Feedback stabilization of multi‐DOF nonlinear stochastic Markovian jump systems

Abstract

Summary A feedback control strategy is designed to asymptotically stabilize a multi‐degree‐of‐freedom (DOF) nonlinear stochastic systems undergoing Markovian jumps. First, a class of hybrid nonlinear stochastic systems with Markovian jumps is reduced to a one‐dimensional averaged Itô stochastic differential equation for controlled total energy. Second, the optimal control law is deduced by applying the dynamical programming principle to the ergodic control problem of the averaged systems with an undetermined cost function. Third, the cost function is determined by the demand of stabilizing the averaged systems. A Lyapunov exponent is introduced to analyze approximately the asymptotic stability with probability one of the originally controlled systems. To illustrate the application of the present method, an example of stochastically excited two coupled nonlinear oscillators with Markovian jumps is worked out in detail.

Dynamical reliability of the stochastic power systems with discrete random variability

“…ii   , H can be considered as a slowly varying process. According to Khasminskii's theorem [35,36], the time interval t → , H converges to a onedimensional diffusion process. Subsequently, an associated Itô equation can be obtained through time averaging.…”

“…Compared to multi-DOF systems, these systems are relatively simpler to analyze and understand. However, with the development of advanced techniques such as the stochastic averaging method [32][33][34][35][36], there has been a growing interest in studying first-passage problems in multi-DOF strongly nonlinear systems. These systems often exhibit complex dynamics and interactions among multiple variables.…”

Reliability of the bi-stable piezoelectric energy harvester under random excitation

“…Equations ( 23) and ( 27) with conditions ( 24) and ( 27) can be generally solved using the finite difference method (Hu et al, 2019).…”

Stochastic stabilization of multi‐degree‐of‐freedom nonlinear random‐time‐delay controlled systems

“…Substituting Equation (20) into Equation ( 19), the following time-delay optimal control can be obtained…”

“…Up to now, a number of methodologies have been developed for modeling, stability analysis, and controller design in the presence of network‐induced RTD. Generally, previous research can be fallen into two broad categories: (1) to design a controller first, and then estimate the network conditions by the largest permissible transfer time interval to ensure the high stability and excellent performance 16–18 ; (2) to utilize given models to describe the network‐induced time delay and then incorporate these models into the process of controller design 19–21 . The strategy proposed in this paper belongs to the latter.…”