Routh-Hurwitz Stability
Routh array and right-half-plane root count from your characteristic polynomial.
e.g. 1, 1, 2, 8 means s³ + s² + 2s + 8
Stability without finding the roots
The Routh-Hurwitz criterion tells you whether a system is stable — whether every root of its characteristic polynomial sits in the left half-plane — without actually solving for the roots. You build a triangular array from the coefficients, and the answer is in its first column:
A necessary first check: for stability every coefficient must be present and the same sign — a missing or sign-flipped term means it's already unstable. Beyond that, the array's first column does the work: no sign changes → stable; each sign change is one root in the right half-plane. Two special cases get flagged: a zero in the first column (handled with the ε-method, often marginal) and an entire row of zeros (roots on the jω axis — a marginally stable, oscillating system). It's the fastest hand check there is, and the basis for finding the range of a gain K that keeps a loop stable.