Adaptive fuzzy sliding mode control applied to inverted pendulum.

The inverted pendulum on a cart represents a classical benchmark of underactuated nonlinear systems characterized by strong coupling, inherent instability, and sensitivity to uncertainties. This paper proposes an Adaptive Fuzzy Sliding Mode Controller (AFSMC) that integrates fuzzy inference with sliding mode principles to simultaneously preserve robustness and eliminate chattering effects. A hierarchical sliding surface is first formulated to coordinate cart position and pendulum angle stabilization using a single control input. While conventional SMC provides strong robustness against external disturbances, it suffers from severe chattering and excessive control effort. FSMC replaces the discontinuous switching action with a fuzzy inference mechanism, producing a smoother control signal, but it shows limited robustness under significant parametric uncertainties. To overcome this limitation, an adaptive fuzzy mechanism is introduced to tune the sliding gain online based on the pendulum error dynamics, increasing robustness during large deviations and reducing control activity near equilibrium. The controllers are evaluated under nominal conditions, impulse disturbances, and parameter variations. Simulation results demonstrate that AFSMC achieves smooth control, strong robustness, and the lowest control norm among all methods, providing an effective balance between robustness, chattering elimination, and energy-efficient control for underactuated nonlinear systems.