Popov stability and nonlinear control of El Borhamy–Rashad–Sobhy Duffing-type system

Abstract

Abstract Numerous electrical and electromechanical systems have oscillators with parametric-excitation and Duffing-type nonlinearity. The suggested model specifically demonstrates close resemblances to switched reluctance machines (SRMs), where nonlinear magnetic effects and time-varying inductance are important factors. In this work, we examine El Borhamy–Rashad–Sobhy oscillator, a model based on RLC circuit that combines periodic modulation and cubic stiffness. It is possible to use traditional frequency-domain tools by reformulating the system in a conservative Luré form. While amplitude-dependent robustness margins are provided by the Circle criterion, global stability conditions are established by the Popov criterion. Three methods are taken into consideration for control design. Transient response and damping are enhanced by a PD controller. Lyapunov-based exponential stability guarantees are provided by a backstepping controller. Robustness against bounded disturbances and parameter variations is guaranteed by a sliding-mode controller. The analysis is supported by numerical investigations such as Nyquist plots, Lyapunov functions, and closed-loop responses. The theoretical results are further validated through high-fidelity MATLAB/Simulink simulations of a 6/4 switched reluctance motor drive, enabling a practical comparison of controller performance. The findings have direct application to SRM-inspired systems and demonstrate how absolute-stability theory and nonlinear control can be used to address parametrically excited Duffing-type oscillators.