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CONTROL ENGINEERING
Course Work – Lab report
LAB REPORT
Student Name
Programme
Student Number
Semester
Experiment No.
2
Name of Experiment
Controllability, Observability and Eigen Values
Date of Experiment
Assessment Methodology
Each exercise will be marked according to the following grading rubrics
Serial No.
1
Component
Weightage
Aim, Objectives and relevant background theory
100*0.20
Software interference and
2
100*0.40
Output (m files and graphs)/Files
3
Result, Discussions, and Conclusion
100*0.30
4
Report structure and references
100*0.10
Total
100%
Notes:
Control Engineering (MHH324719)
Semester B – 2020-21
M&IE Department
-See question paper attached two assignments for Controllability, Observability and Eigen
Values
-Answer sheet report, ensure followings: For introduction & background information, do not use
manual information in the questions, do some research information shall be interested to read
the introduction with meaningful information for at least one page and half.
-Follow up structure reports as below index started from 1 to 5 and cover remaining sections.
-These are two assignments in one lab reports, check both.
-after completion of the report, run plagiarism checker report
– check my answer for Matlab input code
-check results and conclusions and make sure are covered as per requirements.
Aim:
To evaluate the system whether it is completely controllable and completely observable. To
evaluate the Eigen Value of the system.
APPARATUS/SOFTWARE REQUIRED:
MATLAB
Objective:
To learn to use MATLAB:
1. To generate a system transfer function from state space representation.
2. To determine the system controllability.
3. To determine the system observability.
4. To determine the eigen values of the system function from state space representation.
Introduction/Background:
do some research information, make readable introductions, meaning full information (one full
page), after completion Shall you do Plagiarism checker report
Read below and please rewrite it Controllability, Observability, Eigenvalues and
Eigenvectors:
Controllability:
The controllability verifies the usefulness of state variable. In the controllability test we can
find, whether the state variable can be controlled to achieve the desired output. The choice
of state variables is arbitrary while forming the state model. After determining the state
Control Engineering (MHH324719)
Semester B – 2021-22
M&IE Department
model, the controllability of the state variable is verified. If the state variable is not
controllable then we must go for another choice of state variable.
A system is said to be completely state controllable if it is possible to transfer the system
state from any initial state X(t0) to any other desired X(td) in specified finite time by a control
vector U(t).
The controllability of a state model can be tested by Kalman’s test.
Consider a system with state equation,
̇ = +
For this system, a composite matrix CM can be formed such that
= [ 2 … . -1 ]
Where n is the order of the system (n is also equal to number of state variables).
In this can the system is completely state controllable if the rank of the composite matrix, CM
is n.
The rank of the matrix is n if the determinant of (n x n) composite matrix CM is non-zero.
That is if,
| | ≠ 0
Then rank of CM = n than the system is completely state controllable.
The advantage of Kalman’s test is that the calculations are simpler. But the disadvantage is
we can’t find the state variable which is uncontrollable.
Observability:
In Observability test we can find whether the state variable is observable or measurable. The
concept of Observability is useful in solving the problem of reconstructing un-measurable
state variables from measurable ones in the minimum possible length of time. In state
feedback control the estimation of un-measurable state variables is essential in order to
construct the control signal.
A system is said to be completely observable if every state X(t) can be completely identified
by measurements of the output Y(t) over a finite time interval.
The Observability of a system can be tested by Kalman’s method. Consider a system with
state model
Consider a system with state equation,
̇ = +
For this system, a composite matrix CM can be formed such that
Where n is the order of the system (n is also equal to number of state variables).
In this can the system is completely state controllable if the rank of the composite matrix, O
is n.
The rank of the matrix is n if the determinant of (n x n) composite matrix CM is non-zero.
That is if,
|O| ≠ 0
