Routh Stability Criterion Calculator

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Reviewed by David Chen, CFA • Control Systems Specialist

Determine the stability of your dynamic system instantly. This Routh Stability Criterion Calculator analyzes the characteristic equation of a system to identify the number of roots in the right-half of the s-plane, providing a full Routh Array and step-by-step stability analysis.

Routh Stability Criterion Calculator

Example: For s³ + 10s² + 31s + 1030 = 0, enter “1, 10, 31, 1030”

Routh Stability Criterion Calculator Formula

The Routh Array is constructed from the characteristic equation:

$$a_n s^n + a_{n-1} s^{n-1} + \dots + a_1 s + a_0 = 0$$

Elements are calculated using: $$b_i = \frac{a_{n-1} \cdot a_{n-2i} – a_n \cdot a_{n-2i-1}}{a_{n-1}}$$

Source: MIT OpenCourseWare – Routh Stability Notes

Variables:

  • Coefficients ($a_n \dots a_0$): The numerical values of the polynomial’s coefficients.
  • Order ($n$): The highest power of ‘s’ in the equation.
  • Sign Changes: The number of sign changes in the first column of the Routh Array equals the number of unstable roots.

What is the Routh Stability Criterion?

The Routh Stability Criterion is a mathematical test used in control system engineering to determine whether a linear, time-invariant (LTI) system is stable. It identifies if any roots of the system’s characteristic equation lie in the Right-Half Plane (RHP) of the complex s-domain without explicitly solving for the roots.

A system is considered stable if and only if all roots of its characteristic equation have negative real parts. If there are any sign changes in the first column of the Routh array, the system is unstable.

How to Calculate Routh Stability (Example)

  1. Write the characteristic equation coefficients in order.
  2. Arrange the first two rows using alternating coefficients.
  3. Calculate subsequent rows using the cross-multiplication formula.
  4. Observe the first column: if all numbers have the same sign, the system is stable.

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Frequently Asked Questions (FAQ)

What if a zero appears in the first column? If a zero appears, it is replaced by a small positive value $\epsilon$ to continue the calculation.

Can Routh test determine relative stability? No, it only determines absolute stability (stable or unstable).

What does a row of zeros mean? A row of zeros indicates roots symmetric about the origin, such as purely imaginary roots.

Does it work for non-linear systems? No, the Routh Criterion is strictly for linear time-invariant systems.

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