An adaptive order/state estimator for linear systems with non-integer time-varying order

Consensus of Multiagent Systems Described by Various Noninteger Derivatives

“…(ii) For = 2, (13) is a fractional system of timevarying order. This proof follows from Theorem 1 of Tabatabaei et al [18]. That result requires the weaker hypotheses ( , ( )) ≥ 0, ( , ( )) = 0 ⇐⇒ = 0 (which are implied by (17)) and 2 ( , ( )) < 0 in − {0} (which is implied by (18)).…”

“…We suppose that ( ) is a differentiable function for all ≥ 0 and for all operators in this work. Definition 2 (Tabatabaei et al [18]). The modified initialized Caputo fractional derivative of time-varying order ( ) is defined as follows:…”

“…The two Theorems in this section summarize the known results for Lyapunov stability theory for nonlinear systems of (A) Li et al [16] and Wang and Li [20] (Definition 1); (B) Souahi et al [17] (Definition 2); (C) Tabatabaei et al [18] (Definition 3); (D) Taghavian and Tavazoei [19] (Definition 4); (E) Almeida et al [12] (Definition 5) and also extends the Lyapunov stability theory for nonlinear systems defined by operators introduced in Definitions 5 and 6. Specifically, in (A) the Lyapunov direct method for standard Caputo fractional system (13) with = 1 is proved and the definition of Mittag-Leffler stability is introduced.…”

“…(i) For = 1, the proof can be found on Theorem 6.2 of Li et al [16]. (ii) For = 2, the proof is presented in Tabatabaei et al [18], as explained in item (ii) of Theorem 12 proof. (iii) For = 3, the proof is the same as the one of Theorem 4.2 of Taghavian and Tavazoei [19].…”

“…In recent years, Lyapunov stability theory has integrated to fractional calculus; see Li, Chen, and Podlubny [16]; Souahi, Makhlouf, and Hammami [17]; Tabatabaei, Talebi, and Tavakoli [18]; Taghavian and Tavazoei [19]; Wang and Li [20]. One of the objectives of the present article is to take advantage of the similarities between those papers and unify them into a generalized fractional Lyapunov method, useful for systems of differential equations with the fractional derivatives mentioned in the above paragraphs.…”