Revisiting distributed order PID controller

The paper addresses structural properties of distributed order controllers. A Distributed Order PID (DOPID) controller is a control structure in which a continuum of “differintegral” actions of orders between -1 and 1 are integrated together, and where relative contributions of different orders is determined by a weighting function. This stands in sharp contrast to conventional proportional-integral-derivative controllers, or even fractional order PID (FPID) controller and multi-term FPID, in which discrete actions appear only and a finite set of real parameters, controller gains, are sufficient to specify contributions of each action. The paper presents an in-depth analysis of the DOPID controller, emphasizing its theoretical properties and distinctions with integer and fractional order PID. It is shown that DOPID can be considered a generalization of these controllers only if the weighting function is a sequence of Dirac pulses. Some structural deficiencies of DOPID in case of a wide class of weighting functions have been emphasized. A modified DOPID structure — which we refer to as the DOPID of the Second Kind — is proposed and analyzed as well. Among other things, it has been shown that such modified DOPID controller provides better generalization to discrete order controllers (PID and FPID).

This work is an extended and supplemented version of the paper presented at ICFDA 2024 at Bordeaux University, July 2024 (see [27]).