Reliability Estimation of Stochastic Dynamical Systems with Fractional Order PID Controller

In this paper, the reliability of stochastic dynamical systems under Gaussian white noise excitations with fractional order proportional–inegral–derivative (FOPID) controller is estimated. First, the FOPID controller is approximated by a set of combination of displacement and velocity based on the generalized van der Pol transformation. Then, the stochastic averaging method of energy envelope is applied to obtain a diffusive differential equation, from which the Backward Kolmogorov equation, governing the conditional reliability function, and the Generalized Pontryagin equation, governing the statistical moments of first_passage time, are derived from the averaged equation and solved numerically. Finally, in the two examples, the critical parameters in the FOPID controller are shown to be capable of improving the reliability of the stochastic dynamical system apparently, and all numerical results are verified to be efficient and correct by the Monte Carlo simulation.