Category: Theory

Routh-Hurwitz stability criterion

I am often asked why I chose not to cover the Routh-Hurwitz stability criterion in the book. Here are some of the reasons.

  1. First is space. I would need to develop a lot of ideas to be able to do a decent presentation of the Routh-Hurwitz criterion. That takes space.
  2. Second is lack of connection with other ideas in the book. The algebraic polynomial machinery that is behind the criterion is somewhat disconnected from the other topics covered.
  3. Third is utilitarian. For system with numeric coefficients the use of the criterion has been virtually made obsolete by numeric calculators. Finding the roots of high order polynomials in the 21st century is not as hard as it used to be in 1876! For symbolic coefficients, the algebraic rational inequalities produced by the application of the criterion are only useful for relatively small problems. They do not scale very well or are as universal as the root-locus method or the Nyquist criterion. In the end one still has to solve a set of rational inequalities in the system’s coefficients to assess stability.
  4. Fourth is the Pandora’s box effect. If I cover Routh-Hurwtiz, why not the Liénard–Chipart criterion? Or the Hermite criterion? Those are even nicer than the Routh-Hurwitz one!

What do you think?

To disturb or not to disturb?

So here is a recurrent question. You have some model that has a constant showing up on the differential equation and you’re not sure how to handle it. For example, your model looks like:

$$\dot{y}(t)=ay(t)+bu(t)+c$$

where $c$ is a constant. There are multiple ways you can handle this situation.

Continue reading “To disturb or not to disturb?”