A survey on advancement of hybrid type 2 fuzzy sliding mode control

Stabilization of Robots With a Regulator Containing the Sigmoid Mapping

“…Now, we will detail the stabilization of the regulator error. Theorem 1: The stabilization of the regulator error in the close-loop model of a regulator containing the sigmoid mapping (12) and robots (3) is ensured, and the speed regulator error w 2 will complies with:…”

“…as the position regulator error, w 1 ∈ n×1 as the position, w d 1 ∈ n×1 as the constant desired position, w 2 = w 2 ∈ n×1 as the speed regulator error, K p , K d ∈ n×n as positive definite, symmetric and constant matrices, sat(•) as the saturation mapping, b(•) as the sigmoid mapping, K as a constant such as O + h ≤ K , O as in (11), h as in (7), n l as a actuators nonlinearities term. It is important to note that we do not know the behavior of O(w 1 ), h(v) and we utilize their upper bounds O, h. Remark 2: Since w 2 will reach to w d 2 and w d 2 = 0, it yields w 2 = w 2 ∼ = 0; consequently, w 2 is bounded, and since b( w 1 ) and sat( w 2 ) also are bounded, it yields that the regulator law terms v for a regulator containing the sigmoid mapping (12) are bounded. In Figure 2 we show a regulator containing the sigmoid mapping called RSM for the stabilization of robots called Robot.…”

“…Q(w 1 ) as the positive definite matrix of (3) and K p as the positive definite matrix of (12). We take into account w 2 = w 2 and we substitute (12) into (3), we obtain the closed-loop model as:…”

“…Authors addressed regulators for stabilization of microgrids in [9], [10], and [11]. In [12]- [14], and [15], authors focused fuzzy sliding model regulators of robotic exoskeletons and robotic manipulators. Authors employed neural network sliding mode regulators of wheeled aerial vehicles and robotic manipulators in [16]- [18], and [19].…”

“…= 0, and it is (13). Remark 3: Our regulator (12) requires to comply with conditions (11), (7) to be applied for the stabilization of robots (3), (4).…”