Accuracy aspects of iterative correlation-based controller tuning

The recently-proposed method for iterative correlation-based controller tuning that uses instrumental variables is considered. A confidence interval based on the covariance of the criterion function is introduced. An explicit expression for the variance of the controller parameter estimates around the optimal solution is developed and used to construct the optimal instruments. This expression in turn allows constructing a region around the obtained controller that contains the optimal controller with a certain probability level. The accuracy of the corresponding controller transfer function is also investigated. It is shown that the variance of the controller transfer function is proportional to the noise spectrum at the controller output and inversely proportional to the control error spectrum, the factor of proportionality being the ratio of the controller order to the number of data points. This result, which is asymptotic in both the controller order and the data length, is independent of the instrumental variables used.