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.     

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 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.     

First-passage failure of quasi linear systems subject to multi-time-delayed feedback control and wide-band random excitation

A procedure for studying the first-passage failure of quasi-linear systems subject to multi-time-delayed feedback control and wide-band random excitation is proposed. The stochastic averaging method for quasi-integrable Hamiltonian systems is first introduced. The backward Kolmogorov equation governing the conditional reliability function and a set of generalized Pontryagin equations governing the conditional moments of first-passage time are then established. The conditional reliability function, the conditional probability density and moments of first-passage time are obtained by solving the backward Kolmogorov equation and generalized Pontryagin equations with suitable initial and boundary conditions. An example is given to illustrate the proposed procedure and the results from digital simulation are obtained to verify the effectiveness of the proposed procedure. The effects of time delay in feedback control forces on the conditional reliability function, conditional probability density and moments of first-passage time are analyzed.     

一类中立型时滞系统新的鲁棒稳定性准则

本文考虑了一类具有时变时滞和非线性扰动的中立型系统的鲁棒稳定性问题.基于Lyapunov稳定性理论和自由权矩阵方法,得到保证系统鲁棒渐近稳定的新的充分条件.所得结果同时依赖于离散时滞和中立时滞,并用LMIs表示.由于对Lyapunov泛函导数采用了无保守的估计,因此所得结果具有较小的保守性,能够给出时变时滞较大的允许时...     

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.     

一类动力系统的稳定性及镇定

本文定义了一类新的动力系统-HCTD系统,利用矩阵迹的不等式理论研究了这类系统的稳定性,并给出了该系统稳定的充分必要条件。讨论了当Lyapunov方程的解P是HCTD阵时,动力系统稳定的充分条件。提出了用解矩阵迹的不等式设计动力系统镇定控制器的一种新方法。     

Asymptotic Lyapunov stability with probability one of Duffing oscillator subject to time-delayed feedback control and bounded noise excitation

The asymptotic Lyapunov stability with probability one of a Duffing system with time-delayed feedback control under bounded noise parametric excitation is studied. First, the time-delayed feedback control force is expressed approximately in terms of the system state variables without time delay. Then, the averaged Itô stochastic differential equations for the system are derived by using the stochastic averaging method and the expression for the Lyapunov exponent of the linearized averaged Itô equations is derived. It is inferred that the Lyapunov exponent so obtained is the first approximation of the largest Lyapunov exponent of the original system, and the asymptotic Lyapunov stability with probability one of the original system can be determined approximately by using the Lyapunov exponent. Finally, the effects of time delay in feedback control on the Lyapunov exponent and the stability of the system are analyzed. The theoretical results are well verified through digital simulation.     

未知控制方向的非线性级联系统鲁棒自适应输出反馈控制

本文研究了一类具有未知控制方向的非线性级联系统的鲁棒自适应输出反馈问题．通过线性变换将有多个未知控制方向的系统转化为无未知控制方向的系统,并根据线性高增益控制观测器与Nussbaum函数,设计了一种新的鲁棒自适应输出反馈控制器,进一步证明了在该控制器下闭环系统所有信号有界且状态渐进趋于零．进而,通过构造Lyapunov函数,给出了闭环系统渐进稳定的充分条件．最后,利用仿真实例说明了控制算法的有效性．     

一类变时滞非线性Lurie控制系统的绝对稳定性

应用Lyapunov函数方法讨论了一类变时滞非线性控制系统的绝对稳定性。得到Lurie控制系统绝对稳定的充分条件。文末用例了说明此方法的有效性。     

An evaluation of the Lyapunov characteristic exponent of chaotic continuous systems

A procedure to calculate the Lyapunov characteristic exponent of the response of structural continuous systems, discretized using finite element methods, is proposed. The Lyapunov characteristic exponent can be used to characterize the asymptotic stability of the system dynamic response, and it is frequently employed to identify a chaotic behaviour. The proposed procedure can also be used in the stability characterization of fluid–structure interaction systems in which the focus of the analysis is on the behaviour of the structural part. Copyright © 2002 John Wiley & Sons, Ltd.     

