The matching coefficients PID controller

The problem of designing a PID controller is posed in a setting where a selected reference system presents the design requirements. This leads to a simple problem of equating coefficients of like powers in polynomials originating in the reference system transfer function, the transfer function of the system to be controlled as well as the PID coefficients. Effectively, an overdetermined system of equations in the PID coefficients results, which is solved in the minimum least squares sense. We refer to this controller as the Matching Coefficients PID (MC PID). The computation is very simple involving only basic high school mathematics. While there is no explicit criterion for a selection of a reference system that will guarantee closed loop stability, systematic approaches can be designed for modifying the reference system in these cases.