A variance-constrained proportional-integral feedback controller that tunes itself

Proportional-Integral (PI) feedback controllers have been shown to provide a very robust process adjustment strategy. Recent work by Box and Luceno provides PI controller settings that trade-off the variance of the quality characteristic versus the variance of the adjustments, an important concern in manufacturing. A Quality Engineer may desire a particular maximum adjustment variability, perhaps because of safety considerations or because of the resolution of the manufacturing equipment. The graphs and tables provided by these authors were developed by minimizing a weighted sum of adjustment and output variance and will not always provide enough information to guarantee a specific maximum adjustment variance. Specifying a constraint on the adjustment variance is much more natural for a Quality Engineer than specifying the value of the relative weight that satisfies the adjustment constraint, and easier than choosing from a list of possible controller designs. In order to obtain a specific adjustment variance, parameter estimates of the process are needed. Moreover, to allow for rapid control after startup and to minimize scrap, it is desirable that on-line estimation occurs during closed-loop operation. This paper presents a PI controller that tunes its parameters on-line during closed-loop operation for a first order process under the assumption that no prior process information is known before starting the control session. Under these conditions, the proposed PI controller provides the minimum output variance for a specified upper bound in the adjustment variance; the value of the Lagrange multiplier associated with the constraint, similar to the relative weight in Box and Luceno's procedure, is computed internally by the controller and there is no need to specify it a priori. An analysis of the performance of the controller is presented. It is shown that the proposed controller converges rapidly to the desired configuration without sacrificing the performance during the transient phase.