General stabilization method of fractional-order PIλDμ controllers for fractional-order systems with time delay

In this paper, an effective method is proposed to get the stabilizing regions of fractional-order (PID mu)-D-lambda controllers for an arbitrarily given fractional-order systemwith time delay. For each known proportional gain (k(p)), integral gain (k(i)), or derivative gain (k(d)) in the (PID mu)-D-lambda controllers, the stabilizing regionwith respect to the other two control gains is derived respectively. The boundaries of the stabilizing regions are firstly obtained based on singular frequencies. Then, themain results are presented to directly determine the stabilizing region from an analytical viewpoint. The results avoid the time-consuming stability test since the stabilizing region is usually determined by manually choosing lots of test points from all the divided regions by the resultant boundaries. Besides, the stabilizing (k(i), k(d)) regions of the (PID mu)-D-lambda controllers for lambda+mu = 2 can be determined and the linear programming characteristic of the stabilizing (k(i), k(d)) region for the case lambda + mu = 2 is obtained. Furthermore, the robust stabilizing region is analyzed. The results in this paper provide the basis for both the tuning of the (PID mu)-D-lambda controller in practice and the design of the (PID mu)-D-lambda controller satisfying different performance criteria. Numerical examples and an application example are presented to check the validity of the proposed method.