Dynamical analysis and accelerated optimal stabilization of the fractional-order self-sustained electromechanical seismograph system with fuzzy wavelet neural network

This paper investigates dynamical analysis and accelerated optimal stabilization issues of the fractional-order (FO) self-sustained electromechanical seismograph system under energy mechanism. The FO equation governing this system with gyroscopic coupling is established. Its dynamical analysis, based on phase diagrams and Lyapunov exponents, shows that chaotic and periodic behaviors strongly depend on physical parameters and on the values of the FOs. In accelerated feedforward controller, a shaping behavior function (SBF) is used to accelerate tracking error convergence at controllable rate and time, a fuzzy wavelet neural network (FWNN) with transformation is employed to approximate unknown functions of system, and a tracking differentiator is set to solve issue from complexities of SBF and FO under the framework of FO backstepping. In optimal feedback controller, an adaptive dynamic programming strategy is proposed to deal with a zero sum differential game solution problem, wherein the FWNN is employed to approximate the solution of the constrained Hamilton–Jacobi-Isaacs equation online. Furthermore, it is testified that all signals of the closed-loop system are bounded by using barrier Lyapunov function and the constrained conditions are not violated along with the cost function being minimized. Numerical simulation proves the effectiveness and advantages of the proposed scheme.