Code: A-11 Subject: CONTROL ENGINEERING

Time: 3 Hours Max. Marks: 100

NOTE: There are 11 Questions in all.

· Question 1 is compulsory and carries 16 marks. Answer to Q. 1. must be written in the space provided for it in the answer book supplied and nowhere else.

· Answer any THREE Questions each from Part I and Part II. Each of these questions carries 14 marks.

· Any required data not explicitly given, may be suitably assumed and stated.

Q.1 Choose the correct or best alternative in the following: (2x8)

a. Closed-loop transfer function of a unity-feedback system is given by . Steady-state error to unit-ramp input is

(A) (B)

(C) 1 (D)

b. Electrical time-constant of an armature-controlled dc servomotor is

(A) equal to mechanical time-constant.

(B) smaller than mechanical time-constant.

(C) larger than mechanical time-constant.

(D) not related to mechanical time-constant.

c.

In the system of Fig.1, sensitivity of with respect to parameter is

(A)

(B)

(C) 1

(D) None of the above.

d. The open-loop transfer function of a unity feedback system is

The system is unstable for

(A) K>5 (B) K<5

(C) K>0 (D) all the above.

e. Peak overshoot of step-input response of an underdamped second-order system is explicitly indicative of

(A) settling time. (B) rise time.

(C) natural frequency. (D) damping ratio.

f. A unity feedback system with open-loop transfer function is critically damped. The value of the parameter p is

(A) 4. (B) 3.

(C) 2. (D) 1.

g. Consider the position control system of Fig.2. The value of K such that the steady-state error is for input rad/sec, is

(A) 104.5

(B) 114.5

(C) 124.5

(D) None of the above.

h. Polar plot of

(A) crosses the negative real axis.

(B) crosses the negative imaginary axis.

(C) crosses the positive imaginary axis.

(D) None of the above.

PART I

Answer any THREE Questions. Each question carries 14 marks.

Q.2 a. Define the transfer function of a linear time-invariant system in terms of its differential equation model. What is the characteristic equation of the system? (5)

b.

Derive the transfer function of the op amp circuit shown in Fig.3. Also, prove that the circuit processes the input signal by ‘proportional + derivative + integral’ action. (9)

Q.3 The electro hydraulic position control system shown in Fig.4 positions a mass M with negligible friction. Assume that the rate of oil flow in the power cylinder is where x is the displacement of the spool and is the differential pressure across the power piston.. Draw a block diagram of the system and obtain therefrom the transfer function .

The system constants are given below.

Mass M = 1000 kg

Constants of the hydraulic actuator:

= 200 cm²/sec per cm of spool displacement

= 0.5 cm²/sec per gm- t/ cm²

Potentiometer sensitivity K_P = 1 volt/cm

Power amplifier gain K_A = 500 mA/volt

Linear transducer constant K = 0.1 cm/mA

Piston area A = 100 cm² (14)

Q.4 A servo system is represented by the signal flow graph shown in Fig.5. The nominal values of the parameters are .

Determine the overall transfer function and its sensitivity to changes in under steady dc conditions, i.e., s = 0. (14)

Q.5 a. Define the terms:

(i) bounded-input, bounded-output (BIBO) stability,

(ii) asymptotic stability. (2+2)

b. Determine the values of K>0 and a>0 so that the system shown in

Fig.6 oscillates at a frequency of 2 rad/sec. (10)

Q.6 Consider the control system shown in Fig 7 in which a proportional compensator is employed. A specification on the control system is that the steady-state error must be less than two per cent for constant inputs.

(i) Find K_c that satisfies this specification. (5)

(ii) If the steady-state criterion cannot be met with a proportional compensator, use a dynamic compensator . Find the range of that satisfies the requirement on steady-state error. (9)

PART II

Answer any THREE Questions. Each question carries 14 marks.

Q.7 Discuss the compensation characteristics of cascade PI and PD compensators using root locus plots. Show that

(i) PD compensation is suitable for systems having unsatisfactory

transient response, and it provides a limited improvement in

steady-state performance.

(ii) PI compensation is suitable for systems with satisfactory

transient response but unsatisfactory steady-state response. (7+7)

Q.8 a. State and explain the Nyquist stability criterion. (5)

b. Use the Nyquist criterion to determine the range of values of K>0 for the stability of the system in Fig. 8. (9)

Q.9 a. When is a control system said to be robust? (4)

b. A unity-feedback system has open-loop transfer function .

(i) Using Bode plots of , determine the phase margin of the system.

(ii) How should the gain be adjusted so that phase margin is 50°?

(iii) Determine the bandwidth of gain-compensated system.

The –3dB contour of the Nichols chart may be constructed using the following table.

Phase, degrees	0	-30	-60	-90	-120	-150	-180	-210
Magnitude, dB	7.66	6.8	4.18	0	-4.18	-6.8	-7.66	-6.8

(10)

Q.10 The open-loop transfer function of a control system is

(i) Draw the Bode plot and determine the gain crossover frequency, and phase and gain margins.

(ii) A lead compensator with transfer function

is now inserted in the forward path. Determine the new gain crossover frequency, phase margin and gain margin. (6+8)

Q.11 Discretize the PID controller

to obtain PID algorithm in

(i) position form

(ii) velocity form.

What are the advantages of velocity PID algorithm over the position

algorithm? (14)