Tuning fractional order proportional integral controllers based on vector in the complex plane for first order systems

In this paper, an improved fractional order proportional integral controller parameter tuning method has been proposed. This method is to map the controller to get a set of vectors in the complex plane. Following the method of vector and geometric in the complex plane, the complicated derivation and calculation can be simplified in the process of the controller design. In addition, the established geometric relationship equations not only can still be used when the plant is different, but also can be programmed in MATLAB. Therefore, the repetitive process of formula derivation can be omitted. Because of the triangle constraints in the complex plane, we can solve the problem that using MATLAB to calculate the controller parameters has multiple solutions.