Stochastic stability of Duffing oscillator with fractional derivative damping under combined harmonic and white noise parametric excitations

The stochastic stability of a Duffing oscillator with fractional derivative damping of order α (0 < α < 1) under parametric excitation of both harmonic and white noise is studied. First, the averaged Itô equations are derived by using the stochastic averaging method for an SDOF strongly nonlinear stochastic system with fractional derivative damping under combined harmonic and white noise excitations. Then, the expression for the largest Lyapunov exponent of the linearized averaged Itô equations is obtained and the asymptotic Lyapunov stability with probability one of the original system is determined approximately by using the largest Lyapunov exponent. Finally, the analytical results are confirmed by using those from a Monte Carlo simulation of the original system.     

Feedback stabilization of quasi nonintegrable Hamiltonian systems under combined Gaussian and Poisson white noise excitations

A procedure for designing a feedback control to asymptotic Lyapunov stability with probability one of quasi nonintegrable Hamiltonian systems under combined Gaussian and Poisson white noise excitations is proposed. First, a one dimensional partially averaged Itô stochastic differential equation for controlled Hamiltonian is derived from the motion equations of the system by using the stochastic averaging method. Second, the dynamical programming equation for the ergodic control problem of the averaged system with undetermined cost function is set up based on the dynamical programming principle and the jump–diffusion chain stochastic differential rules. The optimal control law is obtained by solving the dynamical programming equation. Third, the analytical expression for the largest Lyapunov exponent of the averaged system is derived. Finally, the asymptotic Lyapunov stability with probability one of the originally controlled system is analyzed approximately by using the largest Lyapunov exponent. The cost function and optimal control forces are determined by the requirements of stabilizing the system. An example is worked out in detail to illustrate the effectiveness of the proposed method for stabilization control, and the control effect of the proposed feedback stabilization varies with the change of parameters is also studied in this paper, such as, the greater the excitation intensity of Gaussian and Poisson white noise, the better the stabilization control effect.     

Stochastic stability of quasi non-integrable Hamiltonian systems under parametric excitations of Gaussian and Poisson white noises

The asymptotic Lyapunov stability with probability one of n-degree-of-freedom (n-DOF) quasi non-integrable Hamiltonian systems subject to weakly parametric excitations of combined Gaussian and Poisson white noises is studied by using the largest Lyapunov exponent. First, an n-DOF quasi non-integrable Hamiltonian system subject to weakly parametric excitations of combined Gaussian and Poisson white noises is reduced to a one-dimensional averaged Itô stochastic differential equation (SDE) for Hamiltonian by using the stochastic averaging method for quasi non-integrable Hamiltonian systems. Then, the expression for the Lyapunov exponent of the averaged Itô SDE is derived and the approximately necessary and sufficient condition for the asymptotic Lyapunov stability with probability one of the trivial solution of the original system is obtained. Finally, one example is worked out to illustrate the proposed procedure and its effectiveness is confirmed by comparing with Monte Carlo simulation. It is found that analytical and simulation results agree well.     

Stochastic stability of linear viscoelastic systems

The stochastic almost-sure stability of a single degree-of-freedom linear viscoelastic system subjected to random fluctuation in the stiffness parameter is investigated. For small damping and weak random fluctuation, asymptotic expressions are derived for the Lyapunov exponent and the rotation number using the method of stochastic averaging. From the sign of the Lyapunov exponent, the condition for asymptotic stability with probability 1 of the trivial equilibrium state is obtained.     

Minimax optimal control of uncertain quasi-integrable Hamiltonian systems with time-delayed bounded feedback

A minimax optimal control strategy for uncertain quasi-integrable Hamiltonian systems with time-delayed bounded feedback control is proposed. First, a quasi-integrable Hamiltonian system with time-delayed bounded control forces and uncertain excitation and system parameters is converted into a set of Itô stochastic differential equations without time delay. Then, the partially averaged Itô stochastic differential equations for the energy processes are derived by using the stochastic averaging method for quasi-integrable Hamiltonian systems. For these equations together with an appropriate performance index, a worst-case optimal control strategy is derived via solving a stochastic differential game problem. The worst-case disturbances and the optimal bounded controls are obtained by solving a Hamilton–Jacobi–Isaacs (HJI) equation. Finally, two examples are worked out in detail to illustrate the application and effectiveness of the proposed method.     

Lyapunov functions for quasi-Hamiltonian systems

A procedure for constructing the Lyapunov functions and studying their asymptotic Lyapunov stability with probability one for quasi-Hamiltonian systems is proposed. For quasi-non-integrable Hamiltonian systems, the Hamiltonian (the total energy) is taken as the Lyapunov function. For quasi-integrable and quasi-partially-integrable Hamiltonian systems, the optimal linear combination of the independent first integrals in involution is taken as the Lyapunov function. The derivative of the Lyapunov function with respect to time is obtained by using the stochastic averaging method for quasi-Hamiltonian systems. The sufficient condition for the asymptotic Lyapunov stability with probability one of quasi-Hamiltonian systems is determined based on a theorem due to Khasminskii and compared with the corresponding necessary and sufficient condition obtained by using the largest Lyapunov exponent. Three examples are worked out to illustrate the proposed procedure and its effectiveness.     

