Performance of ewma versus last observation for feedback control

Abstract

Two forms of feedback regulation that have been used for process adjustment in Automatic Process Control are considered. In the first, recommended originally by Box and Jenkins (1963) and by Box, Jenkins and MacGregor (1974), action is taken when the absolute difference between an exponentially weighted moving average (EWMA) of the past data [Zcirc]_n+1 and the target value T first crosses a threshold value L* In the second, recommended by Taguchi (1981), the action is triggered when the absolute difference between the last observation z_n and the target value T exceeds a given constant L. It has been shown (Box and Jenkins 1963) that if we consider the disturbance as an integrated moving average IMA(0,1,1) model containing parameter θ, then the former is optimal when θ is equal to the smoothing parameter used in the EWMA, and the latter is optimal if θ is zero. Thus some controversy has arisen because, as we show here, for any given scheme of the second kind with parameter L, there is a scheme of the first kind with boundary L^* which for a single crossing of the boundaries gives the same overall cost However, because the adjustment is in one case based on the deviation from target of the EWMA [Zcirc]_n+1−T and in the other on the deviation of the observation z_n−T, and it can be shown that z_n=z_n+1+θa_n where a_n is a random variable independent of [Zcirc]_n, the actual cost of regularly applying the policy based on z_n is considerably more expensive than that based on z_n+1. For example, in practice θ is frequently in the range from about 0.6 to about 0.9, and it is shown here that an increase in the mean square deviation of 64% can occur with θ=0.8.

Keywords:

• action limit
• adjustment
• automatic process control
• average adjustment interval
• cost
• mean square deviation