The Routh-Hurwitz criterion is a mathematical test used to determine the stability of a system. This calculator allows you to input the coefficients of a polynomial and check if the system represented by the polynomial is stable.
Understanding the Routh-Hurwitz Criterion
The Routh-Hurwitz criterion provides a systematic method to check the stability of a linear time-invariant system without solving for the roots of the characteristic equation. The criterion states that a polynomial is stable if and only if all the elements in the first column of the Routh array have the same sign and are non-zero.
Δ(s) = a₀sⁿ + a₁sⁿ⁻¹ + a₂sⁿ⁻² + ... + aₙ
Variables:
- Δ(s) is the characteristic polynomial.
- a₀, a₁, a₂, …, aₙ are the coefficients of the polynomial.
To form the Routh array, the coefficients of the polynomial are arranged in a tabular format, and the elements of the first column are evaluated to determine stability.
What is Stability Analysis?
Stability analysis is a fundamental concept in control systems engineering. It involves determining whether the output of a system will remain bounded and behave predictably over time. Stability ensures that the system responds appropriately to inputs without exhibiting undesirable behaviors like oscillations or divergence.
How to Use the Routh-Hurwitz Criterion?
The following steps outline how to apply the Routh-Hurwitz criterion:
- Write down the characteristic polynomial of the system.
- Form the Routh array using the coefficients of the polynomial.
- Evaluate the elements of the first column of the Routh array.
- Check the signs of the elements in the first column. If they are all positive or all negative and non-zero, the system is stable.
- If any element in the first column is zero or changes sign, the system is unstable.
Example Problem:
Use the following polynomial as an example problem to test your knowledge:
Δ(s) = s³ + 3s² + 5s + 7
FAQ
1. What is a characteristic polynomial?
The characteristic polynomial of a system is derived from its characteristic equation, which is obtained from the system’s differential equations. It is used to analyze system stability.
2. How does the Routh-Hurwitz criterion work?
The Routh-Hurwitz criterion works by arranging the coefficients of the characteristic polynomial into a Routh array and analyzing the signs of the elements in the first column of the array.
3. Why is stability analysis important?
Stability analysis is crucial for ensuring that a system will perform as expected under various conditions. It helps in designing controllers and predicting system behavior.
4. Can this calculator handle high-order polynomials?
Yes, this calculator can be used for polynomials of any order, as long as the coefficients are provided correctly.
5. Is the Routh-Hurwitz criterion applicable to non-linear systems?
The Routh-Hurwitz criterion is primarily used for linear time-invariant systems. For non-linear systems, other methods like Lyapunov’s direct method are used.