Routh-Hurwitz Criterion and Stability Analysis

Introduction

• Routh-Hurwitz Criterion (RH Criterion):
  • A method to determine the stability of a control system.
  • Helps to analyze stability without solving high-order characteristic equations.

Necessary Conditions for Stability

• Poles and Stability:

  • Closed-loop poles are the roots of the characteristic equation.
  • Higher-order equations make it difficult to find roots and comment on stability.

• Conditions:

  1. Same Sign Coefficients:
     • All coefficients of the characteristic polynomial must have the same sign.
  2. Non-Vanishing Coefficients:
     • All powers of s must be present.

• Hurwitz Polynomial:

  • A polynomial satisfying the necessary conditions.
  • If not a Hurwitz polynomial, the system is unstable.

Routh-Hurwitz Criterion (RH Criterion)

• Origins:

  • Developed by Edward Routh and Adolf Hurwitz.
  • Routh proposed an algorithm in 1876, and Hurwitz developed a matrix method in 1895.

• RH Criterion:

  • A mathematical test for the stability of an LTI system.
  • Requires certain algebraic combinations of coefficients to have the same sign.

• Routh Array:

  • Tabulate coefficients in a specific way to form a Routh array.
  • Analyze the array to determine the number of poles in the right half of the s-plane.

Method to Form Routh Array

1. Create a Table with Decreasing Powers of s.
2. Fill Coefficients in Rows:
   • First two rows can be filled directly from the characteristic equation.
   • Fill coefficients in an alternating manner.
3. Calculate Remaining Rows:
   • Use previously filled rows to calculate new entries.
   • Use a determinant-like calculation for further rows.
4. Routh Stability Criteria:
   • All terms in the first column must have the same sign.
   • No sign changes indicate stability.

Examples

• Example 1:

  • Given polynomial: (s^3 + 6s^2 + 11s + 6 = 0)
  • Formed Routh array shows no sign change, indicating stability.

• Example 2:

  • Given polynomial: (s^3 + 4s^2 + s + 16 = 0)
  • Routh array has two sign changes, indicating instability.
  • Number of sign changes corresponds to the number of poles in the right half-plane.

Conclusion

• RH Criterion provides a method to assess system stability without solving high-order equations.
• Necessary conditions must be met, and the Routh Array helps evaluate the sufficient conditions.
• Further examples and special cases to be discussed in the next lecture.