Development of D-decomposition method for computing stabilizing gain ranges for general delay systems

• A graphical method to determine the stabilizing gain and delay ranges for a delay system is proposed.
• This is achieved by determining the boundary functions that may change system's stability.
• The exact and complete set of stabilizing controller gain and delay is produced.
• The proposed method is very general and applicable to any strictly proper process.
• The method significantly relaxes the restrictions with the existing works.

This paper deals with the problem of determining the stabilizing controller gain and plant delay ranges for a general delay system in feedback configuration. Such a problem admits no analytical solutions in general. Instead, the condition of the loop Nyquist plot's intersection with the critical point is employed to graphically determine stability boundaries in the gain-delay space and stability of regions divided by these boundaries is decided with the help of a new perturbation analysis of delay on change of closed-loop unstable poles. As a result, all the stable regions are obtained and each stable region captures the full information on the stabilizing gain intervals versus any delay of the process. The proposed method is applicable to both stable and unstable processes of any order with or without the right-half plane zeros. Several examples are provided for illustration and comparison with the existing methods.