Output fractional-order sliding modes for structural energy shaping: application to robust brushless DC (BLDC) traction systems

This paper presents an observer-based fractional-order sliding-mode control framework for nonlinear Brushless DC (BLDC) traction systems, combining a filtered high-gain observer (FHGO) with a fractional-order sliding-mode controller (FOSMC). The fractional exponent β is interpreted not merely as a tuning parameter, but as a structural energy-shaping and bifurcation variable that continuously modifies the dissipative geometry of the sliding manifold, generating a transition from integral-type regulation at β = 1 to nonlinear fractional dissipation for 0 < β < 1 . A Lyapunov-based analysis is developed to establish exponential convergence under nominal conditions and input-to-state stability in the presence of bounded load disturbances, measurement noise, and approximation residuals associated with the finite-order realization of the fractional operator. Performance is evaluated through both classical tracking indices, namely, RMSE, IAE, and ITAE, and structural energy metrics, including control energy J u , control smoothness J u ̇ , control deviation energy J e u , time-weighted control deviation I T A E u , and the sliding-surface deformation index J s . Numerical simulations on a nonlinear d q Brushless DC (BLDC) traction model show that fractionalization reshapes the closed-loop energy distribution, significantly reducing accumulated tracking error while maintaining bounded actuation effort. The results reveal that an intermediate fractional regime, particularly within β ∈ [ 0.6 , 0.9 ] , provides the most favorable compromise between robustness, smoothness, and structural efficiency. In addition, the proposed FHGO–filtered-surface–fractional-control interconnection leads to a clear attenuation of chattering, although not to its complete elimination. These findings show that fractional sliding control should be understood not as a simple parametric extension of classical sliding-mode control, but as a geometric energy-shaping mechanism for structurally adaptive nonlinear regulation in electric vehicle drives.