Optimal Adaptive Back‐Stepping SMC Synthesis via the Fuzzy Model: Applications to a Two‐Degree‐of‐Freedom Helicopter System

ABSTRACT This paper scrutinizes the issue of optimal adaptive back‐stepping SMC approach for a broad category of Takagi–Sugeno (TS) fuzzy systems with disturbances and unknown parameters, as well as its applications. The key innovation of this work is the implementation of an optimal control scheme for TS fuzzy‐based systems within the ISMC paradigm, compared to other existing methods and approaches. The TS Fuzzy Model extends the principles of linear systems to the domain of non‐linear systems by utilizing a convex skeleton. The switching manifold is shown to be achievable within a finite‐time, and the sliding motion's asymptotic stability is ensured if specific linear matrix inequalities are solvable. Subsequently, the gains for each individual local linear subsystem in the fuzzy rule consequences are optimally designed using a generalized eigenvalue problem (GEVP). Additionally, the maximum limits of both matched and mismatched disturbances are estimated through adaptive rule mechanisms. The multi‐criteria control improves transient performance, with 2‐index optimization defined by new LMI criteria. To further illustrate the benefits and suitability of the suggested back‐stepping SMC approach, a simulation example is provided.