“…Under-actuated systems are a kind of important nonlinear systems in which the independent control variables are less than the system degrees of freedom (Zhao et al, 2023a(Zhao et al, , 2023b(Zhao et al, , 2023c(Zhao et al, , 2023dZhang et al, 2022aZhang et al, , 2022bZhang et al, , 2022c, such systems can be found in many areas, such as four rotors rotorcraft (Zemalache et al, 2005), micro-machined ultrasonic transducers (Darbasi et al, 2022), unmanned aerial vehicles (Ullah et al, 2022), etc. In addition, compared with full-actuated systems (Duan, 2021), under-actuated systems are superior in reducing costs, weight and increasing system flexibility.…”