Stability Analysis in Control Systems
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U
NIT
4
STABILITY ANALYSIS
Subject Name: Control Systems
Subject Code:10ES43
Prepared By: M.Brinda, Sreepriya Kurup,
Robina Gujral
Department: ECE
Date:30/3/2015
UNIT 4
STABILITY ANALYSIS
T
OPICS
Concepts of stability
Necessary conditions for stability
Routh Stability Criterion
Relative stability analysis
More on Routh Stability Criterion
The Concept of Stability
The concept of stability
can be illustrated by a
cone placed on a plane
horizontal surface.
A necessary and
sufficient condition
for a feedback system
to be stable is that
all the poles of the
system transfer
function have
negative real parts.
A system is considered marginally stable if only certain
bounded inputs will result in a bounded output.
S
TABLE
SYSTEM
Response or output is predictable.
A system is said to be stable if for a
bounded disturbing input signal, the output
vanishes ultimately as t
infinity.
A system is unstable if for a bounded
disturbing input signal the output is of
infinite amplitude or oscillatory.
i)
For a bounded i/p, it produces unbounded
o/p.
ii)
In the absence of i/p, o/p may not return to
zero. It shows certain o/p without i/p.
STABLE SYSTEM
UNSTABLE SYSTEM
UNCONTROLLABLE RESPONSE
D
EFINITIONS
OF
STABILITY
Bounded Input, Bounded Output
(BIBO) Stability:
A system is said to be BIBO Stable if
i)
For Bounded i/p, we have Bounded
o/p; o/p – Controllable.
ii)
In the absence of i/p, o/p must tend to
zero irrespective of initial conditions.
Relaxed system:
A System in which
initial conditions are zero.
D
EFINITIONS
OF
STABILITY
Critically or Marginally Stable system:
for a bounded i/p, o/p oscillates with constant
frequency and amplitude. Such oscillations
are called Damped or sustained oscillations.
Conditionally Stable system:
o/p is bounded only for certain condition. If
this condition is violated, o/p is unbounded.
Stability depends on condition of parameter
of the system.
CRITICALLY OR MARGINALLY
STABLE SYSTEM
D
EFINITIONS
OF
STABILITY
Zero input stability:
If the zero input response of the system subjected
to finite initial conditions, reaches to zero as
t
infinity, then the system is zero input stable.
Asymptotic Stability:
As magnitude of zero input response reaches zero
as t approaches infinity, then zero input
stability is also called asymptotic stability.
If in the absence of i/p, the o/p tends to zero or to
the equilibrium state irrespective of initial
conditions.
D
EFINITIONS
OF
STABILITY
Absolutely Stable system:
If the system o/p is stable for all
variations of its parameters then the
system is called absolutely stable
system.
T
RANSFER
F
UNCTION
When order of the denominator polynomial is
greater than the numerator polynomial the
transfer function is said to be ‘
proper
’.
Otherwise ‘
improper
’
14
S
TABILITY
OF
C
ONTROL
S
YSTEM
Roots of denominator polynomial of a transfer function
are called ‘
poles
’.
And the roots of numerator polynomials of a transfer
function are called ‘
zeros
’.
15
S
TABILITY
OF
C
ONTROL
S
YSTEM
Poles of the system are represented by ‘
x
’ and zeros of
the system are represented by ‘
o
’.
System order is always equal to number of poles of the
transfer function.
Following transfer function represents
n
th
order plant.
16
S
TABILITY
OF
C
ONTROL
S
YSTEM
Poles is also defined as “it is the frequency at
which system becomes infinite”. Hence the
name pole where field is infinite.
And zero is the frequency at which system
becomes 0.
17
S
TABILITY
OF
C
ONTROL
S
YSTEM
Poles is also defined as “it is the frequency at
which system becomes infinite”.
Like a magnetic pole or black hole.
18
S
TABILITY
OF
C
ONTROL
S
YSTEMS
The poles and zeros of the system are plotted
in
s-plane
to check the stability of the system.