Effect of derivative process on the almost-sure asymptotic stability of second-order linear stochastic systems

A method of obtaining a sufficient almost-sure (a.s.) asymptotic stability condition for second-order, linear systems with ergodic damping coefficient is presented. The probabilistic property of the derivative process of the damping coefficient is taken into account. A sufficient condition for almost-sure asymptotic stability is derived and numerical results are presented for the case of Gaussian noise coefficients. The results are found to be a significant improvement over previously available results for second-order systems with stochastic damping.     

Effect of derivative process on the almost-sure asymptotic stability of second-order linear stochastic systems

A method of obtaining a sufficient almost-sure (a.s.) asymptotic stability condition for second-order, linear systems with ergodic damping coefficient is presented. The probabilistic property of the derivative process of the damping coefficient is taken into account. A sufficient condition for almost-sure asymptotic stability is derived and numerical results are presented for the case of Gaussian noise coefficients. The results are found to be a significant improvement over previously available results for second-order systems with stochastic damping.     

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.     

异步切换下切换系统的动态输出反馈保成本控制

作为一类重要的混杂系统,切换系统由若干个子系统以及一个协调各子系统之间切换的切换信号构成,在自然科学、工程控制和社会系统等方面有着广泛的应用。在对切换系统的控制问题进行研究时,一般假设子系统和控制器同步运行。然而,在实际工程控制中,控制器的切换相对于子系统的切换存在切换时延,从而产生异步切换。因此,对异步切换下的切换系统研究是十分必要的。针对一类异步切换下切换系统的动态输出反馈保成本控制问题进行了研究。报告了切换系统的研究现状以及异步切换下切换系统的最新研究成果。利用分段李雅普诺夫函数法和平均驻留时间法,得到了使得异步切换下的闭环切换系统稳定的动态输出反馈控制器存在的充分条件。从线性矩阵不等式的角度,提出了异步切换动态输出反馈保成本控制器设计方案,并给出成本上界的优化方法。最后,通过一个数值算例说明了提出的方法的有效性。     

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.     

Stability criteria for neutral systems with time‐varying delays and nonlinear uncertainties

Abstract

The asymptotic stability problem for a class of neutral systems with time‐varying delays and nonlinear uncertainties is investigated in this paper. LMI‐based delay‐dependent criteria are proposed to guarantee the asymptotic stability of the considered systems. New Lyapunov‐Krasovskii functional and Leibniz‐Newton formulae are used to find the delay‐dependent stability results. Finally, some numerical examples are illustrated to show the improved results from using this method.     

A weak energy–momentum method for stochastic instability induced by dissipation and random excitations

This paper focuses on the problem of stochastic instability resulting from the action of dissipation and random excitations. The energy–momentum theorem is extended from deterministic Hamiltonian systems to stochastic Hamiltonian systems, and then a weak energy–momentum method is presented for stochastic instability analysis of random systems suffering destabilizing effects of dissipation and random excitations. The presented method combines the stochastic averaging procedure to formulate the equivalent systems of random systems for obtaining the stochastic instability criteria in probability, and can be applied to a class of systems including random gyroscopic systems with positive or negative definite potential energy. As an example, the stochastic instability conditions of a Lagrange top subjected to random vertical support excitations are formulated to express the stochastic instability induced by dissipation and random excitations.     

First passage failure of SDOF nonlinear oscillator with lightly fractional derivative damping under real noise excitations

The first passage failure of single-degree-of-freedom (SDOF) nonlinear oscillator with lightly fractional derivative damping under real noise excitations is investigated in this paper. First, the system state is approximately represented by one-dimensional time-homogeneous diffusive Markov process of amplitude through stochastic averaging. Then, the backward Kolmogorov equation governing the conditional reliability function and the Pontryagin equation governing the conditional mean of first passage time are established from the averaged Itô equation for Hamiltonian. The conditional reliability function, the conditional probability density and mean of the first passage time are obtained by solving these equations together with suitable initial condition and boundary conditions. Finally, two examples are worked out in detail and the analytical solutions are checked by those from the Monte Carlo simulation of original systems.