Adaptive Control

Lecture 3

Control System’s Stability

Dr. Abusabah I. A. Ahmed

abusabah22@hotmail.com
Lecture Outline
❑Introduction
❑Classification of Control Systems
❑Control System’s Stability
❑Routh-Hurwitz Stability Criterion
❑Practice Problems

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Introduction
❑ This lecture covers the basic control system concepts
and stability conditions.
❑ To apply knowledge of mathematics and engineering to
analyze a control system to meet desired specifications.
❑ Control is the process of causing a system variable to
conform to some desired value.
❑ configuration that will provide a desired response.
❑ Control is called automatic if it is accomplished without
manual (human) intervention.
❑States of the system: describes enough about the system to
determine its future behavior in the absence of any external
forces affecting the system.(representation what the system
is currently doing).
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 Classification of Control Systems
❑ Control systems may be classified according to their ability to
follow step inputs, ramp inputs, parabolic inputs, and so on.
❑ The magnitudes of the steady-state errors due to these
individual inputs are indicative of the goodness of the system.
❑ Consider the unity-feedback control system with the following
open-loop transfer function:

❑ It involves the term sN in the denominator, representing N

poles at the origin.
❑ A system is called type 0, type 1, type 2, ... , if N=0, N=1,
N=2, ... , respectively.

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 Classification of Control Systems
❑ As the type number is increased, accuracy is improved.
❑ However, increasing the type number aggravates the stability
problem.
❑ A compromise between steady-state accuracy and relative
stability is always necessary.
❑ Process is the plant, or system under control. The input and output
relationship represents the cause-and effect relationship of the
process. The process outputs are the variables to be controlled.
The process inputs are the variables that are manipulated by the
controller.

Multiple Input Multiple Single Input Single Output

Output (MIMO)
(SISO)
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 Classification of Control Systems
❑ A transfer function is defined as the ratio of the Laplace
transform of the output to the input with all initial conditions
equal to zero. Transfer functions are defined only for linear
time invariant systems.
❑Transfer functions can usually be expressed as the ratio of
two polynomials in the complex variable, s.
K ( s + z )( s + z ) ... ( s + z )
G(s) = 1 2 m

( s + p )( s + p ) ... ( s + p )
1 2 n

The roots of the numerator polynomial are called zeros.

The roots of the denominator polynomial are called poles.

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Classification of Control Systems
System’s Roots

An Example: You are given the following transfer function. Show the
poles and zeros in the s-plane.

( s + 8)( s + 14)
G( s) =
s( s + 4)( s + 10) j axis

S - plane

origin
o x o x x
-14 -10 -8 -4 0  axis

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 Control System’s Stability
❑ Idea of Bounded Input Bounded Output (BIBO)
stability–Apply step input to system:
✓ Stable: output →steady state
✓ Unstable otherwise.
❑ All poles have negative real parts.

❑ A LTI is stable ⇔all the roots of the transfer function

denominator polynomial lie in the LHP
✓ Stable ⇔no poles in RHP or on jω-axis
✓ Repeated poles on jω-axis ⇒unstable
✓ Non-repeated pole on jω-axis marginally stable

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Control System’s Stability
❑System Stability is one of the most
important performance specification of
a control system, which is utmost
importance in designing and analysis of
feedback control systems.
❑A system is said to be stable if the output response is bounded for
all bounded inputs (BIBO). Otherwise, it is said to be unstable.

❑The physical meaning or concept of the stability can be illustrated by

a cone placed on a plane horizontal surface.

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Control System’s Stability
❑The poles and zeros of the system are plotted in s-plane to check
the stability of the system.
✓ If all the poles of the system
lie in left half plane (LHP) the
system is said to be Stable.
✓ If any of the poles lie in right
half plane (RHP) the system is
said to be unstable.
✓ If pole(s) lie on imaginary
axis the system is said to be
marginally stable.

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Control System’s Stability
Example 3-1
Check the system stability for the following systems!

Solution

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 Routh-Hurwitz Stability Criterion
❑ It is a method for determining continuous system stability.
❑ The Routh-Hurwitz criterion states that “the number of roots of
the characteristic equation with positive real parts is equal to
the number of changes in sign of the first column of the Routh
array”.
❑ This method yields stability information without the need to
solve for the closed-loop system poles.
❑ Using this method, we can tell how many closed-loop system
poles are in the left half-plane, in the right half-plane, and on
the jw-axis. (Notice that we say how many, not where.)
❑ The method requires two steps:
✓ Generate a data table called a Routh table.
✓ interpret the Routh table to tell how many closed-loop system
poles are in the LHP, the RHP, and on the jw-axis.
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Routh-Hurwitz Stability Criterion
Characteristic equation, q(s)

For example

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Routh-Hurwitz Stability Criterion
❑Four Special Cases or Configurations in the First Column Array of
the Routh’s Table:

Case-I: No element in the first column is zero.

Case-II: A zero in the first column but some other elements
of the row containing the zero in the first column
are nonzero.
Case-III: A zero in the first column and the other elements
of the row containing the zero are also zero.
Case-IV: As in the third case but with repeated roots on the
jw -axis.

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Routh-Hurwitz Stability Criterion
Case-I: No element in the first column is zero.
Second-Order System. The characteristic polynomial of a second
order system is given below

The Routh array is written as

Where

The requirement for a stable second order system is simply that all
the coefficient be positive or all the coefficients be negative.

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Routh-Hurwitz Stability Criterion
Third-Order System. The characteristic polynomial of a third
order system is given below

The Routh array is

Where

❑ The requirement for a stable third order system is that the coefficients be
positive and
❑ The condition when results in a marginally stability case (recognized
as Case-3 because there is a zero in the first column) and one pair of roots lies on
the imaginary axis in the s-plane.

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Routh-Hurwitz Stability Criterion
Example 3-2

Find the stability of the continues system having the characteristic

equation of

The Routh table of the given system is computed as;

Since there are no sign changes in the first column of the Routh
table, it means that all the roots of the characteristic equation
have negative real parts and hence this system is stable.
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Routh-Hurwitz Stability Criterion
Example 3-3
Find the stability of the continues system having the
characteristic polynomial of a third order

The Routh array is:

Because TWO changes in sign appear in the first column, we

find that two roots of the characteristic equation lie in the right
hand side of the s-plane. Hence the system is unstable.
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Routh-Hurwitz Stability Criterion
Example 3-4
Find the stability of the system shown below using Routh criterion.

The Routh table

of the system is

System is unstable because there are two sign changes in the

first column of the Routh’s table. Hence the equation has
two roots on the right half of the s-plane.
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Routh-Hurwitz Stability Criterion
Case-II: A Zero Only in the First Column
There are TWO methods in case-II.
1. Stability via Epsilon Method.
2. Stability via Reverse Coefficients (Phillips, 1991).
Case-III: Entire Row is Zero.
• Sometimes while making a Routh table, we find that an entire
row consists of zeros.
• This happen because there is an even polynomial that is a factor
of the original polynomial.
• This case must be handled differently from the case of a zero in
only the first column of a row.

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Routh-Hurwitz Stability Criterion
Case-IV: Repeated roots of the characteristic equation on
the jω-axis.
• If the jω-axis roots are repeated, the system response will
be unstable with a form tsin(ωt + Ф). The Routh-Hurwitz
criteria will not reveal this form of instability.

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Practice Problems
1. Consider the following transfer functions.

a. b.

1\ Find the Poles and zeros of the system

2\ Determine the order of the system
3\ Draw the pole-zero map
4\ Determine the Stability of the system

2. Determine the stability for the following 4th order

system.

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 Thank You

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