Routh–Hurwitz stability criterion calculator
Enter your values. The answer and the full worked solution update as you type.
Result
| column 1 | column 2 | column 3 | |
|---|---|---|---|
| s⁴ | 1 | 3 | 5 |
| s³ | 2 | 4 | 0 |
| s² | 1 | 5 | 0 |
| s¹ | −6 | 0 | 0 |
| s⁰ | 5 | 0 | 0 |
Check the stability of a linear system from its characteristic polynomial. The calculator builds the Routh array with exact fractions, handles a zero in the first column (the ε method) and a row of zeros (the auxiliary polynomial), counts the sign changes and gives the number of roots in the right half-plane, on the imaginary axis and in the left half-plane.
Step-by-step solution
Characteristic polynomial
First two rows
Next rows
Count the sign changes in the first column
Key formulas
- Stability
- all roots of the characteristic polynomial in the left half-plane
- Routh array, s¹ entry of a cubic
- b1 = (a2a1 − a3a0)/a2
- Right half-plane roots
- number of sign changes in the first column
- Cubic s³ + a₂s² + a₁s + a₀
- stable when all a > 0 and a₂·a₁ > a₀
- Characteristic equation
- 1 + K·G(s) = 0
How to use the Routh–Hurwitz criterion
- Write the characteristic polynomial with all powers of s from the highest down. If a coefficient is missing or the signs differ, the system cannot be stable.
- Fill the first row with the 1st, 3rd, 5th, … coefficients and the second row with the 2nd, 4th, 6th, ….
- Each next element is (b₁·a_(j+1) − a₁·b_(j+1))/b₁, using the two rows above it.
- If a first element is zero but the row is not, replace it with a small ε > 0. If a whole row is zero, replace it with the derivative of the auxiliary polynomial formed from the row above.
- The number of sign changes in the first column equals the number of roots in the right half-plane.
Questions and answers
What does a row of zeros mean?
The polynomial has roots that are symmetric about the origin: pairs ±a, ±jω, or quadruples. The auxiliary polynomial from the row above contains exactly those roots, and its roots on the imaginary axis make the system at best marginally stable.
Is the ε method always reliable?
Not when roots lie on the imaginary axis. The calculator therefore also splits off the common factor of P(s) and P(−s), which holds all symmetric roots, and counts its imaginary-axis roots exactly. If that changes the count, the steps say so.
How do I find the range of K for stability?
Put the gain K into the coefficients, build the array symbolically and require every element of the first column to be positive. This calculator works with numbers, so you can use it to check a value of K on either side of the limit you found.
Does a stable Routh array mean a good design?
It only says that all poles are in the left half-plane. How fast and how well damped the response is depends on where the poles are; use the root locus or a Bode plot for that.
Practise it yourself
Practise Stability: Routh–Hurwitz & root locus
A calculator gives the answer. In the exam you need the method: practise it with new problems, hints and worked solutions.