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.