Higher-order stochastic averaging to study stability of a fractional viscoelastic column

Stochastic stability of a fractional viscoelastic column axially loaded by a wideband random force is investigated by using the method of higher-order stochastic averaging. By modelling the wideband random excitation as Gaussian white noise and real noise and assuming the viscoelastic material to follow the fractional Kelvin–Voigt constitutive relation, the motion of the column is governed by a fractional stochastic differential equation, which is justifiably and uniformly approximated by an averaged system of Itô stochastic differential equations. Analytical expressions are obtained for the moment Lyapunov exponent and the Lyapunov exponent of the fractional system with small damping and weak random fluctuation. The effects of various parameters on the stochastic stability of the system are discussed.     

A stochastic averaging method for analyzing vibro-impact systems under Gaussian white noise excitations

A new stochastic averaging method for predicting the response of vibro-impact (VI) systems to random perturbations is proposed. First, the free VI system (without damping and random perturbation) is analyzed. The impact condition for the displacement is transformed to that for the system energy. Thus, the motion of the free VI systems is divided into periodic motion without impact and quasi-periodic motion with impact according to the level of system energy. The energy loss during each impact is found to be related to the restitution factor and the energy level before impact. Under the assumption of lightly damping and weakly random perturbation, the system energy is a slowly varying process and an averaged Itô stochastic differential equation for system energy can be derived. The drift and diffusion coefficients of the averaged Itô equation for system energy without impact are the functions of the damping and the random excitations, and those for system energy with impact are the functions of the damping, the random excitations and the impact energy loss. Finally, the averaged Fokker–Plank–Kolmogorov (FPK) equation associated with the averaged Itô equation is derived and solved to yield the stationary probability density of system energy. Numerical results for a nonlinear VI oscillator are obtained to illustrate the proposed stochastic averaging method. Monte-Carlo simulation (MCS) is also conducted to show that the proposed stochastic averaging method is quite effective.     

Stability Analysis of an Inverted Pendulum Subjected to Combined High Frequency Harmonics and Stochastic Excitations

Stability of vertical upright position of an inverted pendulum with its suspension point subjected to high frequency harmonics and stochastic excitations is investigated. Two classes of excitations, i.e., combined high frequency harmonic excitation and Gaussian white noise excitation, and high frequency bounded noise excitation, respectively, are considered. Firstly, the terms of high frequency harmonic excitations in the equation of motion of the system can be set equivalent to nonlinear stiffness terms by using the method of direct separation of motions. Then the stochastic averaging method of energy envelope is used to derive the averaged Ito stochastic differential equation for system energy. Finally, the stability with probability 1 of the system is studied by using the largest Lyapunov exponent obtained from the averaged Ito stochastic differential equation. The effects of system parameters on the stability of the system are discussed, and some examples are given to illustrate the efficiency of the proposed procedure.     

Quantum stochastic differential equations for boson and fermion systems—method of non-equilibrium thermo field dynamics

A unified canonical operator formalism for quantum stochastic differential equations, including the quantum stochastic Liouville equation and the quantum Langevin equation both of the Itô and the Stratonovich types, is presented within the framework of non-equilibrium thermo field dynamics (NETFD). It is performed by introducing an appropriate martingale operator in the Schrödinger and the Heisenberg representations with fermionic and bosonic Brownian motions. In order to decide the double tilde conjugation rule and the thermal state conditions for fermions, a generalization of the system consisting of a vector field and Faddeev-Popov ghosts to dissipative open situations is carried out within NETFD.     

Lyapunov control of squeezed noise quantum trajectory

The quantum trajectory renders the optimal estimation of quantum state. It is a classical Itô stochastic differential equation. The Lyapunov global stabilization problem is solved for squeezed noise quantum trajectory. Lyapunov control stabilizes the quantum system toward one eigenstate. A two-level bistable quantum system is simulated as an example.     

Trace Formula for Noisy Flows

A trace formula is derived for the Fokker–Planck equation associated with Itô stochastic differential equations describing noisy time-continuous nonlinear dynamical systems. In the weak-noise limit, the trace formula provides estimations of the eigenvalues of the Fokker–Planck operator on the basis of the Pollicott–Ruelle resonances of the noiseless deterministic system, which is assumed to be non-bifurcating. At first order in the noise amplitude, the effect of noise on a periodic orbit is given in terms of the period and the derivative of the period with respect to the pseudo-energy of the Onsager–Machlup–Freidlin–Wentzell scheme.     

