A uniform LMI formulation for tuning PID, multi-term fractional-order PID, and Tilt-Integral-Derivative (TID) for integer and fractional-order processes

In this paper first the Multi-term Fractional-Order PID (MFOPID) whose transfer function is equal to ∑j=1Nkjsαj, where kj and αj are unknown and known real parameters respectively, is introduced. Without any loss of generality, a special form of MFOPID with transfer function kp+ki/s+kd1s+kd2sμ where kp, ki, kd1, and kd2 are unknown real and μ is a known positive real parameter, is considered. Similar to PID and TID, MFOPID is also linear in its parameters which makes it possible to study all of them in a same framework. Tuning the parameters of PID, TID, and MFOPID based on loop shaping using Linear Matrix Inequalities (LMIs) is discussed. For this purpose separate LMIs for closed-loop stability (of sufficient type) and adjusting different aspects of the open-loop frequency response are developed. The proposed LMIs for stability are obtained based on the Nyquist stability theorem and can be applied to both integer and fractional-order (not necessarily commensurate) processes which are either stable or have one unstable pole. Numerical simulations show that the performance of the four-variable MFOPID can compete the trivial five-variable FOPID and often excels PID and TID.