19
S
TABILITY
OF
C
ONTROL
S
YSTEMS
If all the poles of the system lie in left half
plane the system is said to be
Stable
.
If any of the poles lie in right half plane the
system is said to be
unstable
.
If pole(s) lie on imaginary axis the system is
said to be
marginally stable
.
20
If all the poles
S
TABILITY
OF
C
ONTROL
S
YSTEMS
For example
Then the only pole of the system lie at
21
LOCATION OF ROOTS ON S PLANE
LOCATION OF ROOTS ON S PLANE
LOCATION OF ROOTS ON S PLANE
LOCATION OF ROOTS ON S PLANE
CONCLUSIONS BASED ON THE LOCATION OF
ROOTS OF CHARACTERISTIC EQUATION
Roots-LHS – negative real parts – Response –
Bounded- BIBO Stable.
Roots-RHS – Positive real parts –Response –
Unbounded- Unstable.
Repeated roots on Imaginary axis –Response –
Unbounded- unstable.
Single root at origin – Bounded –Unstable.
Repeated roots at origin –Unbounded, unstable.
Non repeated roots on imaginary axis or single
pole at origin- Limitedly or marginally stable
system.
O
BSERVATIONS
All the co efficients –Positive =>
roots –LHS
If any co efficient is zero=> roots-
Imaginary axis or RHS
If any co efficient is negative =>
atleast one root -RHS
N
ECESSARY
CONDITIONS
FOR
STABILITY
All the co efficients of a characteristic
polynomial be positive.
If any co efficient is zero or negative, we can
immediately say that the system is unstable.
But not sufficient condition
s
3
+ s
2
+2s+8 = (s+2) (s- 0.5 – 1.93j) (s-
0.5+1.93j)
Co efficients –positive but roots –RHS
So s/m – Unstable.
R
OUTH
H
URWITZ
C
RITERION
Sufficient conditions for stability.
Hurwitz – investigated stability interms of
determinants.
Routh – in terms of array formulation.
Routh Stability criterion:
The necessary and sufficient condition for
stability is that all the elements in the first
column of the routh array must be positive. If
this condition is not met, the system is
unstable and the no of sign changes in the
elements of the first column of the routh array
corresponds to the no of roots of characteristic
equation in RHS of s plane.
Stability analysis in control systems is crucial for ensuring the predictability and reliability of system responses. It involves assessing various stability criteria such as Routh Stability Criterion and investigating the behavior of systems under different conditions like bounded inputs. Understanding the concepts of stability, from necessary conditions to types of system responses, is essential for designing robust and effective control systems.
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Presentation Transcript
Subject Name: Control Systems Subject Code:10ES43 UNIT 4 STABILITY ANALYSIS Robina Gujral Prepared By: M.Brinda, Sreepriya Kurup, Department: ECE Date:30/3/2015
UNIT 4 STABILITY ANALYSIS
TOPICS Concepts of stability Necessary conditions for stability Routh Stability Criterion Relative stability analysis More on Routh Stability Criterion
The Concept of Stability The concept of stability can be illustrated by a cone placed on a plane horizontal surface. A necessary and sufficient condition for a feedback system to be stable is that all the poles of the system transfer function have negative real parts. A system is considered marginally stable if only certain bounded inputs will result in a bounded output.
STABLESYSTEM Response or output is predictable. A system is said to be stable if for a bounded disturbing input signal, the output vanishes ultimately as t infinity. A system is unstable if for a bounded disturbing input signal the output is of infinite amplitude or oscillatory. For a bounded i/p, it produces unbounded o/p. ii) In the absence of i/p, o/p may not return to zero. It shows certain o/p without i/p. i)
DEFINITIONSOFSTABILITY Bounded Input, Bounded Output (BIBO) Stability: A system is said to be BIBO Stable if i) For Bounded i/p, we have Bounded o/p; o/p Controllable. ii) In the absence of i/p, o/p must tend to zero irrespective of initial conditions. Relaxed system: A System in which initial conditions are zero.
