Rr220206 Control Systems Sep 2007

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Code No: RR220206 Set No. 1

II B.Tech II Semester Supplimentary Examinations, Aug/Sep 2007CONTROL SYSTEMS

( Common to Electrical & Electronic Engineering, Electronics &Communication Engineering, Electronics & Instrumentation Engineering,

Electronics & Control Engineering, Electronics & Telematics andInstrumentation & Control Engineering)

Time: 3 hours Max Marks: 80Answer any FIVE Questions

All Questions carry equal marks

1. (a)

Figure 1a

For the system in the above Figure 1a, obtain transfer functioni. C/R

ii. C/D

(b) Verify the above transfer function using signal flow graph. [8+8]

2. (a) Derive the transfer function of an a.c. servomotor and draw its characteristics.

(b) Explain the Synchro error detector with circuit diagram. [8+8]

3. (a) A unity feedback system has a forward path transfer function G(s)= 9s(s+1)

.Find the value of damping ratio, undamped natural frequency of the system,

percentage overshoot, peak time and settling time.

(b) Measurements conducted on servomechanism show the system response to bec(t) = 1 + 0.2e60t - 1.2e10t when subjected to a unit-step unit. Obtain theexpression for the closed-loop transfer function. [10+6]

4. (a) A unity feedback system has forward transfer function: G(s)= K(s+13)s(s+3)(s+7)

Using R-H criterion, find the range of K for which the closed loop system isstable.

(b) The characteristic equation of a servo system is given by a0s4 + a1s

2 + a3s+a4 = 0. Determine the conditions, which must be satisfied by the coefficientsof the characteristic equation for the system to be stable. [8+8]

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5. (a) Find the angle of arrival and the angle of departure at the complex zeros andcomplex poles for the root locus of a system with open-loop transfer function

G(s)H(s)= K(s2+1)

s(s2+4s+8)

(b) Draw the root locus diagram for a feedback system with open-loop transfer

function G(s)=K(s+5)s(s+3)

. , following systematically the rules for the construction

of root locus. Show that the root locus in the complex plane is a circle. [8+8]

6. (a) Explain the concept of phase margin and gain margin [8]

(b) Draw the Bode Plot for the system having G(s)= 10s(1+0.01s)(1+0.1s)

, H(s) = 1.Determine

i. The gain cross over frequency and corresponding phase margin.

ii. The phase cross over frequency and corresponding gain margin.

[4+4]

7. (a) Plot the polar plot of G(s)20(s2+s+0.5)

s(s+1)(s+10)

(b) Explain the concept of Nyquist stability criterion. [8+8]

8. (a) Reduce the matrix A to diagonal matrix.

(b) Derive the state models for the system described by the differential equationin phase variable form.

...

y +4..

y +5.

y +2y = 2..

u +6.

u +5u [8+8]

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1. (a)

Figure 1a

For the system in the above Figure 1a, obtain transfer functioni. C/R

ii. C/D

(b) Verify the above transfer function using signal flow graph. [8+8]

2. Derive the Transfer Function for the field controlled d.c. servomotor with neatsketch. [8+8]

3. Determine the damping ratio, undamped natural frequency, damped natural fre-quency for the system shown in Figure3. What is the response c(t) of this system

to a unit step function excitation r(t) = u(t) when all initial conditions are zero ?Also find out the tr, tp, ts. [8+8]

Figure 34. (a) The open loop Transfer function of unity feedback system given by

G(s)= Ks(s2+8s+T)

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Using Rouths Criterion Determine the conditions to be satisfied by K andT. If it is required that all the roots of the characteristics equation lie in theregion to the left of a line s = -1

(b) Determine the Kp, Kv, andKa for a unity feedback system with the followingopen loop transfer function.G(s)= 1000

s(s+10)(s+100)

And hence calculate the steady state error for r(t)=(2+t) u(t). [8+8]

5. A unity feedback control system has an open-loop transfer function G(s)= Ks(s2+4s+13)

.Sketchthe root locus plot of the system by determining the following:

(a) Centroid, number and angle of asymptotes

(b) Angle of departure of root loci from the poles

(c) Breakaway points if any

(d) The value of K and the frequency at which the root loci cross the j-axis.[4x4]

6. G(s)H(S) = K10(1s)S(S+2)(S2+2S+25)

with K = 1,

Sketch asymptotic bode plot & find gain margin & phase margin. By what factorshould K be increased or decreased to obtain a gain margin of 40 db. [4x4]

7. (a) Construct the complete Nyquist plot for a unity feed back control system

whose open loop transfer function is G(s)H(s) = Ks(s2+2s+2) . Find maximumvalue of K for which the system is stable.

