Stability crossing surfaces for linear time-delay systems with three delays

In this article, the stability of linear systems with time-delays is studied. The cases where the characteristic equations of the system include three delays are investigated. Using the geometrical relations in a normalised polynomial plane, a graphical method is presented to visualise the stability domains in the space of the time-delays. In this space, the points at which the characteristic equation has a zero on the imaginary axis (the border between stability and instability regions) are identified. These points form several surfaces called the 'stability crossing surfaces'. This work extends the results of the previous works on the 'stability crossing curves' defined in the two-dimensional space (plane) of delays to a higher dimension and provides new geometric interpretations for the stability crossing conditions.