New Generation of Higher-Order Controllers: Leveraging PID Control for Improved System Response

The article extends the performance that can be achieved with the setup for the design of higher-order proportional-integral-derivative (HO PID) controllers using ultra-local integrator-plus-dead-time (IPDT) models. The previously derived reference family of controllers, referred to as 1PID, is complemented by a new family of 2PIDs corresponding to the double integrator plus dead time (DIPDT) model. The individual members of the 1PID and 2PID families are derived using the multiple real dominant pole method (MRDP) and normalized by the parameters of the ultra-local models used. The degree of the considered controller derivatives <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> ∈ [1, 6] significantly outperforms most of the known methods for PID control design, which mainly use <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> ≤ 1 and only exceptionally consider the cases with <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> > 1. Therefore, the obtained results are even comparable to the HO controllers created by approximating fractional-order PID controllers. To implement the mth-order derivative, binomial low-pass filters of order <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> ≥ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> are used in the controllers, specified by an equivalent filter delay added to the process. The traditional controllers with two degrees of freedom, for the separate design of setpoint and disturbance responses, are extended by two additional degrees of freedom provided by <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i>, allowing to modify the speed of the transients together with their shapes and the closed-loop robustness. The choice of 1PIDs and 2PIDs brings another degree of freedom related to the process model used. The multiple controller parameters are set by a new modification of the Performance Portrait Method (PPM), which makes it possible to shape the closed-loop responses by specifying tolerances for deviations of step responses from their ideal shape in the time domain. The illustrative examples, which deal with two stable higher-order processes and an unstable system, show the possibility of a multiple increase in performance compared to previously known methods. They show that HO-2PID controllers are inherently better suited for processes characterized by multiple dominant time constants. The novel design opens the door to wider use of HO controllers enabled by embedded control and programmable device technology, and allows the development of new HO controllers tuned by parallel computation.