Visvesvaraya Technological University (VTU) 2007 B.E Electrical and Electronics Engineering Fifth Semester , 06/ 07 - Question Paper
MODERN CONTROL THEORY
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Fifth Semester B.E. Degree Examination, Dec. 06 / Jan. 07 Electrical and Electronics Engineering Modern Control Theory
3 hrs.] [Max. Marks: 100
Note: 1. Answer any FIVE full questions.
2. Assume any missing data.
What is a controller? Explain P, I, PI and PID controllers. (10 Marks)
Obtain the state space representation model for the following electrical circuit in
Time:
1 a. b.
fig. 1(b). Given R = 1 Ohm and C = 1 Farad.
& f u
(10 Marks)
V,
ti'
U 4- (] f. j -
7
V,
02-
li -V- - 1
Fig-1(b)
Explain the terms: i) State ii) State variable iii) State vector iv) State space - with an example. . (10 Marks)
Obtain the state space representation of the following system and draw it,s phase variable diagram: ,
2 a. b.
3 a.
Y+6Y+11Y+ 6Y = 6u .
(10 Marks)
What is state transition matrix? List out the properties and advantages of state transition matrix. (10 Marks)
Obtain the state transition matrix using:
i) Laplace Transformation method and
ii) Cayley - Hamilton method, for the system describe by,
X(t)= 1 X(0) (10 Marks)
State the conditions for completely controllability and complete observability. Determine the state controllability and observability of the system described by, r.
4 a.
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1 |
0 " |
xi' |
'0 | |
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0 |
1 |
x2 |
+ |
0 |
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-6 |
-11 |
-6 |
_X3. |
1 |
[u]
x.
Y = [4 5 1
(10 Marks)
X2
X,
Explain common physical non-linearities in control systems.
(10 Marks)
Contd.... 2
a. What are singular points? Explain different singular points adopted in non-linear control systems. _ (08 Marks)
b. Find out singular points for the following systems:
i) x+0.5x+2x = 0
ii) y+3y+2y = 0
iii) y+3y-10 = 0. (12 Marks)
a. Obtain the necessary and sufficiency condition for arbitrary pole placement.
b. Obtain the gain matrix for the system:
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Given: oon = 4. |
M
(10 Marks)
a. Determine whether or not following quadratic form is positive definite:
Q(x,, x2) = 10x,2 + 4x1 + x3 + 2x,x2 - 2x2x3 - 4XjX3 (10 Marks)
b. Explain with an example - i) Liapunov Main Stability theorem ii) Liapunov Second method and iii) Krasovskiis theorem. (10 Marks)
8 a. Find the Liapunov function for the system:
X(t) =
X.
0 1
(08 Marks)
-1 -1_
b. Draw the phase-plane trajectory for the following equation using Isocline method:
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x+ 2<u>x+ a>2x = 0
Given, q = 0.5, co = 1, Initial point (0, 6). (12 Marks)
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| Earning: Approval pending. |