最优有界控制下色噪声驱动多时滞拟线性系统瞬态响应

基于Fokker-Planck-Kolmogorov方程瞬态求解研究了受最优有界控制的色噪声驱动的多时滞拟线性系统的瞬态响应。利用等价变换将时滞系统转化为非时滞系统。在弱扰动假设下应用标准随机平均法得到振幅过程的部分平均It?随机微分方程。由动态规划原理和控制力界值条件得到最优有界控制率从而得到完全平均的Fokker-Planck-Kolmogorov方程。通过原系统的退化线性系统导出一组正交基并在该基空间内进行Galerkin变分得到近似瞬态响应。最后将该方法应用到受最优有界控制率和色噪声共同作用的时滞Duffing-Van Der Pol振子进行理论求解并综合讨论了色噪声、时滞、控制力和共振对系统瞬态响应的影响,采用Monte-Carlo模拟验证了所有理论和计算结果的正确性。     

Direct control method for improving stability and reliability of nonlinear stochastic dynamical systems

Optimal control for improving the stability and reliability of nonlinear stochastic dynamical systems is of great significance for enhancing system performances. However, it has not been adequately investigated because the evaluation indicators for stability (e.g. maximal Lyapunov exponent) and for reliability (e.g. mean first-passage time) cannot be explicitly expressed as the functions of system states. Here, a unified procedure is established to derive optimal control strategies for improving system stability and reliability, in which a physical intuition-inspired separation technique is adopted to split feedback control forces into conservative components and dissipative components, the stochastic averaging is then utilized to express the evaluation indicators of performances of controlled system, the optimal control strategies are finally derived by minimizing the performance indexes constituted by the sigmoid function of maximal Lyapunov exponent (for stability-based control)/the reciprocal of mean first-passage time (for reliability-based control), and the mean value of quadratic form of control force. The unified procedure converts the original functional extreme problem of optimal control into an extremum value problem of multivariable function which can be solved by optimization algorithms. A numerical example is worked out to illustrate the efficacy of the optimal control strategies for enhancing system performance.     

Some aspects of chaotic and stochastic dynamics for structural systems

In this paper, the bifurcation behaviour of an externally excited four-dimensional nonlinear system is examined. Throughout this paper, a two-degree-of-freedom shallow arch structure under either a periodic or a stochastic excitation will be considered. For the case when the excitation is periodic, the local and global behaviour is examined in the presence of principalsubharmonic resonance and1:2 internal resonance. The method of averaging is used to obtain the first order approximation of the response of the system under resonant conditions. A standard Melnikov type perturbation method is used to show analytically that the system may exhibit chaotic dynamics in the sense of Smale horseshoe for the 1:2 internal resonance case in the absence of dissipation. In the case of stochastic excitation, the stability of the stationary solution is examined by determining themaximal Lyapunov exponent andmoment Lyapunov exponent in terms of system parameters. An asymptotic method is used to obtain explicit expressions for various exponents in the presence of weak dissipation and noise intensity. These quantities provide almost-sure stability boundaries in parameter space. When the system parameters lie outside these boundaries, it is essential to understand the nonlinear behaviour. The method of stochastic averaging is applied to obtain a set of approximate Itô equations which are then examined to describe the local bifurcation behaviour.     

Generation of non-Gaussian stochastic processes using nonlinear filters

Non-Gaussian stochastic processes are generated using nonlinear filters in terms of Itô differential equations. In generating the stochastic processes, two most important characteristics, the spectral density and the probability density, are taken into consideration. The drift coefficients in the Itô differential equations can be adjusted to match the spectral density, while the diffusion coefficients are chosen according to the probability density. The method is capable to generate a stochastic process with a spectral density of one peak or multiple peaks. The locations of the peaks and the band widths can be tuned by adjusting model parameters. For a low-pass process with the spectrum peak at zero frequency, the nonlinear filter can match any probability distribution, defined either in an infinite interval, a semi-infinite interval, or a finite interval. For a process with a spectrum peak at a non-zero frequency or with multiple peaks, the nonlinear filter model also offers a variety of profiles for probability distributions. The non-Gaussian stochastic processes generated by the nonlinear filters can be used for analysis, as well as Monte Carlo simulation.     

Stochastic control of hysteretic structural systems

A method of stochastic optimal control of hysteretic structural systems under earthquake excitations is presented. Stochastic estimation and control problems are formulated in the form of Itô stochastic differential equations on the basis of the theory of continuous Markov processes. The conditional moment equations given observation data are derived for nonlinear filtering, and are closed by introducing appropriate analytical form of the conditional probability density functions of the state variables. Under the assumption that the admissible controls are expressed as functions of the conditional moment functions the Bellman equation is derived. If the spatial variables of the Bellman equation are defined by a part of the full set of conditional moment functions appearing in the closed moment equations, the resulting Bellman equation is coupled with conditional moment equations both for filtering and for prediction. The Gaussian and non-Gaussian stochastic linearization techniques combined with simple solution techniques to the Bellman equation are examined to solve the Bellman equation or extended Riccati equations without prediction procedures.     

