Routh-Hurwitz Criterion

Definition

The Routh-Hurwitz Criterion is a mathematical test used to determine the stability of a linear time-invariant (LTI) system by analyzing the coefficients of its characteristic polynomial. This criterion provides conditions for stability based on the arrangement of coefficients in a constructed table, allowing for the assessment of whether all roots of the polynomial have negative real parts. It connects deeply with linearization and stability analysis by offering a systematic approach to evaluate system stability without directly computing the roots.

5 Must Know Facts For Your Next Test

1. The Routh-Hurwitz Criterion helps determine stability without needing to find the roots of the characteristic polynomial explicitly, which can be computationally intensive.
2. For a system to be stable, all entries in the first column of the Routh array must be positive, indicating that all roots have negative real parts.
3. The method is applicable for polynomials of any order and can handle cases where complex roots or repeated roots are present.
4. The Routh array is constructed by organizing coefficients into rows based on their polynomial degrees, allowing for easy assessment of stability conditions.
5. If any row of the Routh array consists entirely of zeros, additional steps must be taken to resolve this situation, usually involving auxiliary equations.

Review Questions

• How does the Routh-Hurwitz Criterion provide an alternative method for determining stability compared to calculating the roots of the characteristic polynomial?
  • The Routh-Hurwitz Criterion allows for a systematic assessment of stability through the construction of a Routh array from the coefficients of the characteristic polynomial. By examining this array, one can determine the signs of the first column entries, indicating whether all roots have negative real parts. This method simplifies the process as it avoids direct computation of roots, which can be complicated or impractical for higher-order systems.
• What specific conditions must be met in the Routh array to confirm that a system is stable, and how does this relate to the system's response to disturbances?
  • To confirm stability using the Routh-Hurwitz Criterion, all elements in the first column of the Routh array must be positive. This condition indicates that all roots of the characteristic polynomial lie in the left half-plane, ensuring that any disturbances will die out over time and that the system will return to equilibrium. If any element is zero or negative, it suggests potential instability, leading to unbounded output responses.
• Evaluate how an understanding of the Routh-Hurwitz Criterion can enhance control system design when dealing with non-linear systems and their linearized models.
  • Understanding the Routh-Hurwitz Criterion equips engineers with tools to assess and ensure stability in control systems effectively. When dealing with non-linear systems, linearization around an equilibrium point often leads to a characteristic polynomial that can be analyzed using this criterion. By applying it during design phases, engineers can anticipate potential instabilities before implementing control strategies, ultimately leading to more robust and reliable control systems in practice.

Related terms

Characteristic Polynomial: A polynomial whose roots are the eigenvalues of a system, determining the system's dynamic behavior.

Pole: A value of 's' in the Laplace transform that makes the system's transfer function infinite, representing system behavior in the frequency domain.

Stability: The property of a system where its output remains bounded for bounded inputs, implying that it will return to equilibrium after a disturbance.