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System Stability (Special Cases)
Date: 11th September 2008
Prepared by: Megat Syahirul Amin bin Megat Ali
Email: [email protected]
Introduction Zero Only in First Column Zero for Entire Column Stability via Routh Hurwitz
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.
Two special cases exists when:i. There exists zero only in the first column.ii. The entire row is zero.
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.
Two special cases exists when:i. There exists zero only in the first column.ii. The entire row is zero.
Exercise: For the following closed-loop transfer function T(s), determine the number of poles that exist on RHP.
2006116
200)(
234
sssssT
If the first element of a row is zero, division by zero would be required to form the next row.
To avoid this, an epsilon, , is assigned to replace the zero in the first column.
Example: Consider the following closed-loop transfer function T(s).
35632
10)(
2345
ssssssT
To determine the system stability, sign changes were observed after substituting with a very small positive number or alternatively a very small negative number.
Exercise: For the following closed-loop transfer function T(s), determine the number of poles that exist on RHP.
123232
1)(
2345
ssssssT
An entire row of zeros will appear in the Routh table when a purely even or purely odd polynomial is a factor of the original polynomial.
Example: s4 + 5s2 + 7 has an even powers of s. Even polynomials have roots that are symmetrical
about the origin.i. Roots are symmetrical & real ii. Roots are symmetrical & imaginaryiii. Roots are quadrantal
Example:
Differentiate with respect to s:
5684267
10)(
2345
ssssssT
86)( 24 sssP
0124)( 3 ss
ds
sdP
Example:How many poles are on RHP, LHP and jω-axis for the closed-loop system below?
Exercise: For the following closed-loop transfer function T(s), determine the number of poles that exist on RHP, LHP and the jω-axis
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20)(
2345678
sssssssssT
Example: Find the range of gain K for the system below that will cause the system to be stable, unstable and marginally stable, Assume K > 0.
Closed-loop transfer function:
Ksss
KsT
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23
Example: Find the range of gain K for the system below that will cause the system to be stable, unstable and marginally stable, Assume K > 0.
Forming the Routh table:
Example: Find the range of gain K for the system below that will cause the system to be stable, unstable and marginally stable, Assume K > 0.
If K < 1386:All the terms in 1st column will be positive and since there are no sign changes, the system will have 3 poles in the left-half plane and are stable.
If K > 1386:The s1 in the first column is negative. There are 2 sign changes, indicating that the system has two right-half-plane poles and one left-half plane pole, which make the system unstable.
Example: Find the range of gain K for the system below that will cause the system to be stable, unstable and marginally stable, Assume K > 0.
If K = 1386:The entire row of zeros, which signify the existence of jω poles. Returning to the s2 row and replacing K with 1386, so we have: P(s)=18s2 +1386
Chapter 6i. Nise N.S. (2004). Control System Engineering (4th
Ed), John Wiley & Sons.ii. Dorf R.C., Bishop R.H. (2001). Modern Control
Systems (9th Ed), Prentice Hall.
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