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This is for self-study of N-dimensional system of linear homogeneous ordinary differential equations of the form:

dx/dt=Ax

where A is the coefficient matrix of the system.

I have learned that you can check for stability by determining if the real parts of all the eigenvalues of A are negative. You can check for oscillations if there are any purely imaginary eigenvalues of A.

The author in the book I'm reading then introduces the Routh-Hurwitz criterion for detecting stability and oscillations of the system. This seems to be a more efficient computational short-cut than calculating eigenvalues.

What are the advantages of using Routh-Hurwitz criteria for stability and oscillations, when you can just find the eigenvalues quickly nowadays? For instance, will it be useful when I start to study nonlinear dynamics? Is there some additional use that I am completely missing?

Wikipedia entry on RH stability analysis has stuff about control systems, and ends up with a lot of equations in the s-domain (Laplace transforms), but for my applications I will be staying in the time-domain for the most part, and just focusing fairly narrowly on stability and oscillations in linear (or linearized) systems.

My motivation: it seems easy to calculate eigenvalues on my computer, and the Routh-Hurwitz criterion comes off as sort of anachronistic, the sort of thing that might save me a lot of time if I were doing this by hand, but not very helpful for doing analysis of small-fry systems via Matlab.

Edit: I've asked this at Math Exchange, which seems more appropriate: https://math.stackexchange.com/questions/690634/use-of-routh-hurwitz-if-you-have-the-eigenvalues There is an accepted answer there.

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eric
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  • This is a first order ODE. I usually associate oscillations with 2nd order ODEs (e.g., spring mass systems). This would have exponential characteristics, not trig functions. – duffymo Feb 26 '14 at 00:57
  • @duffymo : I mentioned A is a coefficient matrix (not a scalar). The equation describes an N-dimensional system of equations. See: http://en.wikipedia.org/wiki/Matrix_differential_equation – eric Feb 26 '14 at 01:00
  • This question appears to be off-topic because it is about math rather than programming. – Jim Lewis Feb 26 '14 at 01:05
  • @JimLewis you are right I've moved it over to Math Exchange and edited my post here to indicate as such. Feel free to delete. – eric Feb 26 '14 at 02:51

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This is just legacy educational curriculum which fell way behind of the actual computational age. Routh-Hurwitz gives a very nice theoretical basis for parametrization of root positions and linked to much more abstract math.

However, for control purposes it is just a nice trick that has no practical value except maybe simple transfer functions with one or two unknown parameters. It had real value when computing the roots of the polynomials were expensive or even manual. Today, even root finding of polynomials is based on forming the companion matrix and computing the eigenvalues. In fact you can basically form a meshgrid and check the stability surface by plotting the largest real part in a few minutes.

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