Control Engineering Practical

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LAB SESSION NO. 03

System Stability and Pole-Zero Models

OBJECT:

To construct pole-zero models of a system and analyze the stability of control systems.

List of Equipment/Software

The following equipment/software is required:

• MATLAB

Introduction

What is Stability?

A system is said to be stable if its output is under control. Otherwise, it is said to be unstable. A
stable system produces a bounded output for a given bounded input.

The following figure shows the response of a stable system.

This is the response of first order control system for a unit step input. This response has values
between 0 and 1. So, it is bounded output. We know that the unit step signal has the value of
one for all positive values of t including zero. So, it is bounded input. Therefore, the first order
control system is stable since both the input and the output are bounded.

To verify the stability of a system the following command is used.

B = isstable(sys)

returns a logical value of 1 (true) if the dynamic system model sys has stable dynamics, and a
logical value of 0 (false) otherwise.

Poles and Zeros

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Poles and Zeros of a transfer function are the frequencies for which the value of the denominator
and numerator of transfer function becomes zero respectively. The values of the poles and the
zeros of a system determine whether the system is stable, and how well the system performs.
Control systems, in the simplest sense, can be designed simply by assigning specific values to
the poles and zeros of the system.

Physically realizable control systems must have a number of poles greater than or equal to the
number of zeros. Systems that satisfy this relationship are called Proper.

Zeros

The zeros of the system are the roots of the numerator. The zeros can be found by

zeros = roots(num);

Poles

The poles of the system are the roots of the denominator polynomial (the characteristic
polynomial). The poles can be found by

poles = roots(den);

Graphs

To plot the poles, marked by the symbol ‘x’, use the following code

plot(poles,’x’)

and to plot zeros, marked by the symbol ‘o’, use the following code

plot(zeros,’o’)

To plot the poles and zeros of any transfer function there is a built-in function pzmap in
MATLAB.

pzmap(num,den)

Using the grid command makes the plot easier to read and analyse.

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Task
S+ 4 s
a) G(s) = b) G(s)
2
s - 2s + 2 s +1

c) s-3 d) S+1
G(s) G(s)
3
s - 2s +10 s-1

(a) Find zeros and poles of above transfer functions (if any)

(b) Are the above systems stable? Why?

(c) Use the plot function to plot the poles of the above transfer function marked by the symbol
‘x’. Record these plots with your Roll. No on the top of the plot.

(d) Plot the pz maps of the above transfer functions.

Report:

Complete and attach the results of the above task at the end of this lab session.

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 Control Engineering Practical
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MEHRAN UNIVERSITY OF ENGINEERING & TECHNOLOGY JAMSHORO

DEPARTMENT OF MECHANICAL ENGINEERING

RUBRICS
RUBRIC FOR ASSESSMENT OF PRACTICAL / EXPERIMENT

Subject:____________________________________________________________

Practical / Experiment # :

Objective:___________________________________________________________

Student Name & Roll Number:_________________________________________

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TOTAL POINTS 18
EVALUATION MATRIC
CLO # T. Below Average Average Good Score
Evaluation Level 1 2 3 Points
Content P2 Demonstrating minimal Demonstrating some Demonstrating full
Understanding understanding of the understanding of the understanding of the
relevant objectives and relevant objectives relevant objectives and
concepts. and concepts. concepts.

Modularity P3 Use of appropriate Use of appropriate The use of more

functions but lack functions but may still appropriate functions,
CLO-1

coherence or reusability contain some reusable units and

and contain unnecessary unnecessary unnecessary repetition
repetition. repetition. has been eliminated.
Output results P4 Some of the results from Results from Results from MATLAB
MATLAB code in the MATLAB code in the code in the form of
form of output functions form of output output functions are
are shown. functions are fairly shown accurately.
shown.
TOTAL POINTS OBTAINED ON CLO-1
Level of A2 Student loses attention; Student pays attention Students thoroughly
Engagement inconsistent and even participate, engaged by demos,
intellectual & and but are not discussion &
physical engagement intellectually engaged Participation
Completion & A3 The student completed The student completed
The student completed
Safety the demonstration the demonstration
the demonstration
precautions partially and successfully with all-
poorly and somehow
CLO-3

sometimes was time was mindful of

was mindful of safety.
mindful of safety. safety.
Work as an A4 Often listens to, shares Usually listens to, Almost always listens to,
individual/ team with, and supports the shares, with, and shares with, and supports
member & efforts of others, but supports the efforts of the efforts of others.
Report sometimes is not a good others. The report is The report is fully
team member. The partially documented. documented.
report is minimally
documented.
TOTAL POINTS OBTAINED ON CLO-3

Teacher Signature & Date