# Control System February 2006

1. (a) List the merits and demerits of open loop and closed loop control systems. Give at least one example for each.

(b) For the mechanical system shown in Fig. 1.

i) Draw the mechanical network

ii)    Write the differential equations governing its dynamic behaviour

iii)  Write the Force – Current (F-i) and Force – Voltage (F-V) analogous electric networks

iv)    List ail the analogous quantifies

(2+3+(3+4)+2=14 Marks)

 fA% 4if K, &70

2. (a) Define ‘Transfer function’ of a system For a single loop unity feed back system the unit step response is given by c(t) -1 – 3e~2* 4- 2e~zi Evaluate

i)  Closed loop transfer function

ii)  Open loop transfer function

(b) For the electric circuit shown in Fig. 2 find xrrr using Mason’s ruie.

Vq(*)W*)toofov-1-0-—A/WWV-t-WWW Tofo.p/*FIi‘0P-T■HP

(c)  Obtain a block diagram representation and evaluate the transfer function of an armature controlled D-C motor.

3.  (a) With usual notations, derive an expression for the unit step-response of an under­damped second order system. Therefrom, derive expressions for peak time and percentage peak over shoot.

(b) Step response of a certain control system is shown in Fig. 3. Assuming single loop unity feed back, determine its open loop and closed loop transfer functions.

4.(a) For the block diagram shown in fig. 4

i) What ‘Type’ of system does represent ?

ii) Find

iii)   Find the position, velocity and acceleration error constants

iv)    If r(i) = 10u(i), evaluate C§g

(b) The open loop transfer function of a unity feed back system is <3(,s) =5(5+3) (5 +55+10)

i)    Find the value of K so that the steady state error for the input r(i) = tu(t) is less than or equal to 0,01.

II) For the value of K found in part i), verify whether the closed loop system is stable or not, using R-H criterion.

5.  (a) Using angle criterion , prove that the complex part of the root loci for the loop transfer function is circular. What is the centre ? What is its radius ?

(b) Following all the 10 steps, draw the root locus diagram for the loop transfer function. G(s) B(s) K 5(52+85 + l 7;

From the diagram evaluate the value of K for a system damping ratio of 0.5.

6. (a) State and explain ‘Nyquist stability criterion’.

(b) Sketch the complete Nyquist diagram and find the range of K for closed loop stability for the ioop transfer function

(c) Define the terms ‘Gain Margin’ and ‘phase margin’. Explain how these can be determined from poiar plots.

7.  (a) With usual notations, derive expressions for resonant peak and resonant frequency of a unity feed back second order system.

(b) Find the open ioop transfer function of a unity feed back second order control system for which resonant peak =1.1 units and resonant frequency – li.2rad/sec

(c) Compute analytically the gain margin and phase margin if GH{s) =K-3QS5(5 + 125 + 100)

8.  (a) The open ioop transfer function of a unity feed back system is G(s) = 5(1+0.25) (1+0.055)

Draw the bode asymptotic magnitude plot and phase plot. From the graph

i)    Find gain limit

ii)   Determine the value of K for a gain margin of lOdB. What is the corresponding phase margin.

iii)  Determine the value of K for a phase margin of 40°. What is the corresponding gain margin ?)

(b) Find the transfer function which has the asymptotic bode magnitude plot showirig Fig. 5.