Design of optimal and reliable PI-type compensators

We consider linear time-invariant single-input single-output (SISO) systems that are stabilizable by proportional and integral (PI) compensators. Such systems are also stabilizable by a PI-type compensator proposed by Phelan (1977). We denote Phelan´s PI-type compensator by I/P. We show that the systems controlled by I/P compensators achieve better performance than systems controlled by PI compensators. We then study a linear three-parameter PI-type compensator that features both PI and I/P compensators, and denote it by PI/P. The PI/P compensator outperforms both PI and I/P compensators. Furthermore, PI/P compensators can be tuned, so that the closed-loop feedback system is tolerant to some loop failures in the system. We present an efficient and easy-to-implement algorithm for computing the optimal gains of the PI, I/P, and PI/P compensators, as well as an algorithm for computing the gains of the PI/P compensators that result in reliable systems.