A uniform LMI formulation for tuning PID, multi-term fractional-order PID, and Tilt-Integral-Derivative (TID) for integer and fractional-order processes. (May 2017) - Merrikh-Bayat, Farshad
Abstract: In this paper first the Multi-term Fractional-Order PID (MFOPID) whose transfer function is equal to ∑ j = 1 N k j s α j, 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 d 1 s + k d 2 s 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. Abstract : Highlights: LMIs for tuning the parameters of any controller which is linear in its parameters are presented. Phase and gain margin can directly be adjusted using the proposed LMIs. Sufficient-type effective LMIs for closed-loop stability are obtained based on the Nyquist theorem. The method is applicable to an integer or fractional-order process which is stable or has one unstable pole. It is shown that adding a fractional differentiator term to the transfer function of PID can highly improve its performance.