Question: Can Theorem 6.7 (p. 245) be used in reverse?

This question is raised by Professor Will Cluett at the University of Toronto:

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.)

Answer: No, you are right, it cannot, but..

A rephrase of the question is:

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).

So can we say anything about the possibility for simultaneous stabilization?

Yes, one useful result is given by Morari and coworkers (see Thm.2 in the paper by Garcia and Morari in Ind. Eng. Chem. Process. Des. Dev., 472-484, 1985, and the correction by Morari in Ind.Eng.Chem.Res., 633-634, 1987). They find that: This can be generalized as follows: Remark 1. Of course this result says nothing about the resulting dynamic performance, which may be very poor if G is very different from G1' or 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?

  1. 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.
  2. 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.
Remark 3 ("otherwise there is no hope"). This follows since $\sigma((G'-G)G^-1) > \rho((G'-G)G^-1) = max_i {\lambda_i((G'-G)G^-1) } = max_i {\lambda_i((G'G^-1 - I) } = max_i {\lambda_i((G'G^-1)} - 1 $ which is always greater than 1 if one of the eigenvalues of \lambda_i((G'G^-1) is in the right half plane, so we must at least require that they are "positive" (in the right half plane).