NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Stability of a Dynamic System
D Bishakh
Dr.
Bi h kh Bhattacharya
Bh tt h
Professor, Department of Mechanical Engineering
IIT Kanpur
Joint Initiative of IITs and IISc - Funded by MHRD
Module 2- Lecture 12
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
This Lecture Contains
 Introduction
Simple Test of Stability
Routh Stability Criteria
Special Cases
An Assignment
An Assignment
Joint Initiative of IITs and IISc - Funded by MHRD
Module 2- Lecture 12
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Module 2- Lecture 12
Introduction
•
The response of a Dynamic System may become unbounded while subjected to
a bounded
b
d d input.
i
t Such
S h systems
t
are referred
f
d as unstable
t bl systems.
t
One
O
common example is an inverted pendulum on a rolling cart as shown below:
•
A Control system could be designed such that by controlling the velocity of the
rolling cart one can control the unstable response of the inverted pendulum.
•
However, we need to first carry out a stability analysis of the system.
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NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Module 2- Lecture 12
How to test the stability of a system
•
•
A simple method to test the stability of a system is by checking the poles of the
system
t
transfer
t
f ffunction.
ti
Consider a system which is represented by a generalized transfer function as
follows:
N ( s ) c0 s m  c1s m 1  ....  cm 1s  cm

T (s) 
D( s ) a0 s n  a1s n 1  ...  an 1s  an
•
•
•
Now, equating the denominator polynomial D(s) to zero, one can obtain the
characteristic equation for the system. The roots of this characteristic
equations
q
are the poles
p
of the system.
y
If you obtain one or more poles with positive real part then the system could be
predicted to be an unstable system.
However, it is often tedious to obtain the poles of a complex system before
predicting stability condition of the system
system.
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NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Module 2- Lecture 12
Routh’s
Routh
s Test for Stability
For a characteristic polynomial D(s), the number of poles in the right‐half plane may be determined without actually finding the roots by using the Routh Test.
The Routh array for the polynomial D(s) could be constructed as follows:
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Module 2- Lecture 12
Routh’s Theorem
YYou may have observed that the first row of the Routh table consists of odd coefficients h
b
d th t th fi t
f th R th t bl
i t f dd
ffi i t
of D(s) starting from the first coefficient related to sn . Again, the second row consists of the even coefficients starting from the second coefficient related to sn‐1 .
The coefficients b1 etc. For the third row could be computed as follows. The same pattern could be used for the subsequent rows.
Routh s Theorem: The number of roots of the characteristic polynomial D(s) in the Routh’s
Theorem: The number of roots of the characteristic polynomial D(s) in the
right‐half plane equals the number of sign changes in the first column of the Routh Table.
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Stability as a function of a parameter
1. Use Mason’s rule to find out the determinant of the system
2. Numerator of the determinant will be the characteristic polynomial
3. Use Routh’s test and obtain the array
4. Note the conditions for stability analysing the left coloumn coloumn
Module 2- Lecture 12
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Module 2- Lecture 12
Unusual Case: Left Column Zero
•
•
•
•
Consider D(s)
( ) = 3s4 + 6s3 + 2s2 + 4s + 5
Note appearance of zero in the first column
Rename the row as Row A
C
Create
row B ffrom Row
R
Ab
by sliding
lidi the
h A row to
left until you get a non-zero pivot
• The sign of the row is changed by (-1)
( 1)n where n
is the number of times this row is slided
• The new non-zero row is formed by adding A
and B
• [Reference Benedir and Picinbond, IEEE Trans
on Automatic Control,, 1990]]
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Module 2- Lecture 12
Other approaches for zero left column
Alternate approaches:
pp
Put a parameter, say ε instead of zero in the pivot,
assuming it to be a very small positive number
Continue and find sign changes
OR
Write the polynomial in reverse order such that the
roots of the reverse polynomial will be the
reciprocal
i
l off the
h roots off the
h original
i i l polynomial
l
i l
and follow the same procedure
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Module 2- Lecture 12
Assignment - Unusual Case: A zero row
Consider the polynomial D(s) as
s8 + 3 s7 + 10 s6 + 24 s5 + 48 s4 + 96s
96 3 + 128 s2 + 192 s+ 128
Find out the stability of the system.
Hints: When you encounter a zero row, go back to the last row, construct the
corresponding polynomial (even/odd).
Differentiate the polynomial and obtain the new non-zero row.
row
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Module 2- Lecture 12
Special References for this lecture
 Control Engineering and introductory course, Wilkie, Johnson and Katebi,
PALGRAVE
 Control Systems Engineering – Norman S Nise, John Wiley & Sons
 Modern Control Engineering – K. Ogata, Prentice Hall
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