Solution Analysis and Novel Admissibility Conditions of SFOSs: The 1 < α < 2 Case

Novel admissibility and robust stabilization conditions for fractional‐order singular systems with polytopic uncertainties

“…Remark Novel admissibility conditions for nominal FOSSs with the fractional‐order $$$ 1\le \alpha &lt;2 $$$ and $$$ 0&lt;\alpha &lt;1 $$$ are proposed in Theorems 1 and 2, respectively, both of which are with no conservatism and without any equalities or nonstrict inequalities. By comparison, the existing works are limited by equalities or nonstrict inequalities such as the results in earlier studies [24, 27–29] for $$$ 0&lt;\alpha &lt;1 $$$ case and those in other literature [26, 30, 31] for $$$ 1\le \alpha &lt;2 $$$ case.…”

“… •Novel admissibility conditions for nominal FOSSs are derived in terms of LMIs with no conservatism, which guarantee the regularity, nonimpulsiveness, and stability. Compared with the existing works limited by equalities or nonstrict inequalities such as the results in earlier works [24, 27–29] for $$$ 0&lt;\alpha &lt;1 $$$ case and those in other literature [26, 30, 31] for $$$ 1\le \alpha &lt;2 $$$ case, the results proposed in this paper are without any equalities or nonstrict inequalities. •The robust admissibility conditions for FOSSs with polytopic uncertainties are given by using the elimination lemma, which can decouple the system matrices and the matrix variables and thus weakening the conservatism of the results. By comparison, most existing works on the robust admissibility of FOSSs focus on norm‐bounded uncertainties [28, 32, 33, 38], of which the methods cannot be directly applied to study the robust admissibility of FOSSs with polytopic uncertainties. •The robust stabilization of FOSSs with polytopic uncertainties is resolved via the output feedback control instead of the state feedback control in earlier work [28, 32] as not all the internal states in actual systems can be detected.…”

“…Taking advantage of the regularity and nonimpulsiveness conditions, different conditions for the admissibility of FOSSs are derived with the fractional‐order $$$ 1\le \alpha &lt;2 $$$ case [26] and $$$ 0&lt;\alpha &lt;1 $$$ case [27, 28]. Different from the frequency domain methods, time domain solutions are analyzed, by using which the regularity and nonimpulsiveness conditions are developed in time domain, and then novel admissibility conditions are established [29, 30]. As can be seen, most of the existing works on the admissibility of FOSSs are with equality limits or nonstrict inequality limits, which could lead to some trouble in applying the MATLAB LMI Toolbox.…”

“…For example, Theorem 2(iii) in Marir et al [26], Theorem 10 in Zhang et al [31], Theorem 2.1 in Zhang and Chen [28], and Theorem 6 in Zhang et al [29] are the admissibility conditions of FOSSs with equality limits in the form like $$$ {E}&#x0005E;TP&#x0003D;{P}&#x0005E;TE\ge 0 $$$ or similar. Besides, Theorem 1 in Yu et al [24], Theorems 2(i) and 2(ii) in Marir et al [26], Theorem 2 in Marir et al [27], Theorem 2.2 in Zhang and Chen [28], and Theorem 8 in Zhang et al [30] are the admissibility conditions of FOSSs with equality limits in the form like $$$ ES&#x0003D;0 $$$ or similar. Theorem 5 in Zhang et al [29] and Theorem 7 in Zhang et al [30] are the admissibility conditions of FOSSs with nonstrict inequality limits in the form like $$$ {E}&#x0005E;T SE\ge 0 $$$ or similar.…”

“…Besides, Theorem 1 in Yu et al [24], Theorems 2(i) and 2(ii) in Marir et al [26], Theorem 2 in Marir et al [27], Theorem 2.2 in Zhang and Chen [28], and Theorem 8 in Zhang et al [30] are the admissibility conditions of FOSSs with equality limits in the form like $$$ ES&#x0003D;0 $$$ or similar. Theorem 5 in Zhang et al [29] and Theorem 7 in Zhang et al [30] are the admissibility conditions of FOSSs with nonstrict inequality limits in the form like $$$ {E}&#x0005E;T SE\ge 0 $$$ or similar. Therefore, it is desired to propose novel admissibility conditions of FOSSs without any equality limits or non‐strict inequality limits.…”