Stability criteria in a nutshell

A continuous linear system is stable if and only if all its eigenvalues have negative real parts.

There are various other stability criteria, but they boil down to the statement above. For example, there is the Routh stability criterion and the Hurwitz stability criterion. There’s also a continued fraction criterion.

But these criteria are just algorithms for determining whether the roots of the characteristic polynomial are all negative; they’re not independent criteria. They were more important in the days of hand calculations; now it’s easy enough to simply compute the roots of the characteristic polynomial numerically.

A discrete-time linear system is stable if and only if all its eigenvalues are inside the unit circle. The change of variables

w = (z − 1) / (z + 1)

maps the interior of the unit disk to the open left half plane, and so the roots of the characteristic polynomial in z are inside the unit circle if and only if the roots of the corresponding function of w are in the left half plane.

Related post: Cayley-Hamilton theorem

5 thoughts on “Stability criteria in a nutshell

  1. This post comes just as my grad students are studying for their final. You’re correct to say that for a given numerical matrix, “eig” in MATLAB does the trick admirably. But let me add to the defense of criteria: when the matrix depends on parameters, and the goal is to find a parameter range in which the system is stable, criteria can find analytical relationships between parameters in the stability region. A parameter sweep is still useful in that case, but analytical expressions have an added benefit of elucidating the role that parameters play in the stability.

    Oh and I should add – thank you for all the excellent posts, this is one of my favorite pages online.

  2. I agree that the various criteria are useful when you’re analytically looking for a parameter range. My only quibble is that I think it would be clearer if the Routh and Hurwitz criteria were called algorithms, though of course the names are firmly established,

  3. I agree completely – I made the same distinction on the final day in my graduate ODE class.

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