Research ArticleOn computation of stabilizing loop gain and delay ranges for bi-proper delay systems

•A graphical method is extended to determine the stabilizing gain and delay ranges for a bi-proper delay system.•A bi-proper process is rare but causes great complications for the method because of possibility of infinite intersections of boundary functions within a finite delay range.•The properties of boundary functions from such processes are investigated in great details to show that finite boundary functions are sufficient to determine all stable regions for finite parameter intervals.•The formula is given for calculating this number.•Algorithms are established to find exact stabilizing gain and delay ranges.

A graphical method for exactly computing the stabilizing loop gain and delay ranges was proposed [Le BN, Wang Q-G, Lee T-H. Development of D-decomposition method for computing stabilizing gain ranges for general delay systems. J Process Control 2012] for a strictly proper process by determining the boundary functions which may change system׳s stability. A bi-proper process is rare but causes great complications for the method, due to the new phenomena that do not exist for a strictly proper process, such as a non-zero gain at infinity frequency, which may cause infinite intersections of boundary functions within a finite delay range. This paper addresses such a kind of processes and develops a general method that can produce the exact and complete set of the loop gain and delay for closed-loop stabilization, which is hard to find with analytical methods.