A closed-loop controlled system usually consists of the main structure, sensors, and actuators. The dynamics of sensors and actuators may influence the motion of the main structure. This article presents an analytical study on the first-passage reliability of a nonlinear stochastic controlled system under the consideration of the dynamics of sensors and actuators. The coupled dynamic equations of the controlled systems with sensors and actuators are first given, which are further integrated into a controlled, randomly excited, dissipated Hamiltonian system. By applying the stochastic averaging method for quasi-Hamiltonian systems, a one-dimensional averaged differential equation for the Hamiltonian function is obtained. The backward Kolmogorov equation associated with the averaged equation is then derived for the first-passage reliability analysis, from which the approximate reliability function and probability density of first-passage time are obtained. The accuracy of the proposed procedure is demonstrated by an example. A comparative analysis of the reliability of the system with/without sensors and actuators is carried out, which indicates that ignoring sensors and actuators will make underestimation of the reliability of the closed-loop system with small time. However, when time increases, there appears the opposite trend. Our findings provide a reference for control strategy design.
Publisher’s Note: “Existence and multiplicity of solutions for the fractional p-Laplacian Choquard logarithmic equation involving a nonlinearity with exponential critical and subcritical growth” [J. Math. Phys. 62, 051507 (2021)]