Digitalcontrolsystems

Code No: 07A71002 R07 Set No. 2
IV B.Tech I Semester Examinations,May/June 2012
DIGITAL CONTROL SYSTEMS Electronics
And Instrumentation Engineering
Time: 3 hours Max Marks: 80
Answer any FIVE Questions
All Questions carry equal marks
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1. Find the solution of the following difference equation:
x(k +2) - 1.3 x(k + 1) + 0.4 x(k) = u(k)
where x(0) = x(1) = 0 and x(k) = 0 for k < 0.
For the input function u(k), consider the following two cases:

u(k) = 1, K= 0,1,2 . . . . . . . .
u(k) = 0, k < 0.
[16]

2. For the system defined by
x1 (k + 1) 0 1 x1 (k) 0
= + u(k)
x2 (k + 1) −0.16 −1 x2 (k) 1
x1 (k)
y(k) = 1 0
x2 (k)
assume that the outputs are observed as y(0)=1 and y(1)=2 and the
control signals given are u(0)=2,u(1)=-1.Determine the initial state X(0) and also
X(1) and X(2). [16]

3. What are the two basic transformations used to convert an analog system transfer
function to a digital system transfer function? Explain each procedure. [16]
4. (a) State and explain the jury stability test.
(b) Investigate the stability of the system shown in figure 1 for sampling period
T = 0.4 sec using root locus technique. [8+8]

Figure 1:

5. The block diagram of a digital control system is shown in Figure 2. Design a com-
pensator D(z) to meet the following specifications:
(a) Velocity error constant, Kv ≥ 4 sec.,
(b) Phase margin ≥ 400 and


Figure 2:

(c) Band width =1.5 rad./sec. [16]

6. Using the partial fraction expansion programming method, obtain a state space
representation of the following pulse transfer function
Y (z) 1+6z −1 +8z −2
U (z) = 1+4z −1 +3z −2 .

Hence, obtain its state transition matrix. [16]

7. Obtain the inverse Z-transform of the following :
−1−2
(a) X(Z) = 1+Z −Z
−1
using inversion integral method.
(1−Z )

(b) X(Z) = Z -1(1 − Z−2 )/(1 + Z−2 ) using direct division method.
(c) X (Z) = Z -1 (0.5 - Z - 1 )/(1 - 0.5 Z - 1 )(1 - 0.8 Z - 1 )2 using partial fraction ex-
pansion method. [4+6+6]

8. Consider the digital control system which has the controlled process described by
0.0125(z+0.195)(z+2.821)
G P (z) = z(z−1)(z−0.368)(z−0.8187)
Design a dead beat response controller so that the output sequence C (KT) will
follow a unit step input in minimum time. [16]

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1. Obtain the state equation and output equation for the system defined by
Y (z) z −1 +5z −2
U (z) = 1+4z −1 +3z −2 . [16]
z+1
2. Consider the system defined by G(z) = z2 +z+0.16. Obtain the state space represen-
tations for this system in the following different forms:

(a) Controllable canonical form
(b) Obseravable canonical form and
(c) Diagonal canonical form. [5+5+6]

3. Solve the following difference equation by using Z - transform method
c(k + 2) - 0.1 c(k + 1) - 0.2 c(k) = r(k + 1) + r(k)
the initial conditions are c(0) = 0, c(1) = 1 and r(k) = us(k). [16]

4. Find the inverse Z -transform of:

(a) F(z) = Z ( ZOH ) / ((Z - 1)(Z 2 - Z + 1)).
(b) Z / (Z -1)(Z - 2).
(c) (Z 2 + 8Z + 12) / (Z 2 + 2Z + 3). [6+4+6]
5. (a) Explain constant Frequency loci.
(b) Consider the discrete - data system as shown in figure 3 and assume its sam-
pling period is 0.5 sec, Determine the range of K, for the system to asymptot-
ically stable. [6+10]

*
O >

Figure 3:

6. The open loop pulse transfer function of an uncompensated digital control system
0.0453(z+0.904)
is GH 0 GP (z) = (z−0.905)(z−0.819) . The sampling period T is equal to 0.1 sec. Find
the time response and steady state error of the system to a unit step input. [16]

