Asymptotic tracking a constant reference without a pole at zero?

On a previous post we have discussed how by invoking stability one can solve problems without having to worry about particular inputs. The contrast between a solution that holds for a particular input versus a solution that holds for a class of inputs is highlighted in the present post by revisiting the problem of asymptotic tracking for a simple example.

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Let your inputs go!

As one learns more about feedback and control systems, at some point, comes an important realization that a variety of distinct problems can be answered if we know how to answer the question: “is some closed-loop system asymptotically stable?” Indeed, the study of stability dominates Chapters 6 and 7, arguably the densest chapters in the book. Many students, when asked what they remember from their undergraduate control class, will quickly point at the root-locus or at the Nyquist stability criterion. Yet, in the beginning, it might seem that stability is a side dish and not the main attraction.

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Controllers gone unstable

In order to obtain asymptotic tracking, we need $S(0)=0$, which means we’d need one of the poles of $G K$ to be zero. If $G$ has no poles at zero, then $K$ must be the transfer-function to have the pole at zero. But in order to be asymptotically stable, we need the real parts of the poles to be less than zero, correct? So does it matter if our controller is not asymptotically stable? Is there a way to design a controller that is both asymptotically stable AND provides asymptotic tracking?

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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?