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Code No: RR420201
Set No. 1
IV B.Tech II Semester Regular Examinations, Apr/May 2007
DIGITAL CONTROL SYSTEMS
(Electrical & Electronic Engineering)
Time: 3 hours Max Marks: 80
Answer any FIVE Questions
All Questions carry equal marks

1. (a) With help of schematic diagram explain the principal operation of digital to
analog conversion.
(b) Explain the conditions to be satis ed for reconstruction of sampled signal into
continuous signal.
(c) Explain zero order hold device. [5+5+6]

2. (a) Obtain the z ? transform of the following x(k)
( ) = 9 (2k-1) - 2k + 3 k=0,1,2, ...... Assume that x(k)=0 for k 0.
(b) Obtain the inverse z-transform of the following

i. X(Z)=z (z+2)(z-1)2 and
[8+8]
ii. X(Z)= z- 2
(1-z -1 )3

3. Find the range of K for the system shown in Figure 3 to be stable. [16]

Figure 3
4. The open loop pulse transfer function of an uncompensated digital control system
(z-0.905)(z-0.819)
is Gh0 Gp(z) = 0.0453(z+0.904) . The sampling perio d T is equal to 0.1 sec. Find
the time response and steady state error of the system to a unit step input. [16]

5. The open loop transfer function of a unity - feedback digital control system is given

as ( ) = K(z2 +0.5z+0.2)
(z-1)(z 2 -z +0.5) . Sketch the root loci of the system for 0 8.
Indicate all important information on the ro ot loci. [16]

6. A blo ck diagram of a digital control system is shown in Figure 6. Design a PID
controller D(z), to eliminate the steady-state error due to a step input and simulta-
neously realizing a good transient response, and the ramp-error constant Kv should
equal 5. [16]


Code No: RR420201
Set No. 1

Figure 6
7. (a) Let A be an nxn matrix. Using Cayley-Hamilton theorem, show that any with
k=n can be written as a linear combination of { n-1}
(b) Explain the method to nd the state transition matrix through z-transform
technique. [8+8]

8. (a) Explain the concept of controllability and observability of discrete time control
system.
(b) Examine whether the discrete data system
X(k + 1) = AX(k) + Bu(k)
C(k) = DX(k) where A = 0 1
-2 -2 , B = 1 -1 , D = [1 0] is
i. State controllable
ii. Output controllable and
iii. Observable [7+9]


Code No: RR420201
Set No. 2

1. (a) What are the di erent types of sampling operations? Explain each of them.
(b) What do you mean by the problem of aliasing? How to overcome this?
(c) Explain the advantages and disadvantages of digital control systems.[6+5+5]

2. (a) Find the Z-transform of the following:
i. f(t) = t2,

ii. f(t) =e-at
(b) What are the popular methods are used to nd the inverse z-transform? Ex-
plain brie y each of them. [8+8]
3. A sampler and ZOH are now intro duced in the forward loop (Figure 3). Study the
stability of the sampled-data system via bilinear transformation and show that the
stable linear continuous time system becomes unstable upon the introduction of a
sampler and ZOH. [16]

Figure 3
4. Compare the characteristics of time-domain responses of continuous - time and
discrete - time systems. [16]

5. Draw root locus in the z-plane for the system shown in Figure 5 for 0 8.
Consider the sampling period T = 4 sec. [16]

Figure 5
6. Consider the system shown in Figure 6 and design lead compensator c(z) in w’-
plane for this system to meet the following speci cations:


Code No: RR420201
Set No. 2
(b) Settling time, ts=1.4 sec., and
(c) Velocity error constant Kv = 2 -1. [16]

Figure 6
7. Find state model for the following di erence equation. Obtain di erent canonical
forms. Also draw state diagram for each.
y(k+3) + 5y(k+2) + 7y(k+1) + 3y(k) = 0 [16]

8. (a) State and explain the Liapunov’s stability theorem for linear digital systems.

