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

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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

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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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