Graphical tuning method of FOPID controllers for fractional order uncertain system achieving robust ‐stability

Abstract

SUMMARYThis paper focuses on the graphical tuning method of fractional order proportional integral derivative (FOPID) controllers for fractional order uncertain system achieving robust D-stability. Firstly, general result is presented to check the robust D-stability of the linear fractional order interval polynomial. Then some alternative algorithms and results are proposed to reduce the computational effort of the general result. Secondly, a general graphical tuning method together with some computational efficient algorithms are proposed to determine the complete set of FOPID controllers that provides D-stability for interval fractional order plant. These methods will combine the results for fractional order parametric robust control with the method of FOPID D-stabilization for a fixed plant. At last, two important extensions will be given to the proposed graphical tuning methods: determine the D-stabilizing region for fractional order systems with two kinds of more general and complex uncertainty structures: multi-linear interval uncertainty and mixed-type uncertainties. Numerical examples are followed to illustrate the effectiveness of the method.

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

“…By using the D‐partition method to carry out the stabilizing region, the expression of k i and k d with respect to ω cannot be separately obtained. In this example, the stabilizing region in the case λ + μ = 2 is determined.…”

“…However, for the fractional‐order system with time delay, the stabilization problem of the PI λ D μ controllers has not been well resolved. Recently, several D‐decomposition methods were used to construct the stabilizing regions of the PI λ D μ controller . In the work of Moornani and Haeri, the D‐decomposition method was used to get the stabilizing region for the first‐order system with time delay.…”

“…Recently, several D-decomposition methods were used to construct the stabilizing regions of the PI λ D μ controller. [15][16][17][18][19][20][21][22] In the work of Moornani and Haeri, 20 the D-decomposition method was used to get the stabilizing region for the first-order system with time delay. The D-decomposition method was also used to choose the stabilizing set for the fractional systems with multiple commensurate delays.…”

Fractional Order Modeling And Nonlinear Fractional Order Pi‐Type Control For PMLSM System

“…The closed loop system will be bounded‐input‐bounded‐output stable under Specification 1). Specifications (2)–(3) is for the phase margin requirement, which is mainly for robustness. Phase margin is usually selected as 45 ∘ ~ 60 ∘ .…”

“…s represents Laplace variable. This motivates us to present the following new FOPI‐type controllers, referred to as (PI μ ) λ (FO[FOPI]) controller: where K p , K i , s are defined as equations (1)–(2), λ , μ are fractional orders. It is noted that C ( s ) in equation has more design parameters than C 1 ( s ) and C 2 ( s ), hence, a better control performance may be achieved by C ( s ).…”