*
We have been working on an industrial problem (a wood
chip refiner) and I am trying to apply some of the
results from your new book, in particular Theorem 6.7.
We did a series of identification
experiments over the course of an industrial refiner plate life
(this wearing of the refiner plates is what causes
the time-varying behaviour) to study the process dynamics
and to determine if the dynamics change significantly.
If one were to design a fixed gain (nonadaptive) controller
for the refiner, one might take an 'average' model
obtained from such a set of identification experiments
and use it for designing the controller. In trying to
determine whether such a fixed gain controller would
work (provide stability), we have attempted to apply your Theorem 6.7
in the following way:
We have assumed that the average
model would be used for controller design (call this G(s))
and we treat the series of identified models over the
plate life as different perturbations from the average
model (call these G1'(s), G2'(s), etc.). What we found
is that the det G(0) has the same sign as the det G1'(0),
det G2'(0), etc. From Theorem 6.7, what can we conclude?
(Your preamble to Theorem 6.7 would seem to indicate that we c
ould design a fixed gain
controller for this process that would stabilize the system over the entire
plate life, but the theorem itself only states the negative result,
i.e. if the sign of the det changes, then instability
will occur.) *

Consider a stable plant and assume that det G(0) has the same sign as the det G1'(0), det G2'(0), etc. Can we from Thm. 6.7 conclude that it is possible to simultaneously stabilize these plants using a fixed controller with integral action?

** No**, we cannot as Theorem 6.7 applies in the other direction (so we
can only conclude that we always will get some instability if the signs are
** different**).

- If $ Re (\lambda_i(G'(0) G(0)^{-1})) > 0; \forall i$ (all eigenvalues of G'(0) G(0)^-1 are "positive" (in the right half plane)), then it is possible to design a controller with integral action (the proof is for a certain IMC controller) such that the closed-loop system is stable for both plants G and G'.

- Theorem 6.A. Consider a square stable plant and assume that all eigenvalues of the matrices G1'(0) G(0)^-1, G2'(0) G(0)^-1, etc. are "positive" (in the left half plane). Then it is possible to design a controller with integral action such that the closed-loop system is stable for G, G1', G2', etc.

*Remark 2*. Note that since the determinant is equal to the product of the
eigenvalues, then "positive" eigenvalues of the matrix G'(0)G(0)^-1
implies that det G(0) has the same sign as the det G'(0). This means that
we must at least have determinants of the same sign to get "positive"
eigenvalues.

**Another**, possibly more useful, result is to apply the
robust stability conditions from Chapter 8.
We first fit the uncertainty to some perturbation, for example,
multiplicative output
uncertainty, as given in (8.15). That is , at each frequency we compute

wO = max_G' {\sigma((G'-G) G^{-1}) }

We then apply Theorem 8.4 and (8.55), from which we find that we are guaranteed robust stability (RS) if we can find a controller such that the nominal T=GK(I+GK)^-1 is sufficiently small, more precicely, we must require

\sigma (T) \le 1/wO at all frequencies.

Since we require integral action we have \sigma (T(0))=1, so we must at least require that wO is less than 1 at steady-state (this means that the relative output uncertainty is less than 100% at steady-state). More generally, we want wO small at all frequencies where we want tight control. But, note that output uncertainty may not be the best choice; this depends on how the uncertainty occurs, but it is difficult to find the right structure a priori.

** So how should we go about in practice to determine if a fixed
controller can be used? **

- Compute the relative output uncertainty wO as outlined above. If it is less than 1 at frequencies where we want control then we are fine.
- If wo is larger than 1 then there may still be some hope if we use the right uncertainty structure. To check that this may work you should first make sure that the eigenvalues are "positive"; see Theorem 6.A (otherwise there is no hope, see remark 3 below). You may then try some other uncertainty structure, e.g. input multiplicative uncertainty; see Figure 8.5 in Chapter 8 for some alternatives.