(b) The open loop transfer function of a unity feed back system is G(s)= 1s(1+0.5s)(1+0.1s)

.Find gain and phase margin. If a phase lag element with transfer function of1+2s1+5s

is added in the forward path, find how much the gain must be changed

to keep the margin same. [8+8]

8. (a) Obtain the solution of a system whose state model is given by X = A X(t) +B U(t) ; X(0) =X0 and hence define state Transition matrix.

(b) Obtain the transfer function of a control system whose state model is [8+8]

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1. (a) Derive the transfer function of the following electrical network. Figure 1a

Figure 1a

(b) Briefly explain the terms used in Signal flow graph. [8+8]

2. (a) Derive the Transfer Function for potential divider.

(b) Derive the Transfer Function for a.c. servomotor. [8+8]

3. Determine the damping ratio, undamped natural frequency, damped natural fre-quency for the system shown in Figure3. What is the response c(t) of this systemto a unit step function excitation r(t) = u(t) when all initial conditions are zero ?Also find out the tr, tp, ts. [8+8]

Figure 3

4. (a) Explain how Routh Hurwitz criterion can be used to determine the absolutestability of a system.

(b) For the feedback control system shown in Figure 4b. it is required that :

i. the steady-state error due to a unit-ramp function input be equal to 1.5.

ii. the dominant roots of the characteristic equation of the third-order system

are at -1+j1 and -1-j1. Find the third-order open-loop transfer function

G(s) so that the foregoing two conditions are satisfied. [6+10]

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Figure 4b

5. Sketch the root locus for a unity feedback system with G(s) = k(s+1)s2(s+9)

find the gain

when all roots are real & equal. [8+4+4]

6. (a) Define peak resonance and band width

(b) Sketch bode plot for.

G (s) = 256(1+0.5s)s(1+2s)(s

2

+3.2s+64)[8+8]

7. (a) Explain how Nyquist contour is selected for stability analysis.

(b) Discuss the stability of the following system using Nyquist stability criterionG(s)H(s) = K

(Ts+1)s[6+10]

8. (a) For the given system X = Ax + Buwhere

A =

1 2 10 1 3

1 1 1

B =

10

1

Find the characteristic equation of the system and its roots.

(b) Given

X(t) =

0 12 3

x1 (t)x2 (t)

+

01

u(t)

Find the unit step response when,

X(0)=

11

[8+8]

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1.

Figure 1Using block diagram reduction technique find the transfer function for the systemshown in Figure 1 below and verify the transfer function by masons gain formula.

[8+8]

2. (a) Derive the transfer function of a field controlled d.c. Servomotor and developthe block diagram. Clearly state the assumptions made in the derivation.

(b) What are the effects of feedback on the performance of a system? Brieflyexplain. [8+8]

3. (a) Define transient response specifications.

i. Delay time

ii. Rise time

iii. Peak time

iv. Maximum overshoot

v. Settling time of second order system.

(b) A unity feedback system is characterised by an open loop transfer function.

G(s) = Ks(s+10)

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Determine gain K so that system will have a damping ratio of 0.5. For thisvalue of K determine settling time, peak overshoot and time to peak overshoot for a unit step input. [10+6]

4. (a) Show that the Rouths stability criterion and Hurwitz stability criterion areequivalent.

(b) Consider the servo system with tachometer feedback shown in Figure 4b. Ob-tain the error signal E(s) when both the reference input R(s) and disturbanceinput D(s) are present. Obtain also the steady-state error when the system issubjected to a reference input (unit-ramp input) and disturbance input (stepinput of magnitude). [6+10]

Figure 4b

5. The open loop transfer function of a unity feedback system control system is givenby G(s)= K(s+2)s(s+3)(s+4)(s2+2s+2)

(a) Sketch the root locus diagram as a function of K.

(b) Determine the value of K which makes the relative damping ratio of the closedloop complex poles equal to 0.707. [8+8]

6. (a) Explain the frequency response specifications.

(b) Draw the Bode Plot for the system having G(s)H(s) = 100(0.02s+1)(s+1)(0.1s)(0.01s+1)

. Find

gain and phase cross over frequency. [8+8]

7. (a) The open loop transfer function of a feed back system is G(s)H(s)=K(1+s)(1s)

.Comment on stability using Nyquist Plot.

(b) The transfer function of a phase advance circuit is 1+0.2s1+0.2s

. Find the maximumphase lag. [8+8]

8. (a) Define the terms

i. State variable

ii. State transition matrix. [4+4]

(b) Obtain the state equation and output equation of the electric network show

in Figure8b [8]

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Figure 8b

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Rr220206 Control Systems Sep 2007