Zero Optimizing Tracking and Disturbance Rejecting Controllers - The extended PID controller

The tuning of PID controllers can essentially be posed as the problem of selecting open-loop zeros such as to obtain a desired system response. In this paper, the general case wherein stable open-loop system zeros can be cancelled is considered, allowing more freedom in placing open-loop zeros, as opposed to just two zeros in the case of a PID controller. Similar to the PID controller, integrators are added, giving the controlled system desired input tracking and disturbance rejection properties. Plant poles can be placed in an inner loop if desired. Subsequently, optimal open-loop zeros are computed such as to minimize the deviation from a desired reference impulse response. The controlled system is shown to be stable