scholarly journals Stationary Response of a Class of Nonlinear Stochastic Systems Undergoing Markovian Jumps

2015 ◽  
Vol 82 (5) ◽  
Author(s):  
Rong-Hua Huan ◽  
Wei-qiu Zhu ◽  
Fai Ma ◽  
Zu-guang Ying
Keyword(s):  
Stochastic Systems ◽  
Original System ◽  
Stochastic Averaging ◽  
Kolmogorov Equation ◽  
Markovian Jump ◽  
Nonlinear Stochastic Systems ◽  
Stationary Response ◽  
Set Up ◽  
Jump Systems ◽  
Stationary Probabilities

Systems whose specifications change abruptly and statistically, referred to as Markovian-jump systems, are considered in this paper. An approximate method is presented to assess the stationary response of multidegree, nonlinear, Markovian-jump, quasi-nonintegrable Hamiltonian systems subjected to stochastic excitation. Using stochastic averaging, the quasi-nonintegrable Hamiltonian equations are first reduced to a one-dimensional Itô equation governing the energy envelope. The associated Fokker–Planck–Kolmogorov equation is then set up, from which approximate stationary probabilities of the original system are obtained for different jump rules. The validity of this technique is demonstrated by using a nonlinear two-degree oscillator that is stochastically driven and capable of Markovian jumps.

scholarly journals Asymptotic Stability of Delay-Controlled Nonlinear Stochastic Systems with Actuator Failures

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
N. Zhou ◽  
R. H. Huan
Keyword(s):  
Time Delay ◽  
Asymptotic Stability ◽  
Stochastic Systems ◽  
Sufficient Conditions ◽  
Original System ◽  
Stochastic Averaging ◽  
Largest Lyapunov Exponent ◽  
Control Force ◽  
Nonlinear Stochastic Systems ◽  
Actuator Failures

The problem of asymptotic stability of delay-controlled nonlinear stochastic systems with actuator failures is investigated in this paper. Such a system is formulated as a continuous-discrete hybrid system based on the random switch model of failure-prone actuator. Time delay control force is converted into delay-free one by randomly periodic characteristic of the system. Using limit theorem and stochastic averaging, an approximate formula for the largest Lyapunov exponent of the original system is then derived, from which necessary and sufficient conditions for asymptotic stability are obtained. The validity and utility of the proposed procedure are demonstrated by using a stochastically driven nonlinear two-degree system with time delay feedback and actuator failure.

scholarly journals Optimal Vibration Control of a Class of Nonlinear Stochastic Systems with Markovian Jump

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
R. H. Huan ◽  
R. C. Hu ◽  
D. Pu ◽  
W. Q. Zhu
Keyword(s):  
Optimal Control ◽  
Stochastic Systems ◽  
Stochastic Averaging ◽  
Control Force ◽  
Dynamical Programming ◽  
Markovian Jump ◽  
Finite Set ◽  
Time Optimal ◽  
Set Up ◽  
Ito Equation

The semi-infinite time optimal control for a class of stochastically excited Markovian jump nonlinear system is investigated. Using stochastic averaging, each form of the system is reduced to a one-dimensional partially averaged Itô equation of total energy. A finite set of coupled dynamical programming equations is then set up based on the stochastic dynamical programming principle and Markovian jump rules, from which the optimal control force is obtained. The stationary response of the optimally controlled system is predicted by solving the Fokker-Planck-Kolmogorov (FPK) equation associated with the fully averaged Itô equation. Two examples are worked out in detail to illustrate the application and effectiveness of the proposed control strategy.

scholarly journals Stochastic averaging using elliptic functions to study nonlinear stochastic systems

1993 ◽  
Vol 4 (4) ◽  
pp. 373-387 ◽  
Author(s):  
Win-Min Tien ◽  
N. Sri Namachchivaya ◽  
V. T. Coppola
Keyword(s):  
Stochastic Systems ◽  
Elliptic Functions ◽  
Stochastic Averaging ◽  
Nonlinear Stochastic Systems

scholarly journals H∞ Control for Nonlinear Infinite Markov Jump Systems

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Yueying Liu ◽  
Ting Hou
Keyword(s):  
Stochastic Systems ◽  
Design Method ◽  
Infinite Horizon ◽  
Mean Square ◽  
State Control ◽  
Markov Jump ◽  
Nonlinear Stochastic Systems ◽  
Markov Jump Systems ◽  
Jump Systems ◽  
Infinite State

In this paper, we discuss the infinite horizon H∞ control problem for a class of nonlinear stochastic systems with state, control, and disturbance dependent noise. The jumping parameters are modelled as an infinite-state Markov chain. Based on the solvability of a set of coupled Hamilton-Jacobi inequalities (HJIs), the exponential mean square H∞ controller for the considered nonlinear stochastic systems is obtained. A numerical example is given to show the effectiveness of the proposed design method.

scholarly journals Stochastic Bifurcations of a Fractional-Order Vibro-Impact System Driven by Additive and Multiplicative Gaussian White Noises

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-10 ◽  
Author(s):  
Yong-Ge Yang ◽  
Ya-Hui Sun ◽  
Wei Xu
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Stochastic Systems ◽  
Original System ◽  
Stochastic Averaging ◽  
Impact Factors ◽  
Stochastic Bifurcation ◽  
Fractional Order Systems ◽  
Impact System ◽  
White Noises

Stochastic fractional-order systems or stochastic vibro-impact systems can present rich dynamical behaviors, and lots of studies dealing with stochastic fractional-order systems or stochastic vibro-impact systems are available now, while the discussion on the stochastic systems with both vibro-impact factors and fractional derivative element is rare. This paper is concerned with the stochastic bifurcation of a fractional-order vibro-impact system driven by additive and multiplicative Gaussian white noises. Firstly, we can remove the discontinuity of the original system with the help of nonsmooth transformation and obtain the equivalent stochastic system. Then, we adopt the stochastic averaging method to get the approximately analytical solutions. At last, an example is discussed in detail to assess the reliability of the developed approach. We also find that the coefficient of restitution factor, fractional derivative coefficient, and fractional derivative order can induce the stochastic bifurcation.

scholarly journals Robust ReliableH∞Control for Nonlinear Stochastic Markovian Jump Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Guici Chen ◽  
Yi Shen
Keyword(s):  
Stochastic Systems ◽  
Sufficient Conditions ◽  
Lyapunov Stability Theory ◽  
Disturbance Attenuation ◽  
Markovian Jump Systems ◽  
Markovian Jump ◽  
Stochastic Disturbance ◽  
Markovian Jump System ◽  
Actuator Failures ◽  
Jump Systems

The robust reliableH∞control problem for a class of nonlinear stochastic Markovian jump systems (NSMJSs) is investigated. The system under consideration includes Itô-type stochastic disturbance, Markovian jumps, as well as sector-bounded nonlinearities and norm-bounded stochastic nonlinearities. Our aim is to design a controller such that, for possible actuator failures, the closed-loop stochastic Markovian jump system is exponential mean-square stable with convergence rateαand disturbance attenuationγ. Based on the Lyapunov stability theory and Itô differential rule, together with LMIs techniques, a sufficient condition for stochastic systems is first established in Lemma 3. Then, using the lemma, the sufficient conditions of the solvability of the robust reliableH∞controller for linear SMJSs and NSMJSs are given. Finally, a numerical example is exploited to show the usefulness of the derived results.

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