General stabilization method of fractional‐order PI^λD^μ controllers for fractional‐order systems with time delay

Abstract

Summary In this paper, an effective method is proposed to get the stabilizing regions of fractional‐order PIλDμ controllers for an arbitrarily given fractional‐order system with time delay. For each known proportional gain (kp), integral gain (ki), or derivative gain (kd) in the PIλDμ controllers, the stabilizing region with respect to the other two control gains is derived respectively. The boundaries of the stabilizing regions are firstly obtained based on singular frequencies. Then, the main 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 (ki, kd) regions of the PIλDμ controllers for λ + μ = 2 can be determined and the linear programming characteristic of the stabilizing (ki, kd) region for the case λ + μ = 2 is obtained. Furthermore, the robust stabilizing region is analyzed. The results in this paper provide the basis for both the tuning of the PIλDμ controller in practice and the design of the PIλDμ controller satisfying different performance criteria. Numerical examples and an application example are presented to check the validity of the proposed method.

Stabilizing region of fractional‐order proportional integral derivative controllers for interval delayed fractional‐order plants

“…From ( 27), (28) and the maximum modulus theorem [37], the following inequality should be satisfied.…”

“…Moreover, the works [24–26] have proposed some numerical nonlinear optimization methods to design FOPID controllers. Also, [27–31] have proposed various methods to tune FOPID controllers for systems having time delay based on gain and phase margins. In addition, the stabilizing region of FOPI and FOPID controllers which can satisfy the $$$ {H}_{\infty } $$$ constraint of the sensitivity function has been studied in [32, 33].…”