Adaptive Fuzzy Variable Structure Control of Fractional-Order Nonlinear Systems with Input Nonlinearities

Stabilization of delayed fractional‐order T‐S fuzzy systems with input saturations and system uncertainties

“…Definition 3. If a function y(t) satisfies fractional equation (20) and is guaranteed to be continuous on…”

“…(28) To ensure the stabilization of the FOTDS (20), the following robust adaptive controller is devised.…”

“…According to the above analysis, the stability of FOTDS can be verified, which is proved by the following theorem. (20) with input saturations and unknown terms, where B i ∈ R n×n , C i ∈ R n×n , ΔB i ∈ R n×n , and ΔC i ∈ R n×n . Based on the controller (30) and adaptive laws ( 31)- (34), where R i ∈ R n×n and G i ∈ R n×n , the state variable x(t) converges into an arbitrarily small neighborhood of the origin, and all closed-loop signals are bounded.…”

“…So the above-mentioned fuzzy adaptive control method will improve control capability. Finally, under the fuzzy control scheme, the state variable x(t) of FOTDS (20) tends to a sufficiently small region near the origin.…”

“…It is well known that fuzzy logic systems (FLSs) are usually utilized for uncertain nonlinear systems and have achieved great success in recent decades. Recently, adaptive fuzzy control (AFC) has been used to control FONSs, and many interesting results have been stated, such as in Ha et al [20] and Tang et al [21]. However, a problem with ordinary AFC is that during the operation of the controller, a large number of fuzzy rules need to be used, which will put an extremely large calculation burden to the control system.…”