Monitoring control performance via structured closed-loop response subject to output variance/covariance upper bound

Owing to the process time delays, the closed-loop response can be divided into feedback control invariant part and feedback controller dependent part. If the latter part is replaced by a user specified response trajectory, we refer to the resultant closed-loop response as structured closed-loop response. The user specified structured closed-loop response has been used as an achievable control against which one can assess performance of control loops. In the control performance monitoring literature, the user specified response is often given as a first-order transfer function with some specified performance requirement, such as time constant. In this paper, we solve this problem from a systematic approach, i.e., in viewpoint of a variance/covariance upper bound on the outputs. With available closed-loop routine operating output data and process time delay/interactor matrix, the desired structured closed-loop response can be obtained directly via estimated closed-loop time series model. A significant feature is that the output variance/covariance upper bound constraint can be explicitly specified according to the product specifications and is always satisfied when the problem is feasible. This desired structured closed-loop response can thus be served as a benchmark against which the existing controller performance can be compared. We also show that two approaches, linearizing change of variables and Frank and Wolfe algorithm, are suitable for solving this problem, which result in a full order and a reduced order structured closed-loop response, respectively. Both approaches are illustrated by two case studies.