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.