System Stability
DNT 354 - CONTROL PRINCIPLE
Date: 4th September 2008
Prepared by: Megat Syahirul Amin bin Megat Ali
Email: [email protected]
CONTENTS
Introduction
 System Stability
 Routh-Hurwitz Criterion
 Construction of Routh Table
 Determining System Stability

INTRODUCTION

Stability is the most important system specification. If a system
is unstable, the transient response and steady-state errors are
in a moot point.
c(t )  c forced (t )  cnatural(t )

Definition of stability, for linear, time-invariant system by using
natural response:



A system is stable if the natural response approaches zero as time
approaches infinity.
A system is unstable if the natural response approaches infinity as time
approaches infinity
A system is marginally stable if the natural response neither decays nor
grows but remains constant or oscillates.
INTRODUCTION

Definition of stability using the total response bounded-input,
bounded-output (BIBO):
i.
ii.
A system is stable if every bounded input yields a bounded
output.
A system is unstable if any bounded input yields an unbounded
output.
ABSOLUTE & RELATIVE STABILITY

Absolute Stability:
i.
ii.

The absolute stability indicates whether the system is stable or
not.
This is indicated by the presence of one or more poles in RHP.
Relative Stability:
i.
ii.
iii.
Relative stability refers to the degree of stability of a stable
system described by above.
This depends on the transfer function of the system, which is
represented by both the numerator (that yields the zeros) and
denominator (that yields the poles).
This can then be referred to in the study of system response
either in time or frequency domain.
SYSTEM STABILITY

Stable systems have closed-loop transfer functions with poles
only in the left half-plane.
SYSTEM STABILITY

Unstable systems have closed-loop transfer functions with at
least one pole in the right half-plane and/or poles of multiplicity
greater than 1 on the imaginary axis.
SYSTEM STABILITY

Marginally stable systems have closed-loop transfer functions
with only imaginary axis poles of multiplicity 1 and poles on the
imaginary axis.
DETERMINING SYSTEM STABILITY

To determine stability of a given system, we have to consider
the manner in which the system is operating, whether openloop or closed-loop.
i.
ii.
iii.
iv.
v.
If the system is operating in closed-loop, first find the closed
loop transfer function.
Find the closed-loop poles.
If the order of the system is 2 or less, factorise the denominator
of the transfer function. This will provide the roots of the
polynomial, or the closed-loop poles of the system.
If the system order is higher than 2nd-order, use construct Routh
table and apply Routh-Hurwitz Criterion.
Any poles that exist on the RHP will indicate that the system is
unstable.
ROUTH–HURWITZ CRITERION




Routh-Hurwitz Stability Criterion: The number of roots of the
polynomial that are in the right half-plane is equal to the
number of changes in the first column.
Systems with the transfer function having all poles in the LHP is
stable.
Hence, we can conclude that a system is stable if there is no
change of sign in the first column of its Routh table.
However, special cases exists when:
i.
ii.
There exists zero only in the first column.
The entire row is zero.
ROUTH–HURWITZ CRITERION

If a polynomial is given by:
T (s)  an s n  an 1s n 1  .....  a1s  a0  0
Where,
an, an-1, …, a1, a0 are constants
n = 1, 2, 3,…, ∞

The necessary conditions for stability are:
i.
ii.
All the coefficients of the polynomial are of the same sign. If
not, there are poles on the right hand side of the s-plane.
All the coefficient should exist accept for a0.
ROUTH-HURWITZ CRITERION

For the sufficient condition, we must form a Routh-array.
an
b1  
an 1 an 3 an 1an  2  an an 3

an 1
an 1
an
b2  
an  2
an  4
an 1 an 5 an 1an  4  an an 5

an 1
an 1
ROUTH-HURWITZ CRITERION

For the sufficient condition, we must formed a Routh-array.
an 1 an 3
c1  
b1
b2
b1

b1an 3  an 1b2
b1

b1an 5  an 1b3
b1
an 1 an 5
c2  
h1 h2
j1  
i1
i2
i1

i1h2  h1i2
i1
k1  i2
b1
b3
b1
CONSTRUCTION OF ROUTH TABLE
Equivalent closed-loop transfer function
Initial layout for Routh table
Completed Routh table
DETERMINING SYSTEM STABILITY

Example: How many roots exist on RHP?
FURTHER READING…

Chapter 6
i.
ii.
Nise N.S. (2004). Control System Engineering (4th Ed), John
Wiley & Sons.
Dorf R.C., Bishop R.H. (2001). Modern Control Systems (9th Ed),
Prentice Hall.
“If we knew what we were doing, it would not be called research,
would it?…"
THE END…