Short-term dynamic instability of a nonlinear stochastic SDOF system

An intensity-modulated Gaussian process {x(t)} is defined as the product of a non-negative stationary random process {σx(t)} and a stationary Gaussian process {z(t)} that is statistically independent of {σx(t)}. When such an intensity-modulated process is the excitation to a linear time-invariant system, the response process {y(t)}, conditioned on the excitation-modulating function σx(t), is a generally nonstationary Gaussian process. This property is used to develop an approximation to the probability density function (PDF) of the system response by assuming that the time-varying conditional response variance σ2y(t), conditioned on the excitation-modulating function σx(t), is governed by the gamma PDF. The resulting response process PDF is characterized by two parameters: the mean and variance of the conditional response variance σ2y(t). An alternative approximation to the response PDF, characterized by these same two parameters, is developed by using a Taylor’s series representation in the variable σ2y of the response PDF conditioned on σ2y. This latter PDF approximation is shown to be identical to the Gram–Charlier series of type A through terms involving the Hermite polynomial of degree 6. The two response PDF approximations are shown to give essentially identical results for values of the coefficient of variation of σ2y of one-half or less. Simple, easily evaluated expressions for the mean and variance of σ2y are derived in terms of measurable quantities by utilizing the assumption that fluctuations in σx(t) occur slowly in comparison with those of z(t).

In this work, a novel shape control approach of the probability density function (PDF) for nonlinear stochastic systems is presented. First, we provide the formula for the PDF shape controller without devising the control law of the controller. Then, based on the exact analytical solution of the Fokker-Planck-Kolmogorov (FPK) equation, the product function of the polynomial and the exponential polynomial is regarded as the stationary PDF of the state response. To validate the performance of the proposed control approach, we compared it with the exponential polynomial method and the multi-Gaussian closure method by implementing comparative simulation experiments. The results show that the novel PDF shape control approach is effective and feasible. Using an equal number of parameters, our method can achieve a similar or better control effect as the exponential polynomial method. By comparison with the multi-Gaussian closure method, our method has clear advantages in PDF shape control performance. For all cases, the integral of squared error and the errors of first four moments of our proposed method were very small, indicating superior performance and promising good overall control effects of our method. The approach presented in this study provides an alternative for PDF shape control in nonlinear stochastic systems.

Nonlinear stochastic seismic dynamic response analysis of submerged floating tunnel subjected to non-stationary ground motion

A generalized method for the stationary probabilistic response of nonlinear dynamical system

The joint probability density function (PDF) of different structural responses is a very important topic in the stochastic response analysis of nonlinear structures. In this paper, the probability density evolution method, which is successfully developed to capture the instantaneous PDF of an arbitrary single response of interest, is extended to evaluate the joint PDF of any two responses. A two-dimensional partial differential equation in terms of the joint PDF is established. The strategy of selecting representative points via the number theoretical method and sieved by a hyper-ellipsoid is outlined. A two-dimensional difference scheme is developed. The free vibration of an SDOF system is examined to verify the proposed method, and a frame structure exhibiting hysteresis subjected to stochastic ground motion is investigated. It is pointed out that the correlation of different responses results from the fact that randomness of different responses comes from the same set of basic random parameters involved. In other words, the essence of the probabilistic correlation is a physical correlation.

Direct probability integral method for stochastic response analysis of static and dynamic structural systems

Adaptive fault tolerant tracking control for a class of stochastic nonlinear systems with output constraint and actuator faults

Stochastic ship roll motion via path integral method

New statistical theory was developed by the authors in the previous paper, which gives probability density function (p. d. f.) and extreme values of total second order responses of moored vessels in random seas. It assumes the response represented in the form of two term Volterra functional series and is based on the generalized Laguerre polynomial expansion of p. d. f. of which the first term is a gamma p. d. f. consisting of three parameters.This paper shows non-normality of surge responses of moored tanker and semi-submersible measured in a wave basin, and determines that the theory estimates it very well. Furthermore effect of time varying wave drift damping and other slow motion hydrodynamic force coefficients on p. d. f. and extreme values are discussed by time domain simulations. This effect cannot be represented in the form of two term Volterra functional series.

