Visvesvaraya Technological University (VTU) 2006 B.E Electrical and Electronics Engineering 5th SEM MODERN CONTROL THEORY - Question Paper
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Page No. 1
EE52
USN
NEW SCHEME Fifth Semester B.E. Degree Examination, July 2006 Electrical and Electronics Engineering
Time: 3 hrs.} [Max. Marks: 100
Note: 1. Answer any FIVE full questions.
1 a. Define the concept of i) State ii) State variables iii) State space (06 Marks)
b. A temperature control system has the block diagram given in fig. 1(b). The input signal is a voltage and represents the desired temperature 9r. Find the steady-state
error of the system when 9r is a unit step function and i) Z)(.s) = l ii) D(s) = 1 +
iii)Z)(.y} = 1 + 0.3s. What is the effect of the integral term in the PI controller and the derivative term in PD controller on the steady state error? (08 Marks)
 |
|
Fig.1(b) |
c. Fig. 1(c) shows the block diagram of a speed control system with state variable feedback. The- drive motor is an armature controlled dc motor with armature resistance Ra, armature inductance La, motor torque constant Kr, inertia referred to motor shaft J, viscous friction coefficient referred to the motor shaft B. back emf constant and tachometer Kt. The applied armature voltage is controlled by a three phase full-converter. ec is control voltage. e;, is armature voltage, er is the reference voltage corresponding to the desired speed. Taking X| = od (speed) and X2 = ia (armature current) as the state variables, u = er as the input, and y = oo as the output, derive a state variable model for the feed back system. (06 Marks)
4
 |
tl-*\ TS-tbV-T- |
I
Fig. 1(c) r
a. For the RLC network shown in fig.2(a) write the state model in matrix notation choosing Xi(t) = Vc(t) + R,(t) and X:(t) =' Vc(t) where X,(t) and X:(t) are state
variables, Vc(t) is output, V(t) is input.
R.
-VVVN-
(08 Marks)
L
STSJP-
03-
VC (*)
Vot)
c
Con?*!... 2
Fig.2(a)
Page Aro. 2
EE52
I'or a transfer function given bN G(.v) = --write the state model in
s2 + 3s + 2
i) Phase variable form ii) Diagonal form. (08 Marks)
Compare classical control theory against modern control theory. (04 Marks)
State the properties of transition matrix. (05 Marks)
b.
c.
a.-
|
1 1 0 1 |
"3 |
0" | ||
|
X = |
_2 -L |
x + |
3 |
b. Given the svstem X =
U . Find the input vector U(t) to give the
11
(10 Marks)
-c")
|
"1 " |
f m 1 <N CN 1 1 | ||
|
c. The vector |
-> |
is an eigen vector of A- |
2 1 -6 |
|
-1 |
-1 -2 0 |
Find the eieen value of A
corresponding to the vector given. (05 Marks)
The following is the state space representation of a linear system w'hose eigen values are -3, -2, -1.
"0
|
X = |
0 O 1 |
1 0 ~ 0 1 | |
|
-6 |
-11 -6 |
0
u
Given that u=0. X(0)=[001]T. Find X(t)
Find the transition matrix 0(1) for a system whose system
- 5 - 1 .
by the following techniques:
(10 Marks)
matrix is <ziven bv
A =
-1
J)
i) Laplace transform ii) Infinite series iii) Cayley-Hamliton.
(10 Marks)
Define controllability and observability. (06 Marks)
Show that the characteristic equation and eigen values of a system matrix are
invariant under linear transformation. (08 Marks)
State the properties of Jordan matrix. (06 Marks)
What are inherent nonlinearities? Explain any three of them. (06 Marks)
Sketch the following nonlinearities :
i) ideal relay ii) relay with dead zone iii) relay w'ith dead zone and hysterisis
iv) relay with hysterisis v) dead zone. (04 Marks)
ft
c. A linear second order servo is described by the equation C+2Ca>n C+co~C = 0 where
#
= 0.15, con = 1 rad/sec. C(0) = 1.5 and C = 0. Determine the singular point. Construct the phase trajectory', using the method of isoclines. (10 Marks)
r(s) io
a. Consider a linear svstem described by the transfer function 7 = ?-77-r.
C/(S) (S-rlXS + 2)
Design a feedback controller with a state feedback so that closed loop poles are placed at -2,-1 =j 1. (10 Marks)
#
b. Consider the system described by the state model X AX\ Y = CX where
-1 1 ~
a.
b
a.
b
C = [1 0]. Design a full order state observer. The desired eigen
A =
1 -2
values for the observer matrix are ux = -5 ; jj.2 = -5 . (10 Marks)
State and explain Liapunov theorems on i) asymptotic stability ii) global asymptotic stability iii) instability. (10 Marks)
Define singular point on a phase plane. Explain different types of singular points.-
(10 Marks)
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a.
b.
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Attachment: |
| Earning: Approval pending. |
