A possible two dimensional system equivalent to one dimensional root-loci

The root locus diagram for a one dimensional system shows graphically how the closed loop poles vary as the gain increases. It provides a powerful design tool, allowing the designer to see immediately how a particular choice of gain will affect closed loop stability and dynamic response. The situation for two dimensional systems is more complex in that singularities in the transfer function (the poles) take the form of surfaces in 4-D space. To display the variation of such curves with gain hence requires, at the least, four dimensions and, because of the potential complexity, probably more. This paper demonstrates a graphical test of stability for open loop, two dimensional systems, and shows how this may be extended to investigate the stability in the closed loop case