DEFINITIONSOFSTABILITY Critically or Marginally Stable system: for a bounded i/p, o/p oscillates with constant frequency and amplitude. Such oscillations are called Damped or sustained oscillations. Conditionally Stable system: o/p is bounded only for certain condition. If this condition is violated, o/p is unbounded. Stability depends on condition of parameter of the system.
CRITICALLY OR MARGINALLY STABLE SYSTEM
DEFINITIONSOFSTABILITY Zero input stability: If the zero input response of the system subjected to finite initial conditions, reaches to zero as t infinity, then the system is zero input stable. Asymptotic Stability: As magnitude of zero input response reaches zero as t approaches infinity, then zero input stability is also called asymptotic stability. If in the absence of i/p, the o/p tends to zero or to the equilibrium state irrespective of initial conditions.
DEFINITIONSOFSTABILITY Absolutely Stable system: If the system o/p is stable for all variations of its parameters then the system is called absolutely stable system.
TRANSFER FUNCTION When order of the denominator polynomial is greater than the numerator polynomial the transfer function is said to be proper . Otherwise improper 14
STABILITYOF CONTROL SYSTEM Roots of denominator polynomial of a transfer function are called poles . And the roots of numerator polynomials of a transfer function are called zeros . 15
STABILITYOF CONTROL SYSTEM Poles of the system are represented by x and zeros of the system are represented by o . System order is always equal to number of poles of the transfer function. Following transfer function represents nth order plant. 16
STABILITYOF CONTROL SYSTEM Poles is also defined as it is the frequency at which system becomes infinite . Hence the name pole where field is infinite. And zero is the frequency at which system becomes 0. 17
STABILITYOF CONTROL SYSTEM Poles is also defined as it is the frequency at which system becomes infinite . Like a magnetic pole or black hole. 18
STABILITYOF CONTROL SYSTEMS The poles and zeros of the system are plotted in s-plane to check the stability of the system. j LHP RHP Recall = + j s s-plane 19
STABILITYOF CONTROL SYSTEMS If all the poles of the system lie in left half plane the system is said to be Stable. If any of the poles lie in right half plane the system is said to be unstable. If pole(s) lie on imaginary axis the system is said to be marginally stable. j LHP RHP If all the poles s-plane 20
STABILITYOF CONTROL SYSTEMS For example C = , if = = and C = ( ) , G s A B 1 3 10 + As B Then the only pole of the system lie at = pole 3 j LHP RHP X -3 s-plane 21
CONCLUSIONS BASED ON THE LOCATION OF ROOTS OF CHARACTERISTIC EQUATION Roots-LHS negative real parts Response Bounded- BIBO Stable. Roots-RHS Positive real parts Response Unbounded- Unstable. Repeated roots on Imaginary axis Response Unbounded- unstable. Single root at origin Bounded Unstable. Repeated roots at origin Unbounded, unstable. Non repeated roots on imaginary axis or single pole at origin- Limitedly or marginally stable system.
OBSERVATIONS All the co efficients Positive => roots LHS If any co efficient is zero=> roots- Imaginary axis or RHS If any co efficient is negative => atleast one root -RHS
NECESSARYCONDITIONSFORSTABILITY All the co efficients of a characteristic polynomial be positive. If any co efficient is zero or negative, we can immediately say that the system is unstable. But not sufficient condition s3+ s2+2s+8 = (s+2) (s- 0.5 1.93j) (s- 0.5+1.93j) Co efficients positive but roots RHS So s/m Unstable.
ROUTH HURWITZ CRITERION Sufficient conditions for stability. Hurwitz investigated stability interms of determinants. Routh in terms of array formulation. Routh Stability criterion: The necessary and sufficient condition for stability is that all the elements in the first column of the routh array must be positive. If this condition is not met, the system is unstable and the no of sign changes in the elements of the first column of the routh array corresponds to the no of roots of characteristic equation in RHS of s plane.
