The Entire Set of First Order Controllers Guaranteed Stability and Gain/Phase Margins for a LTI Plant with Time Delay

The problem of determining the entire set (say S₀) of the first order controllers (FOC) that stabilize a linear time invariant (LTI) plant with delay free has been recently solved. In this paper, we extend this solution to the case of LTI plant with time delay. The main result shows that all the stabilizing FOC for a given LTI plant including time delay are represented in a linear transformation of S₀ at every fixed frequency. It is also shown that the whole set of FOC satisfying the given gain and/or phase margins for the plant is determined by the slightly modified linear transformation of S₀ at every fixed frequency. Examples are given for illustration