The Routh-Hurwitz Criterion
Answer whether any pole is unstable without finding a single one — and get the usable range of gain out of it.
Skip to the animationA tabular test that reports how many closed-loop poles lie in the right half-plane without locating a single one — which makes it usable on a characteristic polynomial containing an unknown gain, and therefore able to produce the range of gain a loop tolerates.
The question it answers
Closed-loop poles are roots of 1 + GH = 0. Past a cubic there is no practical formula, and with an unknown K in the equation there is nothing to factorise. But the question was never *where* the roots are — only whether any lies on the wrong side.
Two conditions can be checked by eye: all coefficients present, and all the same sign. Failing either proves instability. Passing both proves nothing — s³ + s² + 2s + 8 satisfies them and has two right-half-plane roots.
The array
- 1Write alternate coefficients along the first row, the remaining ones along the second.
- 2Compute each subsequent row from the two above it — a small determinant divided by the leading element.
- 3Continue until the
s⁰row. - 4Count the sign changes in the first column: that is exactly the number of right-half-plane roots.
The count is exact rather than an estimate, and the process is mechanical — it needs no insight and always terminates.
What it is actually used for
Because K can stay symbolic throughout, requiring every first-column entry to be positive yields inequalities in K — a usable design limit rather than a verdict about one gain. At the boundary value the loop sits on the imaginary axis and oscillates, which is a number you can verify on the real plant.
Two special cases
- A zero in the first column
- Replace it with a small ε, continue, and take the limit at the end.
- An entire row of zeros
- Means roots symmetric about the origin — typically a conjugate pair on the imaginary axis. The auxiliary equation formed from the row above gives their frequency, which is the oscillation frequency at marginal stability.
That frequency is exactly what Ziegler-Nichols tuning measures on a real plant, which is a satisfying link between an algebraic method and a wrench-in-hand procedure.
What it cannot say
It gives a yes/no plus a count, and nothing about margin. A design one per cent inside the stability boundary passes identically to one comfortably inside — which is why root locus and Bode analysis follow it. It also needs a polynomial, so a transport delay must be approximated first.
The numbers you will be asked for
- Characteristic equation
1 + G(s)H(s) = 0
Its roots are the closed-loop poles.
- Necessary conditions
all coefficients present and same sign
Necessary, never sufficient.
- Routh element
b₁ = (a₁a₂ − a₀a₃) / a₁
A determinant over the leading element.
- The criterion
sign changes in column 1 = right-half-plane roots
Exact count.
Advantages and disadvantages
Advantages
- Answers stability without finding any root.
- Works with a symbolic gain, giving a design range directly.
- The count of unstable roots is exact.
- A row of zeros hands you the oscillation frequency.
Disadvantages
- Gives no stability margin at all.
- Says nothing about transient shape, damping or settling time.
- Requires a polynomial, so delays must be approximated.
- Two degenerate cases need special handling.
Watch it work
Check yourself
question 1 / 4
One question at a time. Pick an answer to see why it is right or wrong, then move on — there is no score to keep and nothing is saved.