Corrigendum to “Disturbance observer based adaptive fuzzy sliding mode control: A dynamic sliding surface approach” [Automatica 129 (2021) 109606] (Automatica (2021) 129, (S0005109821001266), (10.1016/j.automatica.2021.109606))

In Sai and Xu (2022), an error in Lemma 1 in Zhang, Chen, Shen, Sun, and Xia (2021) is pointed out. In the proof of Theorem 2 in Zhang et al. (2021), the Lemma 1 is invoked to deal with the two inequalities [Formula presented] with [Formula presented] and [Formula presented], and [Formula presented], but the condition [Formula presented] in Lemma 1 is not applicable for the inequality with [Formula presented]. Fortunately, the result of Lemma 1 is still true if the condition [Formula presented] is modified as [Formula presented], and the proofs of Lemma 1 and Theorem 2 should be modified slightly, while the main result of Theorem 2 remains unchanged. In the following, a corrected version of Lemma 1 is presented and a complete proof is given. Lemma 1’ For [Formula presented], [Formula presented], and [Formula presented] with [Formula presented] and [Formula presented] being positive odd integers, there exist [Formula presented] and [Formula presented] such that the following inequality holds, [Formula presented] Proof It is clear that [Formula presented], thus the inequality (1) always holds for [Formula presented]. Next, we only consider the case that [Formula presented]. Define [Formula presented], then inequality (1) can be rewritten as [Formula presented] If [Formula presented] and setting [Formula presented], [Formula presented], then the inequality (2) holds. Otherwise, if [Formula presented], it is easy to obtain that [Formula presented], thus the inequality (2) holds for any [Formula presented], and [Formula presented]. If [Formula presented], we have [Formula presented], thus the inequality (2) holds for any [Formula presented], and [Formula presented]. Now, we consider the case that [Formula presented], and define [Formula presented] If [Formula presented], [Formula presented], thus inequality (2) holds for any [Formula presented], [Formula presented]. Finally, we consider the case that [Formula presented]. Let [Formula presented], [Formula presented] If [Formula presented], we can verify that [Formula presented] otherwise, if [Formula presented], we can also have [Formula presented] Therefore, the inequalities (3) and (4) imply that [Formula presented], and thus [Formula presented] can be guaranteed if [Formula presented], [Formula presented]. Based on the above discussions, inequality (1) is true for [Formula presented], [Formula presented], and the proof is completed.■ According to Lemma 1’, there exist [Formula presented] and [Formula presented] ([Formula presented]) such that [Formula presented]. Then, the derivative of [Formula presented] before Remark 5 can be modified as [Formula presented] [Formula presented] where [Formula presented], [Formula presented] are defined as [Formula presented] [Formula presented] [Formula presented] It follows from the above inequality that Theorem 2 is still correct as Lemma 1’ is invoked.