Calculator Input
Enter coefficients in descending powers, from highest power to constant term.
Example Data Table
These examples show common outcomes for continuous-time characteristic polynomials.
| Example | Coefficient List | Polynomial | Expected Result |
|---|---|---|---|
| Stable cubic | 1, 6, 11, 6 | s³ + 6s² + 11s + 6 | Stable |
| Marginal case | 1, 1, 1, 1 | s³ + s² + s + 1 | Marginally Stable |
| Unstable cubic | 1, 2, 3, 4 | s³ + 2s² + 3s + 4 | Unstable |
| Stable fourth order | 1, 8, 24, 32, 16 | s⁴ + 8s³ + 24s² + 32s + 16 | Stable |
Formula Used
Characteristic polynomial
P(s) = ansn + an-1sn-1 + ... + a1s + a0
Routh array setup
First row: an, an-2, an-4, ...
Second row: an-1, an-3, an-5, ...
General Routh element
bij = [(ri-1,1 × ri-2,j+1) - (ri-2,1 × ri-1,j+1)] / ri-1,1
Decision rule
A continuous-time polynomial is stable when every root has a negative real part. In the Routh test, the number of sign changes in the first column equals the number of right half-plane roots.
How to Use This Calculator
- Enter coefficients in descending order, starting with the highest power.
- Keep normalization enabled unless you want to inspect the raw polynomial scale.
- Set epsilon and tolerance only when you need finer numerical control.
- Press Analyze Stability to generate the result above the form.
- Review the stability status, Routh sign changes, and computed roots together.
- Use the pole plot to see whether roots fall left, right, or on the imaginary axis.
- Export the final report through the CSV or PDF buttons.
FAQs
1) What does this calculator test?
It checks the stability of a continuous-time characteristic polynomial using the Routh-Hurwitz table and numerical root estimation. You get a classification, root distribution, sign changes, and a pole plot.
2) What does stable mean here?
Stable means all polynomial roots have strictly negative real parts. If any root has a positive real part, the system is unstable. Roots on the imaginary axis indicate marginal stability.
3) Why show both roots and the Routh table?
The Routh table gives a fast analytical stability test, while roots provide direct numerical confirmation. Seeing both helps you verify results and understand why the classification was assigned.
4) What happens when a full zero row appears?
A full zero row usually signals symmetric root behavior and possible imaginary-axis roots. The calculator applies the auxiliary polynomial derivative method so the analysis can continue correctly.
5) Does normalization change the answer?
No. Dividing every coefficient by the leading coefficient only rescales the polynomial. The root locations stay the same, so the stability classification does not change.
6) Can I use this for discrete-time stability?
This page is designed for continuous-time polynomials in s. Discrete-time stability usually uses z-plane tests such as Jury criteria or unit-circle root checks.
7) Why do some roots look almost zero?
Numerical methods use finite precision, so tiny rounding values can appear. Very small real or imaginary parts often represent values that are theoretically zero within the chosen tolerance.
8) How should I enter coefficients?
Enter numbers in descending powers, separated by commas, spaces, or semicolons. For example, s³ + 6s² + 11s + 6 becomes 1, 6, 11, 6.