1. 1
ME451: Control Systems
ME451: Control Systems
Dr.
Dr. Jongeun
Jongeun Choi
Choi
Department of Mechanical Engineering
Department of Mechanical Engineering
Michigan State University
Michigan State University
Lecture 10
Lecture 10
Routh
Routh-
-Hurwitz stability criterion
Hurwitz stability criterion
2
Course roadmap
Course roadmap
Laplace transform
Laplace transform
Transfer function
Transfer function
Models for systems
Models for systems
•
• electrical
electrical
•
• mechanical
mechanical
•
• electromechanical
electromechanical
Linearization
Linearization
Modeling
Modeling Analysis
Analysis Design
Design
Time response
Time response
•
• Transient
Transient
•
• Steady state
Steady state
Frequency response
Frequency response
•
• Bode plot
Bode plot
Stability
Stability
•
• Routh
Routh-
-Hurwitz
Hurwitz
•
• Nyquist
Nyquist
Design specs
Design specs
Root locus
Root locus
Frequency domain
Frequency domain
PID & Lead
PID & Lead-
-lag
lag
Design examples
Design examples
(
(Matlab
Matlab simulations &) laboratories
simulations &) laboratories
3
Stability summary (review)
Stability summary (review)
ƒ
ƒ (BIBO, asymptotically) stable
(BIBO, asymptotically) stable if
if
Re(s
Re(si
i)<0 for all i.
)<0 for all i.
ƒ
ƒ marginally stable
marginally stable if
if
ƒ
ƒ Re(s
Re(si
i)<=0 for all i, and
)<=0 for all i, and
ƒ
ƒ simple root for
simple root for Re(s
Re(si
i)=0
)=0
ƒ
ƒ unstable
unstable if
if
it is neither stable nor
it is neither stable nor
marginally stable.
marginally stable.
Let
Let s
si
i be
be poles
poles of
of
rational G. Then, G is
rational G. Then, G is …
…
4
Routh
Routh-
-Hurwitz criterion
Hurwitz criterion
ƒ
ƒ This is for LTI systems with a
This is for LTI systems with a polynomial
polynomial
denominator (without sin,
denominator (without sin, cos
cos, exponential etc.)
, exponential etc.)
ƒ
ƒ It determines if all the roots of a polynomial
It determines if all the roots of a polynomial
ƒ
ƒ lie in the open LHP (left half
lie in the open LHP (left half-
-plane),
plane),
ƒ
ƒ or equivalently, have negative real parts.
or equivalently, have negative real parts.
ƒ
ƒ It also determines the number of roots of a
It also determines the number of roots of a
polynomial in the open RHP (right half
polynomial in the open RHP (right half-
-plane).
plane).
ƒ
ƒ It does
It does NOT
NOT explicitly compute the roots.
explicitly compute the roots.

2. 5
Polynomial and an assumption
Polynomial and an assumption
ƒ
ƒ Consider a polynomial
Consider a polynomial
ƒ
ƒ Assume
Assume
ƒ
ƒ If this assumption does not hold, Q can be factored as
If this assumption does not hold, Q can be factored as
where
where
ƒ
ƒ The following method applies to the polynomial
The following method applies to the polynomial
6
Routh
Routh array
array
From the given
From the given
polynomial
polynomial
7
Routh
Routh array
array
(How to compute the third row)
(How to compute the third row)
8
Routh
Routh array
array
(How to compute the fourth row)
(How to compute the fourth row)

3. 9
Routh
Routh-
-Hurwitz criterion
Hurwitz criterion
The number of roots
The number of roots
in the open right half
in the open right half-
-plane
plane
is equal to
is equal to
the number of sign changes
the number of sign changes
in the
in the first column
first column of
of Routh
Routh array.
array.
10
Example 1
Example 1
Routh
Routh array
array
Two sign changes
Two sign changes
in the first column
in the first column
Two roots in RHP
Two roots in RHP
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Example 2
Example 2
Routh
Routh array
array
If 0 appears in the first column of a
If 0 appears in the first column of a
nonzero row in
nonzero row in Routh
Routh array, replace it
array, replace it
with a small positive number. In this
with a small positive number. In this
case, Q has some roots in RHP.
case, Q has some roots in RHP.
Two sign changes
Two sign changes
in the first column
in the first column
Two roots
Two roots
in RHP
in RHP
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Example 3
Example 3
Routh
Routh array
array
If zero row appears in
If zero row appears in Routh
Routh array, Q
array, Q
has roots either on the imaginary axis
has roots either on the imaginary axis
or in RHP.
or in RHP.
No sign changes
No sign changes
in the first column
in the first column
No roots
No roots
in RHP
in RHP
But
But some
some
roots are on
roots are on
imag
imag. axis.
. axis.
Take derivative
Take derivative of an
of an auxiliary polynomial
auxiliary polynomial
(which is a factor of
(which is a factor of Q(s
Q(s))
))

4. 13
Example 4
Example 4
Routh
Routh array
array
No sign changes
No sign changes
in the first column
in the first column
Find the range of K
Find the range of K s.t
s.t.
. Q(s
Q(s) has all roots in the left
) has all roots in the left
half plane. (Here, K is a design parameter.)
half plane. (Here, K is a design parameter.)
14
Simple & important criteria for stability
Simple & important criteria for stability
ƒ
ƒ 1
1st
st order polynomial
order polynomial
ƒ
ƒ 2
2nd
nd order polynomial
order polynomial
ƒ
ƒ Higher order polynomial
Higher order polynomial
15
Examples
Examples
All roots in open LHP?
All roots in open LHP?
Yes / No
Yes / No
Yes / No
Yes / No
Yes / No
Yes / No
Yes / No
Yes / No
Yes / No
Yes / No
16
Summary and Exercises
Summary and Exercises
ƒ
ƒ Routh
Routh-
-Hurwitz stability criterion
Hurwitz stability criterion
ƒ
ƒ Routh
Routh array
array
ƒ
ƒ Routh
Routh-
-Hurwitz criterion is applicable to only
Hurwitz criterion is applicable to only
polynomials (so, it is not possible to deal with
polynomials (so, it is not possible to deal with
exponential, sin,
exponential, sin, cos
cos etc.).
etc.).
ƒ
ƒ Next,
Next,
ƒ
ƒ Routh
Routh-
-Hurwitz criterion in control examples
Hurwitz criterion in control examples
ƒ
ƒ Exercises
Exercises
ƒ
ƒ Read
Read Routh
Routh-
-Hurwitz criterion in the textbook.
Hurwitz criterion in the textbook.
ƒ
ƒ Do Examples.
Do Examples.