system will be stable

ii) If a zero at S=-4 is added to the forward transfer function(8 marks)
**4 (c)** Using R-H criterion, find the stability of a unity feedback system having closed loop transfer function $$ G(s)= \dfrac {e^{-s1}}{S(s+2)} $$(6 marks)
**5 (a)** State the different rules for the construction of root locus.(8 marks)
**5 (b)** A feedback control system has open loop transfer function: $$ G(S)H(S)= \dfrac {k}{S(s+4)(s^2+4s+20)} $$ Plot the root locus for K=0 to ? indicate all point on it.(12 marks)
**6 (a)** Explain co-relation between time domain and frequency domain for second order systems.(6 marks)
**6 (b)** The open loop transfer function of unity feedback control system is given by $$ G(s)H(s)= \dfrac {k}{s(1+0.00 ls)(1+0.25s)(1+0.1s)} $$ Determine the value of K, so that the system will have a phase margin of 40°, what will be the gain margin. Use code plot.(14 marks)
**7 (a)** State and explain Nyquist stability criterion.(6 marks)
**7 (b)** Using Nyquist stability criterion, find the range of K for closed-loop stability $$ G(s)H(s)= \dfrac{K}{S(s^2+2s+2)}K>0 $$(14 marks)
**8 (a)** Explain properties and significance of state transition matrix.(10 marks)
**8 (b)** A linear time invariant system is characterized by the homogeneous state equation: $$ \begin{bmatrix} \dot{x_1} \\ \dot{x_2} \end{bmatrix} = \begin{bmatrix} 1 &0 \\1 &1 \end{bmatrix} \begin{bmatrix}x_1\\x_2\end{bmatrix} $$ Compare the solution of homogeneous equation assume the initial state vector.(10 marks)