Computer algebra for exact complex stability margin computation

The multivariable stability margin (k_M) problem can be formulated as solving polynomial systems by using symbolic computation and stratified Morse theory. Once the solutions are found, the stability margin problem can be easily solved. For the complex k_M problem, no matter how many uncertainties, there is only one one-dimensional polynomial system which needs to be solved in order to find all singularities to determine whether the boundary of the Horowitz template intercepts the origin or not. The objective of the paper is to describe how to use the Groebner basis method to solve this polynomial system. Due to the continuity property of complex μ, numerical solutions are good enough for complex μ computation. In addition, we can sample this one-dimensional polynomial system into several zero-dimensional polynomial systems. There are many efficient algorithm to solve these zero-dimensional polynomial systems. Therefore, we have an efficient way of singularity related method to compute exact complex k_M