Delay-independent stability of moments of a linear oscillator with delayed state feedback and parametric white noise

The stability of a linear oscillator with delayed state feedback driven by parametric Gaussian white noise is studied in this paper. The first and second order moment equations of the system response are derived by using moment method and Itô differential rule. Based on the moment equations, the delay-independent stable conditions of both moments are proposed: For the first order moment, the sufficient and necessary condition that guarantee delay-independent stability is identified to that of the deterministic system; for the second order moment, the sufficient condition that ensure delay-independent stability depends on noise intensity. The theoretical results are also illustrated with numerical simulations.     

Asymptotic properties of the spectrum of neutral delay differential equations

Spectral properties and transition to instability in neutral delay differential equations are investigated in the limit of large delay. An approximation of the upper boundary of stability is found and compared to an analytically derived exact stability boundary. The approximate and exact stability borders agree quite well for the large time delay, and the inclusion of a time-delayed velocity feedback improves this agreement for small delays. Theoretical results are complemented by a numerically computed spectrum of the corresponding characteristic equations.     

Time-delay stochastic optimal control and stabilization of quasi-integrable Hamiltonian systems

Innovative procedures for the time-delay stochastic optimal control and stabilization of quasi-integrable Hamiltonian systems subject to Gaussian white noise excitations are proposed. First, the problem of time-delay stochastic optimal control of quasi-integrable Hamiltonian systems is formulated and converted into the problem of stochastic optimal control without time delay. Then the converted control problem is solved by applying the stochastic averaging method for quasi-integrable Hamiltonian systems and the stochastic dynamical programming principle. The time-delay feedback stabilization of quasi-integrable Hamiltonian systems is formulated as an ergodic control problem with an un-determined cost function which is determined later by minimizing the largest Lyapunov exponent of the controlled system. As an example, a two-degree-of-freedom quasi-integrable Hamiltonian system with time-delay feedback control forces is investigated in detail to illustrate the procedures and their effectiveness.     

Small noise expansion of moment lyapunov exponents for two-dimensional systems

We construct an approximation for the moment Lyapunov exponent, the asymptotic growth rate of the moments of the response of a two-dimensional linear system driven by real or white noise. A perturbation approach is used to obtain explicit expressions for these exponents in the presence of small intensity noise. As an example, we study the moment stability of the stationary solution of nonlinear structural and mechanical systems subjected to real noise excitation. The usefulness of the moment Lyapunov exponent in predicting parameter values at which qualitative changes in the probability density function occur (stochastic bifurcation) is also illustrated.     

Solutions of the FPK equation for time-delayed dynamical systems with the continuous time approximation method

This paper presents a method of finite-dimensional Markov process (FDMP) approximation for stochastic dynamical systems with time delay and numerical solutions of probability density functions of the systems. Solutions of probability density functions of time-delayed systems are rare in the literature. The FDMP method preserves the standard state space format of the system, and allows us to apply all the existing methods and theories for analysis and control of stochastic dynamical systems and to compute the probability density functions efficiently. The solutions of the FPK equation for a linear time-delayed stochastic system are presented. The effects of different spectral differentiation schemes for the FDMP method on the probability density functions are compared.     

Stochastic stability of SDOF linear viscoelastic system under wideband noise excitation

The stochastic moment stability and almost-sure stability of a single-degree-of-freedom (SDOF) viscoelastic system subject to parametric fluctuation is investigated by using the method of higher-order stochastic averaging. The stochastic parametric excitation is modeled as a wideband noise, which is taken as Gaussian white noise and real noise. The viscoelastic material is assumed to follow ordinary Maxwell linear constitutive relation. For small damping and weak stochastic fluctuation, analytical expressions are derived for the moment Lyapunov exponent and the Lyapunov exponent, which indicate moment stability and almost-sure stability respectively. The effects of various system and loading parameters on the stochastic stability are discussed. Both analytical and simulation results show that higher-order stochastic averaging improves the accuracy compared with the first-order stochastic averaging. However, results of the third-order averaging are almost overridden by those of second-order averaging and the third-order averaging involves far more calculation. It is advisable to consider a balance between accuracy achievement and calculation endeavor when using higher-order stochastic averaging.     

Stability,control and reliability of a ship crane payload motion

This paper investigates the stochastic dynamics, stability and control of a ship-based crane payload motion, as well as the first time passage type of failure. The simplified nonlinear model of the payload motion is considered, where the excitation of a suspension point is imposed due to the heaving motion of waves. The latter enters the system parametrically, leading to a Mathieu type nonlinear equation. The stability boundaries are numerically calculated, using the Lyapunov exponent approach. The control strategy, based on the feedback bang–bang control policy, is implemented to minimize the load's swinging motion. Finally, the first time passage problem is addressed employing Monte-Carlo sampling of the failure process.