This paper investigates the stochastic distribution tracking problem while the probability density function (PDF) of the stochastic non-linear system output can be controlled to desired distribution. To achieve the control objective, a data- driven approach is proposed in which no information of the system model is required. The output PDF can be estimated by kernel density estimation (KDE) based on the collected system output data. Using the estimated PDF, the probability states can be obtained by sampling operation which can be used to re-characterise the PDF of the system output. Thus, the tracking performance can be achieved by PID control. The parametric selection of the controller has been analysed following the identified PDF dynamic model to assure the convergence of the system output. The effectiveness of the presented algorithm is illustrated by a numerical example.

Constructing transient response probability density of non-linear system through complex fractional moments

Fixed-time adaptive neural self-triggered decentralized control for stochastic nonlinear systems with strong interconnections

Due to limited computational power, performing uncertainty quantification analyses with complex computational models can be a challenging task. This is exacerbated in the context of stochastic simulators, the response of which to a given set of input parameters, rather than being a deterministic value, is a random variable with unknown probability density function (PDF). Of interest in this paper is the construction of a surrogate that can accurately predict this response PDF for any input parameters. We suggest using a flexible distribution family−the generalized lambda distribution−to approximate the response PDF. The associated distribution parameters are cast as functions of input parameters and represented by sparse polynomial chaos expansions. To build such a surrogate model, we propose an approach based on a local inference of the response PDF at each point of the experimental design based on replicated model evaluations. Two versions of this framework are proposed and compared on analytical examples and case studies.

This note is devoted to study the stabilization problem of stochastic nonlinear delay systems with exogenous disturbances and the event-triggered feedback control. By introducing the notation of input-to-state practical stability and an event-triggered strategy, we establish the input-to-state practically exponential mean-square stability of the suggested system. Moreover, we investigate the stabilization result by designing the feedback gain matrix and the event-triggered feedback controller, which is expressed in terms of linear matrix inequalities. Also, the lower bounds of interexecution times by the proposed event-triggered control method are obtained. Finally, an example is given to show the effectiveness of the proposed method. Compared with a large number of results for discrete-time stochastic systems, only a few results have appeared on the event-triggered control for continuous-time stochastic systems. In particular, there have been no published papers on the event-triggered control for continuous-time stochastic delay systems. This note is a first try to fill the gap on the topic.

The problem of parametric synthesis of an electronic control unit for a brake system of a car is considered, which consists in the process of choice of values of the varied parameters of electronic CU. The selected parameters should provide the maximum margin of stability and maximum velocity of the closed-loop system of automatic control of a given direction of car movement in braking mode. This is important because modern high-velocity cars, operating in conditions of low-quality road surfaces of highways, often become objects of road traffic accidents, despite the fact that they are equipped with anti-lock braking and traction control systems. In this regard, the design development of the latest models of automotive equipment is aimed at creating directional stability systems, increased vehicle dirigibility in case of loss of wheel adhesion with the road surface. The problem is solved using the method of frozen coefficients, which is based on the iterative process of sequential selection of numerical values of the varied parameters of the electronic unit. In particular, it is found that the characteristic equation of a closed-loop stability control system contains three variable parameters, and the system itself is a two-channel one. The first control channel for the inclination angle of the car body contains variable parameters, and the second control channel for side skid contains a variable parameter. It is shown that 5–6 steps of the iterative process are sufficient to obtain optimal values of the varied parameters that provide the maximum safety margin and maximum performance of the closed-loop vehicle stability program. The work proves the effectiveness of the developed algorithms for parametric synthesis of the electronic control unit for a car, the simplicity of their implementation, as well as the high quality of stabilization processes of the car body. The optimal values of the varied parameters are In this case, the margin of stability of the system is