Control performance index minimal tuning of set-point weighted PID-controllers for LTI plants based on convex optimisation

This paper addresses the problem of control performance index (CPI) minimal tuning of proportional integral differential (PID) feed-back-controllers with set-point weighting. Common CPIs, such as the integrated absolute error (IAE), are considered as objective functions to be minimised by parameter tuning. Many CPIs are convex functions, if their domain is convex and thus their only minimum is the global minimum. In order to exploit this for optimal tuning, we formulate parameter tuning as an optimisation problem with controller parameters as variables. Convexity of the IAE as CPI is shown exemplarily, since it can be used for PIDparameter and set-point weight tuning. Furthermore, we show the necessary convexity of the CPI's domain for the fixed structure set-point weight and PID-parameter tuning. For setpoint weight tuning, convexity of the domain is shown straight forward from the standard definition of a convex set. For PID parameter tuning, a convex representation of the domain is found that can be used equivalently to the original domain. To this end, we utilise the fact that CPIs merely quantify input-output behaviour which allows to relax structural constraints. Furthermore, it is shown how readily available iterative algorithms can be used to automate tuning and find the global optimal controller parameters. In a simulation example, the proposed techniques are applied to tune a set-point weighted PI-controller for a first order lag plus delay plant.