System Stability (Special Cases) Date: 11 th September 2008 Prepared by: Megat Syahirul Amin bin Megat Ali

 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.

 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).

 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.

 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: s 4 + 5s 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 & imaginary iii. Roots are quadrantal

 Example:  Differentiate with respect to s:

 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

 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:

 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 1 st 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 s 1 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 s 2 row and replacing K with 1386, so we have: P(s)=18s

 Chapter 6 i. 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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