Fractional-Order State Feedback Controller to Suppress the Nonlinear Self-Excited Oscillations: Accurate Analytical and Numerical Solution

Abstract

In this work, an accurate analytical solution is developed for a harmonically excited self-excited nonlinear oscillator under fractional-order state feedback control using a modified version of the Traditional Multiple Scales Method (TMSM), termed the Detuned Multiple Scale Method (DMSM). In contrast to the TMSM, which perturbs the nonlinear system about its linear natural frequency, the DMSM perturbs the system implicitly about its amplitude-dependent response frequency. Based on this formulation, reduced-order amplitude–phase modulation equations for the considered fractional-order system are derived and compared with those obtained by TMSM. It is found that the DMSM yields exactly the same backbone curve as that obtained using the first-order Harmonic Balance Method (HBM), whereas the TMSM provides only its leading-order approximation. In addition, the DMSM reveals that the effective linear and nonlinear damping and stiffness coefficients depend on the amplitude-dependent response frequency, unlike the TMSM, where they depend on the fixed linear natural frequency. Furthermore, it is shown that the TMSM detuning term represents only the first-order approximation of the exact detuning term obtained by the DMSM. Accordingly, the considered fractional-order system is analyzed using the DMSM in comparison with the TMSM through Frequency Response Curves (FRCs), bifurcation diagrams, and stability charts. The system dynamics are investigated for different fractional-order derivatives under weak, moderate, and strong feedback gains. Moreover, a Runge–Kutta Grünwald–Letnikov (RK–GL) algorithm is developed and validated for fractional-order simulations, and all obtained FRCs are numerically verified. The numerical results clearly demonstrate that the proposed DMSM maintains excellent agreement with the numerical results not only near the primary linear resonance condition, but also over a wide range of excitation frequencies under weak, moderate, and strong feedback gains. In contrast, the TMSM fails to preserve this level of accuracy and may produce misleading predictions, especially under strong feedback and large detuning conditions. Extracting reduced-order amplitude–phase equations using the DMSM provides highly accurate analytical predictions along with deep physical insight into system dynamics, which, despite their accuracy in steady-state solutions, offer limited insight into transient behavior and the overall evolution of the response.