7. Consider the system defined by
   
0 1 0 0
X (k + 1) =  0 0 1  X (k) +  0  U (k).
−0.5 −0.2 1.1 1
Determine the state feed back gain matrix K such that when the control signal is
given by u(k)=-K x(k), the closed loop system will exhibit the dead beat response
to any initial state X(0). [16]

8. Explain the following with respect to digital control system configuration:

(a) A/D and D/A conversion.
(b) Sample and hold circuit.
(c) Transducer.
(d) Different types of sampling operations. [16]

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1. (a) Discuss the necessity of an observer in control systems.
(b) Draw the block diagram of a reduced order observer and explain its advan-
tages over full-order observer. [6+10]

2. What are the popular methods used to find the inverse Z -transform? Explain
briefly each of them with suitable example. [16]

3. With suitable diagram explain any two methods of digital to analog conversion.
[16]

4. The block diagram of a sampled data system is shown below in figure 4. Obtain a
discrete time state model for the system. [16]

+

Figure 4:

5. The block diagram of a discrete - data control system is shown in Figure 5, in which
Gp(s) = 2(s+1) and T = 0.5 sec. Compute and plot the unit step response c*(t) of
s(s+2)
the system. Find c max and the sampling instant at which it occurs. [16]

O
-

Figure 5:

6. (a) Explain constant Frequency loci.
(b) Consider the discrete - data system shown in figure 6 and assume that its sam-
pling period is 0.5 sec. Determine the range of K for the system to be asymptot-


O

Figure 6:

7. Solve the following difference equation by Z transform method.
x(k + 2) = x(k + 1) + x(k)
Given that x(0) = 0 and
x(1) = 1. [16]

8. (a) Explain the Duality between controllability and observability.
(b) Consider that a digital control system is described
 by  state 
the equation.
1 −2 0 1 0
x(k + 1) = A x(k) + B u(k) Where A =  3 2 1  B =  −1 1 
−1 1 4 0 1
Determine the controllability of the system. [6+10]

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1. (a) Explain the design procedure of digital PID controllers.
(b) Derive the pulse transfer function of PID controller. [8+8]

2. (a) Obtain the Z-transform of:
i. f(t) = t2 .
ii. f(t)= e(−at) sinωt.
(b) Explain the limitations of Z-transforms. [12+4]

3. (a) Explain the concept of observability.
(b) Given the system.
x (k+1)=Ax (k)+Bu(k)
y(k)= Cx(k)
0 1 1
where A = , B= , C= 1 1 .
−1 −3 2
determine the state controllability of the system. [6+10]

4. (a) Write short notes on mapping of the left half of the s - plane into the z - plane.
(b) Consider the discrete -data system shown in figure 7 and assume its sam-
pling period is 0.5 sec, Determine the range of K for the system to be asymptot-

Figure 7:

5. (a) Obtain a state space representation of the following system
Y (z) z −1 +2z −2
U (z) = 1+0.7z −1 +0.12z −2
Assume any if necessary.
(b) State and explain the properties of the state transition matrix of discrete time
system. [10+6]

6. Given the difference equation y(k+2) - 1.3y(k+1) + 0.4y(k) = u(k) with y(k) = 0 for
k<0 and y(0) = -1, y(1) = 1, obtain the solution if

(
0 if k < 0
(a) u(k) =
1 if k ≥ 0
(
1 if k = 0
(b) u(k) = [16]
0 if k = 0

7. (a) Explain the digital implementation of analog controllers in detail.
(b) Describe the three digital integration rules used for the digital implementation
of controllers and explain bilinear transformation briefly. [8+8]

8. (a) A state feed back control system has following system equations
X(k+1) = GX(k) + HU(k)
Y(k) = CX(k)
U(k) = -KX(k)
where K is state feed back gain matrix.
Draw the necessary block diagram for the control system and derive the ob-
server error equation.
(b) Briefly explain the design of digital control systems that must follow changing
reference inputs, applying observed-state feed back method. Draw necessary
block diagram. [8+8]

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Digitalcontrolsystems

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