Routh-Hurwitz Stability Calculator: Routh Array, Sign Changes and Right Half-Plane Roots
Stability of a control system without finding the roots.
Engineering student calculators
Free textbook calculators with every step shown, so you can check your own working. Pick one, type in your numbers and see the formulas worked out. Or switch to Ask a question for an explained answer. No account needed.
Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.
How this calculator works
Type the coefficients of the characteristic polynomial from the highest power down.
The calculator builds the Routh array.
It counts the sign changes and gives the verdict.
Formulas used
- R_{i,j} = \frac{R_{i-1,1}R_{i-2,j+1} - R_{i-2,1}R_{i-1,j+1}}{R_{i-1,1}}
- N_{RHP} = \text{sign changes in the first column}
Worked example
With the default numbers in the calculator above:
- Sign changes in the first column: 2
- Roots in the right half of the s-plane: 2 of 4
- Verdict: Unstable: 2 root(s) have positive real parts
- The Routh array (first row is the highest power): s^4: 1, 3, 5 | s^3: 2, 4 | s^2: 1, 5 | s^1: -6 | s^0: 5
Change the numbers in the calculator to see the full working, every step, for your own case.
A second worked example
A different case, with these numbers: Coefficients of the characteristic polynomial, highest power of s first 1 2 3 4 5 6.
- Sign changes in the first column: 2
- Roots in the right half of the s-plane: 2 of 5
- Verdict: Unstable: 2 root(s) have positive real parts
- The Routh array (first row is the highest power): s^5: 1, 3, 5 | s^4: 2, 4, 6 | s^3: 1, 2 | s^2: eps, 6 | s^1: -1.0e+6 | s^0: 6
Common questions
What does the Routh-Hurwitz criterion tell me?
Whether a polynomial has roots in the right half of the s-plane, and how many. If it does, the system is unstable.
What if a zero appears in the first column?
A very small positive number is put in its place and the array is finished. A whole row of zeros means there are roots in matching pairs, such as on the imaginary axis.
Is a positive set of coefficients enough for stability?
No. All coefficients must have the same sign, but that alone is not enough for a polynomial above degree 2.
Related calculators
Got a question about this? Ask an engineering question and get a plain-language answer with references.
Further reading: Any control systems text (the Routh-Hurwitz stability criterion).
Need a calculator we do not have?
Embed this calculator on your own page
Copy this code into your page. The calculator runs inside the frame and links back to Drafting Central.
Need this drawn, designed or checked?
Drafting Central is a mechanical engineering and drafting business working with customers across Australia and around the world. A calculator gives you the numbers; we can turn them into drawings and designs.
Need help with a real project?
Talk to Drafting Central →