Satisfaction of gain and phase margin constraints using proportional controllers

Gain and phase margins (GM and PM) are frequently used as robustness indicators for linear time invariant systems. In many cases, the design problem is reduced to determination of a loop gain to satisfy several design criteria including gain and PM specifications. Therefore finding the set of fixed order compensators that satisfy given gain and PM specifications attracted the attention of a considerable part of the scientific community recently. It is possible to show that it may not be possible to satisfy the required gain and/or PM specifications using only a proportional controller. As a result, finding the limits of the proportional controllers for a given plant (i.e. finding the maximum achievable gain and PMs, MAGM and MAPM) is also important. After providing alternative methods (to those that already exist in the literature) for determining all stabilising proportional controllers that satisfy gain and PM constraints, a new method for calculating MAGM and MAPM using proportional controllers is given in this article. A formulation to calculate maximum gain that results in MAGM is also provided.