Convolution Sum Based Adaptive Control (1964)

Convolution Sum Adaptive Controller

The Adaptive Control Odyssey paper briefly introduces an early adaptive control design based on Convolution Sum Modeling, like much modern Model Predictive Control. The modeling is described in Appendix I of the paper, with the controlled (process output) variable Ci, control (actuator) input Mi-j, process step response samples pj and Ui. This work continued from work studying the identification properties of logical models. From this perspective, a strict time domain input/output modeling approach seemed the most natural.

Convolution Sum

When adaptive identification was carried out, in moving windows, in became clear that only segments of a transient response contained meaningful identification data. The strategy taken to recognize which segments were useful was based on a calculation of worst case identification sensitivity to unit pulse errors in the measured output.

Adaptation results

As seen in the above figure, the input/output data fed both the online identification and sensitivity calculations. The sensitivity result could be applied to a function whose output ranged from 0 (for extreme sensitivity and erroneous identification) to 1 (for complete insensitivity and identification accuracy. The output could be used to determine the weighting in a continuous averaged updating of the identification results.

The resulting averaged identification was fed to the feedback model in a Model Predictive style process model and to a pre-programmed computation of the best forward controller element settings for the given identification. The tunings in turn completed the adaptive loop. The adjacent figure shows the experimental performance of the control and adaptive loops. An early instance of my tendency to later have to compete with my earlier designs!

The adaptation was based on a fixed window for which identification of both process and upset could be identified. (However such a widow could itself be easily adapted.) The tests proceeded testing the robustness to the deign to window errors. Like many adaptive designs implemented since to be both well-founded and highly responsive, the system was extremely robust. The system worked well even when the window was so narrow that the final step response in the window was only half the steady state value.

As the window were set progressively more inappropriately narrow, the identification values should ever greater tendency to jump all over while the control stayed steady. Such robustness might have been a call to commercialize the design. But the reverse reasoning lead to the pattern recognition design: if the design worked so well under circumstances of such theoretical inaccuracy then the theory was unnecessarily complex; it was actually performing pattern matching. A much simpler design should serve as well; and it did!