Computation of all stabilizing first order controllers for fractional-order systems

This paper presents an effective solution to the problem of stabilizing a given but arbitrary fractional-order system using a first order controller c(s)=(x
₁
s + x
₂
)/(s + x
₃
). The problem is solved by determining the global stability region in the controller parameter space [x
₁
, x
₂
, x
₃
] using D-decomposition technique. Analytical expressions are derived for the purpose of obtaining the stability boundaries of this region which are described by real root boundary, infinite root boundary and complex root boundary. Thus, the complete set of stabilizing first order controller parameters is obtained. The algorithm has a simple and reliable result which is illustrated by several examples, and hence is practically useful in the analysis and design of fractional-order control systems.