Name: C
Reg No:
APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY
07 THRISSUR CLUSTER
FIRST SEMESTER M.TECH DEGREE EXAMINATION, DECEMBER 2017
Chemical Engineering
(Process Control)
07CH6105 Modern Control Theory
Time: 3 hours Max. Marks: 60
Answer all six questions. Part of each question is compulsory.
Answer either part or part of each question
Q.no. Module 1 Marks
1a
Obtain the state model of the system whose transfer function is given as
3 2
10
4 2 1
Y s
U s s s s


4
Answer b or c
b
Explain the basic elements of a state diagram? Draw the block diagram of
the system described by state model,
.
1
1 2 1
.
2 2 3
.
3
3
0 1
1 0 1 0
0 1 0 0
x
a a x
x x u y x
x
x






5
c Explain state vector, input vector, output vector. Write the state model of a
system containing m input variables and p output variables.
5
Q.no. Module 2 Marks
2a
State and prove Cayley Hamilton theorem.
4
Answer b or c
b Explain Quadratic forms and sign definiteness of Quadratic forms. 5
Page 1 of 3
c
Find the state transition matrix of
4 1 2
1 0 2
1 1 3
A





5
Q.no. Module 3 Marks
3a
Given
2 2 3
1 1 1
1 3 1




. Find the state transition matrix.
4
Answer b or c
b Derive any one method of solution of non homogeneous state equations? 5
c Obtain the time response of the following system
.
1 1
.
2
2


0 1 2
2 3 0
0
1
0 t
x x
u
x
x
x
x
u t e t







5
Q.no. Module 4 Marks
4a
A discrete time system has the transfer function
3 2
2
4 12 13 7

Y z z z z
U z z z



.Determine the state model of the system in
Jordan Canonical form.
4
Answer b or c
b Explain stability, asymptotic stability and instability using suitable physical
models.
5
c
State and explain the Lyapunov stability theorems
Page 2 of 3
5
Q.no. Module 5 Marks
5a Choose a quadratic scalar form and determine the stability of the system
described by





















2
1
2
1
3 4
0 1 .
.
x
x
x
x
5
Answer b or c
b
Given a linear time independent system Ax x
.
where







12 8
4 0
A .
Determine its stability by the Lyapunov direct method.
7
c Test the stability of the system described by
2
2
1 1 1 2
.
x x x x 2 2
.
x x
by use of the variable gradient method
7
Q.no. Module 6 Marks
6a Test the observability of the following system
u
x
x
x
x
1
0
2 1
1 1
2
1
2
.1
.




























5
Answer b or c
b Test the observability and controllability of the following system
X AX BU
.
Y CX











3
2
1
x
x
x
X












6 11 6
0 0 1
0 1 0
A











1
0
0
B 1 5 4 C
7
c
Describe the features of an optimal control problem.
7
Page 3 of 3