Asymptotic Stability of Controlled Nonlinear Stochastic Systems Considering the Dynamics of Sensors and Actuators

A closed-loop controlled system usually consists of the main structure, sensors, and actuators. In this paper, asymptotic stability of trivial solutions of a controlled nonlinear stochastic system considering the dynamics of sensors and actuators is investigated. Considering the inherent and intentional nonlinearities and random loadings, the coupled dynamic equations of the controlled system with sensors and actuators are given, which are further formulated by a controlled, randomly excited, dissipated Hamiltonian system. The Hamiltonian of the controlled system is introduced, and, based on the stochastic averaging method, the original high-dimensional system is reduced to a one-dimensional averaged system. The analytical expression of Lyapunov exponent of the averaged system is derived, which gives the approximately necessary and sufficient condition of the asymptotic stability of trivial solutions of the original high-dimensional system. The validation of the proposed method is demonstrated by a four-degree-of-freedom controlled system under pure stochastically parametric excitations in detail. A comparative analysis, which is related to the stochastic asymptotic stability of the system with and without considering the dynamics of sensors and actuators, is carried out to investigate the effect of their dynamics on the motion of the controlled system. Results show that ignoring the dynamics of sensors and actuators will get a shrink stable region of the controlled system.

Optimal Position and Attitude Control of Quadcopter Using Stochastic Differential Dynamic Programming with Input Saturation Constraints

This paper considers the position and attitude control of a quadcopter in the presence of stochastic disturbances. Basic quadcopter dynamics is modeled as a nonlinear stochastic system described by a stochastic differential equation. Subsequently, the position and attitude control is formulated as a nonlinear stochastic optimal control problem with input saturation constraints. To solve this problem, a continuous-time stochastic differential dynamic programming (DDP) method with input saturation constraints is newly proposed. Finally, numerical simulations demonstrate the effectiveness of the proposed method by comparing it with the linear quadratic Gaussian and the deterministic DDP with input saturation constraints.

Existence, Uniqueness, and Almost Sure Exponential Stability of Solutions to Nonlinear Stochastic System with Markovian Switching and Lévy Noises

This paper is concerned with existence, uniqueness, and almost sure exponential stability of solutions to nonlinear stochastic system with Markovian switching and Lévy noises. Firstly, the existence and uniqueness of solutions to the system is studied. Then, the almost sure exponential stability of the system is derived. Finally, an example is presented to illustrate the results.

Stochastic Response of SDOF Self-Centering System

As a new type of seismic resisting device, the self-centering system is attractive due to its excellent re-centering capability, but research on such a system under random seismic loadings is quite limited. In this paper, the stochastic response of a single-degree-of-freedom (SDOF) self-centering system driven by a white noise process is investigated. For this purpose, the original self-centering system is first approximated by an auxiliary nonlinear system, in which the equivalent damping and stiffness coefficients related to the amplitude envelope of the response are determined by a harmonic balance procedure. Subsequently, by the method of stochastic averaging, the amplitude envelope of the response of the equivalent nonlinear stochastic system is approximated by a Markovian process. The associated Fokker–Plank–Kolmogorov (FPK) equation is used to derive the stationary probability density function (PDF) of the amplitude envelope in a closed form. The effects of energy dissipation coefficient and yield displacement on the response of system are examined using the stationary PDF solution. Moreover, Monte Carlo simulations (MCS) are used for ascertaining the accuracy of the analytical solutions.

Vibrations of a nonlinear stochastic system with a varying mass under near resonant excitation

Dynamics of a nonlinear second order system with a stochastically varying mass is considered in the article. The system is under an external time-harmonic loading, which is assumed to be near resonant. Damping, excitation strength, nonlinearity, and mass variations are considered to be of the same order of smallness. Under these assumptions, an explicit approximate solution of the problem is obtained using the multiple scales perturbation method. The leading (zero) order component of the solution is not affected by noise, whereas the first order component is stochastic and is energetically unstable because of mass variations. This implies that the considered oscillator will not vibrate in the near resonant regime because of mass variations. Instead, its vibrations will feature a considerable, but limited in amplitude, stochastic component. These vibrations will be stable, which illustrates a qualitative difference of the phenomenon from the motion instability described previously for linear stochastic oscillators for lower values of damping. The presence of the nonlinearity does not considerably affect the stability. To avoid the phenomenon, damping in the system should be increased so as to be much larger than mass variations. The obtained results were validated by a series of numerical experiments. The oscillator can be considered as a simplified model of machines used for processing of granular materials, such as vibrating screens, and the results are relevant for various applications, including mining industry.