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

In this paper first the Multi-term Fractional-Order PID (MFOPID) whose transfer function is equal to [Formula: see text] , where k_j and α_j are unknown and known real parameters respectively, is introduced. Without any loss of generality, a special form of MFOPID with transfer function k_p+k_i/s+k_d1s+k_d2s^μ where k_p, k_i, k_d1, and k_d2 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.