## Control Systems - Dec 2015

### Electronics & Telecom Engineering (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.

### Solve any one question from Q1 and Q2

**1 (a)** Consider the R-L-C network shown in Fig. 1:

i) Obtain transfer function if V_{i} and V_{o} are input and output voltage respectively.

ii) Find the location of poles in terms of R, L and C.

iii) If R=1 MΩ, C=1 μF, L=1 mH. Is the location of poles of transfer function given in (i) are real? If yes, find the location.

(6 marks)
**1 (b)** If $ G(s)= \dfrac {K}{s(s+64)} $ with H(s)=1, determine value of K so that damping factor is 0.5. For this value of 'K' determine:

i) Rise time, and

ii) Settling time.

Assume unit step input.(6 marks)
**2 (a)** Find $ \dfrac {C(s)}{R(s)} $ for the system shown in Fig. 2 using Block diagram rules.

(6 marks)
**2 (b)** The open loop transfer function of unity feedback system is:

$ G(s) = \dfrac {K}{s(\tau s+1)}, K, \tau > 0 $ with a given value of K, the peak overshoot was found to be 80%. Suppose peak overshoot is decreased to 20% by decreasing gain K. Find the new value of K (say K_{2}) in terms of the old value.(6 marks)

### Solve any one question from Q3 and Q4

**3 (a)** Comment on stability of a system using Routh's criteria, if characteristics equation is D(s)=s^{4}+5s^{3}+s^{2}+10+1. How many poles lies in Right of s-plane?(4 marks)
**3 (b)** Construct Bode Plot and calculate GM, PM, W_{gc} and W_{pc} if $ G(s) = \dfrac {200(s+20)}{s(2s+1)(s+40)} $ and H(s)=1.(8 marks)
**4 (a)** Open loop transfer function of unity feedback system is $ G(s) = \dfrac {K}{s(s+2)(s+10)}. $ Sketch the complete root locus and comment on stability of system.(8 marks)
**4 (b)** For unity feedback system with $ G(s) = \dfrac {100}{s(s+5)} $.

Determine:

i) Resonance peak

ii) Resonance frequency.(4 marks)

### Solve any one question from Q5 and Q6

**5 (a)** Enlist any two advantages of state space approach over transfer function. Obtain a state space representation in controllable and observable canonical form for the system $ G(s) = \dfrac {s+3}{s^2 + 3s +2} $(6 marks)
**5 (b)** Obtain the state space representation of system whose differential equation is: $$ \dfrac {d^2 y}{dt^3}+ 2 \dfrac {d^2 y}{dt^2}+ 3 \dfrac {dy}{dt}+ 6y = \dfrac {d^2u}{dt^2} - \dfrac {du}{dt}+ 2u. $$ Also find controllability and observability of the system. Assume zero initial conditions.(7 marks)
**6 (a)** Obtain state transition matrix if: $$ i) \ \dfrac {dx}{dt} = \begin{bmatrix}
0 &1 \\-1
&0
\end{bmatrix}x \\
ii) \ \dfrac {dx}{dt} = \begin{bmatrix}
0 &1 \\0
&0
\end{bmatrix} x $$ using Laplace transformation.(6 marks)
**6 (b)** Write a short note on 'state transition matrix and its properties'.(4 marks)

### Solve any one question from Q7 and Q8

**7 (a)** Advantage of digital control system over analog control systems.(4 marks)
**7 (b)** Application of PLC (Programmable Logic Controller) in Elevator/List.(4 marks)
**7 (c)** PID controllers and its operational characteristics.(5 marks)
**8 (a)** Obtain pulse transfer function of the system shown in Fig. 3 with a=1.

(6 marks)
**8 (b)** Obtain pulse transfer function of system shown in Fig. 4

(7 marks)