Importance sampling for randomly excited dynamical systems

An importance sampling technique for linear and non-linear dynamical systems subjected to random excitations is presented. Applying a transformation of probability measures, controls are introduced in the system of Itô stochastic differential equations such that the sample trajectories can be influenced in a predetermined way. As is shown, there exist controls resulting in unbiased zero-variance estimators. However, these optimal controls are in general not accessible and have to be replaced by sub-optimal ones derived from an optimization procedure analogous to the first order reliability method known from time-invariant problems. The efficiency of the proposed Monte Carlo simulation technique is demonstrated by estimating first-passage probabilities of typical oscillators under external white-noise excitation.     

Stochastic optimal control of partially observable nonlinear quasi-integrable Hamiltonian systems

The stochastic optimal control of partially observable nonlinear quasi-integrable Hamiltonian systems is investigated. First, the stochastic optimal control problem of a partially observable nonlinear quasi-integrable Hamiltonian system is converted into that of a completely observable linear system based on a theorem due to Charalambous and Elliot. Then, the converted stochastic optimal control problem is solved by applying the stochastic averaging method and the stochastic dynamical programming principle....     

An inverse model of MR damper using optimal neural network and system identification

Magnetorheological (MR) damper is one of the more promising new devices for vibration control of structures. External energy required by the adjustable fluid damper is minuscule while speed of its response is in the order of milliseconds. The MR damper is a semi-active control device and has been characterized by a set of non-linear differential equations which represent a forward model of the MR damper, i.e., the model can generate a force to a given displacement and applied voltage.This paper presents an inverse model of the MR damper, i.e., the model can predict the required voltage so that the MR damper can produce the desired force for the requirement of vibration control of structures. The inverse model has been constructed by using a multi-layer perceptron optimal neural network and system identification, which are Gauss-Newton-based Levenberg-Marquardt training algorithm, optimal brain surgeon strategy and autoregressive with exogenous variables (ARX) model. Based on the data from numerical simulation of the MR damper, the trained optimal neural networks can accurately predict voltage. If the inverse model is used in a control system, the semi-active vibration control can be implemented easily by using the semi-active MR damper.     

Semi-active control of 1D continuum vibrations under a travelling load

The paper presents a method for computing the response of a 1D elastic continuum supported by a set of semi-active viscous dampers and induced by a load travelling over it. The magnitude of the moving force has been assumed to be constant by neglect of the inertia forces. Full analytical solution is based on the power series method and is given in an arbitrary time interval. The time-marching scheme allows us to continue a solution in successive layers with initial conditions taken from the end of previous stages. The semi-active open loop control strategy is proposed. Shapes of damping functions are defined as a form of piecewise constant function. The control strategy is suboptimal and it outperforms the passive case. Numerical results are presented for the cases of a string and a Bernoulli-Euler beam.     

起落架磁流变减摆器模糊PID控制算法的研究

为了减小飞机机轮的摆振,提高飞机乘坐的舒适性和驾驶的安全性,将磁流变控制技术应用于飞机起落架减摆器,实现减摆器阻尼力的实时智能控制。针对磁流变减摆器,建立了飞机起落架摆振的半主动控制非线性动力学模型,设计了模糊PID控制算法,并使用Matlab/Simulink建立了半主动控制仿真模型。通过调节流过磁感线圈的电流大小改变磁流变减摆器的阻尼力,从而减小机轮摆动实现半主动控制。通过动力学仿真,在给定速度下分别对比未安装减摆器、被动控制下以及半主动控制下机轮摆角、侧向位移、侧滑角随时间变化的曲线,结果表明了模糊PID控制算法的正确性和可行性,该控制策略可以较好的抑制机轮的摆振,同时也表明模糊PID控制算法具有良好的可控性,减摆效果也明显优于传统的被动控制。     

Fresnel Type Path Integral for the Stochastic Schrödinger Equation

A path integral representation is obtained for the stochastic partial differential equation of Schrödinger type arising in the theory of open quantum systems subject to continuous nondemolition measurement and filtering, known as the a posteriori or Belavkin equation. The result is established by means of Fresnel-type integrals over paths in configuration space. This is achieved by modifying the classical action functional in the expression for the amplitude along each path by means of a stochastic Itô integral. This modification can be regarded as an extension of Menski's path integral formula for a quantum system subject to continuous measurement to the case of the stochastic Schrödinger equation.     

Fractional backward Kolmogorov equations

This paper derives the fractional backward Kolmogorov equations in fractal space-time based on the construction of a model for dynamic trajectories. It shows that for the type of fractional backward Kolmogorov equation in the fractal time whose coefficient functions are independent of time, its solution is equal to the transfer probability density function of the subordinated process X(Sα(t)), the subordinator Sα(t) is termed as the inverse-time α-stable subordinator and the process X(τ) satisfies the corresponding time homogeneous Ito stochastic differential equation.     

