Robust Linear Adaptation with Process Nonlinearity

I know, from many years of feedback and multivariable adaptive work, that design robustness requires experimentation with the effects of process nonlinearity on linear adaptation. An adaptation dependent on linear convergence can lose this convergence as different process states give different temporary "linearized" data under nonlinearity. The adaptation often changes inconsistently under different local partially adapted conditions. Thus the adaptation is unable to converge even when it would with a purely linear process, and may diverge.

Two strategies can contribute to resolving this problem: The first is to create separate pigeon holes for the identification results based on the different process state, so that convergences in different process states is separated from each other. The second requires that the most rapidly convergent methods be favored over slower ones.

A most striking example occurred when demonstrating an immediate precursor to the commercial EXACT design. A heat exchanger process test was being carried out, changing the controller setpoint first up than down and then back, etc. The results tended to be irregular. EXACT is not model based, but based on convergent feedback of pattern/tuning data. In this case the up and down adaptations called for different controller parameters, destroying convergence. The problem was corrected by keeping separate feedback data for the upward and downward going setpoint changes. As a result the convergence settled down, and resulted in .1% repeatability in adaptation for any given condition.

While the particular condition was not expected to be common in operation, the mechanism was included and generalized in the product because this kind of test was likely to be common with customers. Moreover it might incidentally address other similar problems. The "Process System Adaptation Beyond Math Modeling" paper shows a similar situation with the feedforward/multivariable decoupler model based Moment Projection design.

Adaptive Convergence

This design also preserved identification data for upward and downward going disturbances, with a nonlinear process (Figure 10). In this case, the nonlinearity in the different directions achieved different kinds of linear dynamic compensation. The downward going disturbance achieved a much tighter dynamic compensation because the nonlinearity combined with the permitted compensation to even better match the process. But the upward and downward disturbances converged separately to the best control possible for the particular circumstances.

Simplex Modeling

Early work, addressing this problem generally, proposed a method called Simplex Modeling. This was very briefly illustrated in "Pattern Recognition: An Alternative to Parameter Identification in Adaptive Control"*. Simplex Modeling would separately adapt and store parameters for different process states in a general nonlinear static model. In principle, this model could be of arbitrarily high dimension and accuracy, at a cost of computer science complexity and memory. I spent a lot of time with programing assistance developing database and interpolation software for such a system capable of incorporating nonlinear data, and nonlinear models, in a practially optimally compressed form.

Simplex Modeling was elegant but far too complex if it could be avoided. The simplicity and robustness of the Moment Projection called for a nonlinear counterpart. The simple up/down storage of the (actually later) EXACT mechanism was far more efficient. It was simply not worthwhile to store millions of data points which would not be used, let alone not get stale before their use was needed.

EXACT Remembers

I proposed a much simpler generalization of the above "up/down" mechanism, called "EXACT Remembers". In this case, adapted parameters were cached in a small table, each table entry corresponding to an associated process state and including additional data on currency and frequency of process visits to the state. The states themselves were related in no particular structure except that inherent as the process moved along its trajectory. The cache was designed to include only those process states which were found to occur commonly, as on a repeated trajectory. Entries (including both state data and parameters) would be deleted as they failed to repeat, replaced with more useful data. Together with the inherent robustness of the Moment Projection Method, this bypassed the necessity of something as complex as Simplex Modeling.

* with minor clarifying modifications, from Automatica, Vol. 13, March 1977, pp. 197-202, inself a revised version of "Pattern Recognition as an Alternative to Parameter Identification in Adaptive Control", IFAC Sixth Triennial World Congress, Cambridge, Massachusetts, August 24-30, 1975