Csl14 16 f15

ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Control Systems
Stability and Routh Hurtwitz criterion

Stability introduction
• Requirements for design of a control system
– Transient response
– Stability
– Steady state errors
• Stability – most important parameter for
design
• Total response
𝑐 𝑡 = 𝑐𝑓𝑜𝑟𝑐𝑒𝑑 𝑡 + 𝑐 𝑛𝑎𝑡𝑢𝑟𝑎𝑙(𝑡)

System Stability Definition
Types of stability based on Natural response
definition:
1. A system is STABLE if the natural response
approaches zero as time approaches infinity
2. A system is UNSTABLE if the natural response
approaches infinity as time approaches infinity
3. A system is MARGINALLY STABLE if the natural
response neither decays nor grows but remains
constant or oscillates
BIBO Definition
1. A system is stable if every bounded input yields a
bounded output
2. A system is unstable if any bounded input yields an
unbounded output

How to define stability
H(s)
G(s)
R(s) C(s)+
-
Stability with respect to G(s)?
All poles in the left half plane
Stability with respect to
𝑮(𝒔)
𝟏+𝑮 𝒔 𝑯(𝒔)
?
Poles of 1+G(s)H(s) in the
left half.

System Stability Definition – Stable
System
Time approaches
infinity the natural
response approaches
zero
Bounded input
yields bounded
output
Stable system
have poles
only in the left
hand plane

System Stability Definition – Unstable
System
Time approaches
infinity the natural
response
approaches
infinity
Bounded input
yields an
unbounded
output
Unstable
system have
at least one
pole in the
right hand
plane And/or poles of multiplicity greater
than one on imaginary axis

System Stability Definition
Stable system –
closed loop
transfer function
poles only in the
left half plane
Unstable system –
closed loop transfer
function poles with at
least one pole in the
right half and/or poles of
multiplicity greater than
1 on the imaginary axis
𝑨𝒕 𝒏
𝒄𝒐𝒔(𝝎𝒕 + ∅)
Marginally stable –
closed loop transfer
function with only
imaginary axis poles
of multiplicity 1 and
poles in the left half
plane.

j
1
-1













Routh-Hurwitz Stability Criterion
Method to know how many closed-loop
system poles are in the left hand plane, how
many are in the right hand plane and how
many are on the imaginary axis
Step:
1. Generate Routh Table
2. Interpret Routh Table

Routh-Hurwitz Stability Criterion –
Generate Routh Table
Given Routh Table

Generate Routh Table
Routh Table
The value in a
row can be
divided for
easy
calculation

Generate Routh Table Example
Given

Generate Routh Table Example
10303110 23
 ssS

Interpret Routh Table Example
The number of roots of the
polynomial that are in the
right-half plane is equal to
the number of sign
changes in the first column

Interpret Routh Table Example
Two sign changes = two right half plane poles, therefore unstable system

Routh-Hurwitz Stability Criterion –Example
How many roots are in the right-half plane
and in the left-half plane?
62874693)( 234567
 ssssssssP

Example
Determine the value of gain K to make the
system stable

Routh-Hurwitz Criterion – Special Cases
Special cases:
1. Zero in the first column
2. Zero in the entire row

Routh-Hurwitz Criterion – Zero in the
first column case
35632
10
)( 2345


sssss
sT
How many poles?
Five poles

first column case
How many sign
changes?
Two sign changes
Two poles are on
the right half
plane
The system is
unstable

first column case
Alternative method Reverse the coefficients
35632
10
)( 2345


sssss
sT
35632 2345
 sssss 123653 2345
 sssss

Routh-Hurwitz Criterion – Zero in
the first column case
123653 2345
 sssss
How many sign changes?
Two sign changes
Same as previous result

Routh-Hurwitz Criterion – Zero in
the entire row
5684267
10
)( 2345


sssss
sT
0 0 0
What to do?
86)( 24
 sssP ss
ds
sdP
124
)( 3

4 12 0

entire row
How many sign changes?
No sign changes

entire row
What can we learn when the entire
row is zero?
An entire row of zero will appear in
the Routh Table when a purely
even or a purely odd polynomial is
a factor of original polynomial
Even polynomial only has
roots symmetry about the
origin
If we do not have row of
zeros, we don’t have roots on
imaginary axis

entire row
20384859392212 2345678
 ssssssss
0 0 0 0
23)( 24
 sssP ss
ds
sdP
64
)( 3

4 6 0 0

entire row
20384859392212 2345678
Apply only
to even
polynomial
Apply to
original
polynomial

entire row
20384859392212 2345678
No sign
changes
No right
half plane
poles.
Because
symmetry,
no left-half
poles.
Two sign
changes
Two right
half poles

Routh-Hurwitz Criterion – Example

Ksss
K
sT


7718
)( 23
K < 1386, The system is stable
K > 1386, The system is unstable

K = 1386, the system is marginally stable
K = 1386

Csl14 16 f15

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