Robust stabilizability of fractional order proportional integral controllers for fractional order plants with uncertain parameters: A new value set based approach

Abstract

In this paper, a new value set based approach is presented to study the robust stability of a fractional order plant with parametric uncertainty of the interval type by using a fractional order proportional integral controller. The concept of interval uncertainty means that the parameters of the system transfer functions are uncertain parameters that each adopts a value in a real interval. Based on the zero exclusion principle, it is necessary to check whether the value set of the characteristic polynomial for the infinite frequency range [Formula: see text] includes the origin or not. To this end, in this paper, an auxiliary function is defined. The sign of this auxiliary function within a finite frequency range verifies whether the origin is included in the value set of the characteristic polynomial or not. The upper bound and the lower bound of this finite frequency range are obtained using the triangle inequality. Finally, the results of the numerical examples are provided, which confirm the effectiveness of the paper results.

Robust stability analysis of uncertain incommensurate fractional order quasi-polynomials in the presence of interval fractional orders and interval coefficients

“…Remark 3: Consider the interval uncertainties for only the coefficients of the fractional order quasi-polynomial (1) such as the fractional order polynomial (4) studied in Gao (2015) and Ghorbani and Tavakoli-Kakhki (2019). Theorem 1 confirms that (4) is robust stable, which is the same as the obtained result in Gao (2015) and Ghorbani and Tavakoli-Kakhki (2019).…”

“…Generally, this method imposes a computational burden and is difficult to be used. Also, consider Example 1 in Gao (2015) and Ghorbani and Tavakoli-Kakhki (2019), which is…”

“…To analyze the robust stability of the closed loop system, the approach presented in Ghorbani and Tavakoli-Kakhki (2019) is used. Based on Remark 2 in Ghorbani and Tavakoli-Kakhki (2019), at first the stability of one member of the characteristic polynomial in (28) has to be checked.…”

“…To analyze the robust stability of the closed loop system, the approach presented in Ghorbani and Tavakoli-Kakhki (2019) is used. Based on Remark 2 in Ghorbani and Tavakoli-Kakhki (2019), at first the stability of one member of the characteristic polynomial in (28) has to be checked. Then, the sign of the auxiliary function Λ ( ω ) defined in Ghorbani and Tavakoli-Kakhki (2019) needs to be checked for ω ∈ [ 0 . 0312 , 2 . 0149 ] rad/sec…”

“…In Gao (2015), an auxiliary function has been proposed to verify the robust stability of a designed fractional order controller in a general form. In Ghorbani and Tavakoli-Kakhki (2019), another auxiliary function has been introduced to evaluate the robust stability of Fractional Order Proportional Integral (FOPI) controller, which is much simpler than the procedure proposed in Gao (2015). In several studies (Moornani and Haeri, 2010c; Tan et al, 2009), it has been shown that the value set of an interval fractional order polynomial would be a polygon shape in the complex plane.…”