Semi-active friction tendons for vibration control of space structures

Semi-active vibration control systems are becoming popular because they offer both the reliability of passive systems and the versatility of active control without high power demands. In this work, a new semi-active control system is proposed and studied numerically. The system consists of variable-friction dampers linked to the structure through cables. Auxiliary soft springs in parallel with these friction dampers allow them to return to their initial pre-tensioned state. Using cables makes the system suitable for deployable, flexible and lightweight structures, such as space structures (spacecraft). A control system with three control laws applied to a single-degree-of-freedom structure is studied. Two of these laws are derived by using Lyapunov theory, whereas the third one is developed heuristically. In order to assess the performance of the control system, a parametric study is carried out through numerical simulations. An application of the proposed method to multi-degree-of-freedom structures is also presented and demonstrated through a numerical example. The system in semi-active mode is more effective than in passive mode and its effectiveness is less sensitive to loss of pre-tension.     

Stochastic averaging of quasi-integrable Hamiltonian systems with delayed feedback control

A stochastic averaging method for quasi-integrable Hamiltonian systems with time-delayed feedback control is proposed. First, a quasi-integrable Hamiltonian system with delayed feedback control subjected to Gaussian white noise excitations is formulated and then transformed into Itô stochastic differential equations without time delay. Then, the averaged Itô stochastic differential equations for the system are derived and the stationary solution of the averaged Fokker–Planck–Kolmogorov (FPK) equation associated with the averaged Itô equations is obtained for both non-resonant and resonant cases. Finally, three examples are worked out in detail to illustrate the application and effectiveness of the proposed method and the effect of time delayed feedback control on the response of the systems.     

Langevin equations and stochastic integrals

The ambiguity of stochastic integrals involved in Langevin equations is removed by the postulate of invariance with respect to nonlinear transformations of the coordinates. The Stratonovich sense of the integrals, which is imposed thereby, is also strongly suggested by stability considerations requiring small changes of the solutions whenever the perturbations are changed by a small amount. The associated Fokker-Planck equation must include the spurious drift which arises from the transition from the Stratonovich to the Itô sense of the Langevin equations and describes one aspect of the systematic motion due to nonconstant fluctuations.     

Control of noisy chaotic motion in a system with nonlinear excitation and restoring forces

In this study we examine the complex and chaotic oscillations of a dynamical system with nonlinear excitation and restoring forces for the purpose of controlling these oscillatory states. The physical system, modeled as a system of first-order nonlinear ordinary differential equations, takes into account a geometric nonlinearity in the restoring force, a quadratic viscous drag, and a harmonic excitation force. It is controlled using small perturbations about a selected unstable cycle and control is instigated for periodic cycles of varying periodicities. The controller, when applied on the dynamical system with additive random noise in the excitation, successfully controls the system with noise levels in excess of 5% of the total energy, giving the first evidence that (stochastic) control of these systems is possible. (c) 1997 American Institute of Physics.     

The validity of nonlinear Langevin equations

In the presence of internal noise the variables describing a system are intrinsically stochastic. If they constitute a Markov process the expansion enables one to extract a deterministic macroscopic equation and to compute the fluctuations in successive approximations. In the lowest or linear noise approximation the fluctuations can be represented by a Langevin equation, provided it is handled appropriately. Higher orders cannot be described by any white noise Langevin equation. The question whether the equation has to be interpreted according to Itô or Stratonovich concerns these higher orders, for which the equation is not valid anyway.     

Averaging Principle for the Higher Order Nonlinear Schrödinger Equation with a Random Fast Oscillation

This work concerns the problem associated with averaging principle for a higher order nonlinear Schrödinger equation perturbed by a oscillating term arising as the solution of a stochastic reaction–diffusion equation evolving with respect to the fast time. This model can be translated into a multiscale stochastic partial differential equations. Stochastic averaging principle is a powerful tool for studying qualitative analysis of stochastic dynamical systems with different time-scales. To be more precise, under suitable conditions, we prove that there is a limit process in which the fast varying process is averaged out and the limit process which takes the form of the higher order nonlinear Schrödinger equation is an average with respect to the stationary measure of the fast varying process. Finally, by using the Khasminskii technique we can obtain the rate of strong convergence for the slow component towards the solution of the averaged equation, and as a consequence, the system can be reduced to a single higher order nonlinear Schrödinger equation with a modified coefficient.