Then rank of O = n than the system is completely state controllable.
The advantage of Kalman’s test is that the calculations are simpler. But the disadvantage is
we can’t find the state variable which is unobservable.
Eigenvalues and Eigenvectors:
Control Engineering (MHH324719)
Semester B – 2021-22
M&IE Department
Consider multiplying a square 3×3 matrix by a 3×1 (column) vector. The result is a 3×1
(column) vector. The 3×3 matrix can be thought of as an operator – it takes a vector,
operates on it, and returns a new vector. There are many instances in mathematics and
physics in which we are interested in which vectors are left “essentially unchanged” by the
operation of the matrix. Specifically, we are interested in those vectors v for which Av=kv
where A is a square matrix and k is a real number. A vector v for which this equation hold is
called an eigenvector of the matrix A and the associated constant k is called the eigenvalue
(or characteristic value) of the vector v. If a matrix has more than one eigenvector the
associated eigenvalues can be different for the different eigenvectors.
Geometrically, the action of a matrix on one of its eigenvectors causes the vector to stretch
(or shrink) and/or reverse direction.
To find the eigenvalues of a nxn matrix A (if any), we solve Av=kv for scalar(s) k.
Rearranging, we have Av-kv=0. But kv=kIv where I is the nxn identity matrix
Input: INPUT TO THE MATLAB PROGRAM: Matlab codes
Assignment 1
MATLAB codes input
A=[1 1 -1; 4 3 0; -2 1 10]
B=[0; 0; 1]
C = [20 30 10];
D=[0];
% a) system transfer function
[num, den] = ss2tf (A, B, C, D);
g = tf (num, den)
Control Engineering (MHH324719)
Semester B – 2021-22
M&IE Department
Rank_A=rank(A)
% b) system controllability
Cm=ctrb(A,B)
Rank=rank(Cm)
% c) System Observability
Om=obsv(A,C)
Rank=rank(Om)
MATLAB output
A =
1
4
-2
1
3
1
-1
0
10
B =
0
0
1
g =
10 s^2 – 60 s – 70
———————–s^3 – 14 s^2 + 37 s + 20
Continuous-time transfer function.
Rank_A =
3
Cm =
0
0
1
-1
0
10
-11
-4
102
Rank =
3
Control Engineering (MHH324719)
Semester B – 2021-22
M&IE Department
Om =
20
120
440
30
120
560
10
80
680
Rank =
3
Since,
Rank_A = Rank_Cm
System is controllable.
Rank_A = Rank_Om
System is observable.
Assignment 2
MATLAB codes input
A = [3 2 4 ; 2 0 2; 4 2 3];
eig (A)
)
Output
MATLAB codes input
ans =
-1.0000
-1.0000
8.0000
Eign values shows the poles of a system. Two poles present at -1 (left of the plane) and one pole exist
at 8 (right of the plane). So, this system is not stable because one poles is in the right side of the plane.
Methods:
Control Engineering (MHH324719)
Semester B – 2021-22
M&IE Department
locate important information
ss2tf()
In the first assignment, system transfer function is obtained from state variable matrixes by using the
built in function ss2tf(). This function gives transfer function of the system from state equations
matrixes.
Ctrb()
This function is used to determine that the given system is controllable or not.
Obsy()
This function is used to determine that the given system is observable or not.
Eig()
This function is used to calculate the eigen values from the given matrix.
Results:
In the first assignment, system is controllable because the rank of the controllable matrix is equal to the
rank of the state variable matrix. System is also observable because the rank of the controllable matrix
is equal to the rank of the state variable matrix.
In the second assignment, two eigen values are -ve that shows the two poles are in the left side of
plane and one eigen value is +ve that show one pole is right side of plane, and thus this system is not
stable.
Conclusion:
This assignment consists of two assignments, in one assignment I have determined the transfer
function from given state equation and I have determined that the system is controllable and observable
by using the matlab coding. In the second assignment, I have evaluated the eigen values that show this
system is not stable.
Reference:
write the in-text citation and prepare the reference in CCE-Harvard referencing style

https://en.wikipedia.org/wiki/Controllability
https://www.ece.rutgers.edu/~gajic/psfiles/chap5traCO.pdf

http://www.math.iitb.ac.in/~neela/CIMPA/notes/CIMPA_RKG.pdf
Control Engineering (MHH324719)
Semester B – 2021-22
M&IE Department

https://intra.engr.ucr.edu/~enozari/teaching/ME120_Fall20/Lecture%203%20%20Stability,%20Controllability%20&%20Observability.pdf
https://mathworld.wolfram.com/Eigenvalue.html
https://www.mathworks.com/help/matlab/math/eigenvalues.html
https://www.geeksforgeeks.org/eigenvalues-and-eigenvectors-in-matlab/
Control Engineering (MHH324719)
Semester B – 2021-22

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