(b) Given X(k + 1) = 0 5 1
-1 -1 X(k)
Solve for ‘P’ - matrix and justify, by using the Liapunov’s theorem (Direct
method). Show that the system is asymptotically stable. [6+10]


Code No: RR420201
Set No. 3

1. (a) Describe the following parameters:
i. Acquisition time
ii. Aperture time and
iii. settling time
(b) State and explain the sampling theorem.
(c) Derive the transfer function of zero order hold device. [6+5+5]

2. (a) Find the Z-transform of the following:
i. ( ) = 1
s2 (s+1)
ii. f(t)=t sin t
(b) Find the inverse Z-Transform of the following:
i. F(Z)= 5z
z2 +2z +2
[8+8]
ii. F(Z)= 3z2 +2z+1
(z 2 -3z+2)

3. Consider the sample- data system shown in Figure 3 and assume its sampling perio d
is 0.4 Sec.

Figure 3
Find the range of K, so that the closed - loop system for which stable. [16]

4. The block diagram of a discrete - data control system is shown in Figure 4, in which
p( ) = 20 s(s+5) and T = 0.5 sec. Compute and plot the unit step response c*(t) of
the system. Find the step, ramp, and parabolic error constants. Also nal value of
c(kT). [16]

Figure 4


Code No: RR420201
Set No. 3
5. The open loop transfer function of a unity - feedback digital control system is given
(z-1)(z 2 -z +0.5) . Sketch the root loci of the system for 0 8.
as ( ) = K(z2 +0.5z+0.2)
Indicate all important information on the ro ot loci. [16]

6. A block diagram of a digital control system is shown in Figure 6. Design a com-
pensator D(z) to meet the following speci cations:

(a) Velocity error constant, Kv = 4 Sec.,
(b) Phase margin = 400 and

(c) band width =1.5 rad./sec.

Figure 6
y(k+3) + 5y(k+2) + 7y(k+1) + 3y(k) = 0 [16]

8. (a) Derive the necessary condition for the digital control system
X(k + 1) = AX(k) + Bu(k)
Y(k) = CX (k) to be Controllable.
(b) Examine whether the discrete data system
X(k + 1) = AX(k) + BU(k)
Y(k) = CX(k) where A = 1 -2
1 -1 , B = 1 0 0 -1 , C = 1 0 0 1 is
i. State controllable and
ii. Observable. [10+6]


Code No: RR420201
Set No. 4

1. (a) Explain clearly the advantages and disadvantages of digital control systems.
(b) Draw the schematic diagram of basic digital control scheme and explain the
same.
(c) With suitable diagram explain the weighted r esistor digital to analog con-
verter. [5+5+6]

2. (a) Obtain the z transform of the curve x(t) shown in Figure 2a. Assume that
T=sec.

Figure 2a

(b) Find the inverse z-transform of the following function , F(z)= z 2 (z-1) (z - 0.2) [8+8]
3. The block diagram of a digital control system is shown in Figure 3, where
p( ) = K(s+5) s2 .

Figure 3
Determine the range of K for the system to be asymptotically stable. [16]

4. The block diagram of a discrete - data control system is shown in Figure 4, in which
s(s+2)
p( ) = 2(s+1) and T = 0.5 sec. Compute and plot the unit step response c*(t) of
the system. Find * max and the sampling instant at which it occurs. [16]


Code No: RR420201
Set No. 4

Figure 4
5. (a) State and explain the Nyquist stability criterion in z-domain.
(b) Explain the construction of Bode diagram for discrete time systems. [8+8]

6. A blo ck diagram of a digital control system is shown in Figure 6. Design a PID
controller, D(z) to meet the following speci cations:

(a) velo city error constant, Kv = 10.,
(b) Phase margin = 600 and

(c) band width =8 rad./sec. [16]

Figure 6
y(k+3) + 5y(k+2) + 7y(k+1) + 3y(k) = 0 [16]

8. (a) State and explain the Liapunov’s stability theorem for linear digital systems.

(b) Given X(k + 1) = 0 5 1
-1 -1 X(k)
Solve for ‘P’ - matrix and justify, by using the Liapunov’s theorem (Direct
method). Show that the system is asymptotically stable. [6+10]

D I G I T A L C O N T R O L S Y S T E M S J N T U M O D E L P A P E R{Www

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