Poisson white noise parametric input and response by using complex fractional moments

In this paper the solution of the generalization of the Kolmogorov–Feller equation to the case of parametric input is treated. The solution is obtained by using complex Mellin transform and complex fractional moments. Applying an invertible nonlinear transformation, it is possible to convert the original system into an artificial one driven by an external Poisson white noise process. Then, the problem of finding the evolution of the probability density function (PDF) for nonlinear systems driven by parametric non-normal white noise process may be addressed in determining the PDF evolution of a corresponding artificial system with external type of loading.     

Pseudo-force method for a stochastic analysis of nonlinear systems

Nonlinear systems, driven by external white noise input processes and handled by means of pseudo-force theory, are transformed through simple coordinate transformation to quasi-linear systems. By means of Itô stochastic differential calculus for parametric processes, a finite hierarchy for the moment equations of these systems can be exactly obtained. Applications of this procedure to the first-order differential equation with cubic nonlinearity and to the Duffing oscillator show the versatility of the proposed method. The accuracy of the proposed procedure improves by making use of the classical equivalent linearization technique.     

Stochastic averaging of quasi-linear systems driven by Poisson white noise

The averaged generalized Itô and Fokker–Planck–Kolmogorov (FPK) equations for single-degree-of-freedom (SDOF) quasi-linear systems driven by Poisson white noise are derived and the approximate stationary solutions of the averaged generalized FPK equations are obtained by using the perturbation method for four typical quasi-linear systems, i.e., van der Pol oscillator, Rayleigh oscillator, system with energy-dependent damping, and system with power law damping. The effectiveness and accuracy of the perturbation solution are assessed by performing appropriate Monte Carlo simulations. It is found that analytical and numerical results agree well and the effect of non-Gaussianity of the excitation process is not negligible for predicting the probability densities of total energy and displacement of quasi-linear systems in most cases.     

Optimal bounded control of harmonically and stochastically excited strongly nonlinear oscillators

A procedure for designing optimal bounded control to minimize the response of harmonically and stochastically excited strongly nonlinear oscillators is proposed. First, the stochastic averaging method for controlled strongly nonlinear oscillators under combined harmonic and white noise excitations using generalized harmonic functions is introduced. Then, the dynamical programming equation for the control problem of minimizing response of the systems is formulated from the partially completed averaged Itô equations by using the dynamical programming principle. The optimal control law is derived from the dynamical programming equation and control constraint without solving the dynamical programming equation. Finally, the stationary probability density of the amplitude and mean amplitude of the optimally controlled systems are obtained from solving the reduced Fokker–Planck–Kolmogorov equation associated with fully completed averaged Itô equations. An example is given to illustrate the proposed procedure and the results obtained are verified by using those from digital simulation.     

Path integral solution for nonlinear systems under parametric Poissonian white noise input

In this paper the problem of the response evaluation in terms of probability density function of nonlinear systems under parametric Poisson white noise is addressed. Specifically, extension of the Path Integral method to this kind of systems is introduced. Such systems exhibit a jump at each impulse occurrence, whose value is obtained in closed form considering two general classes of nonlinear multiplicative functions. Relying on the obtained closed form relation liking the impulses amplitude distribution and the corresponding jump response of the system, the Path Integral method is extended to deal with systems driven by parametric Poissonian white noise. Several numerical applications are performed to show the accuracy of the results and comparison with pertinent Monte Carlo simulation data assesses the reliability of the proposed procedure.     

Semi-active control of wind excited building structures using MR/ER dampers

A semi-active control strategy for building structures subject to wind loading and controlled by MR/ER dampers is proposed. The power spectral density (PSD) matrix of the fluctuating part of wind velocity vector is diagonalized in the eigenvector space. Each element of the diagonalized PSD matrix is modeled as a set of second-order linear filter driven by white noise. A Bingham model for MR/ER dampers is used. The forces produced by MR/ER dampers are split into passive and active parts and the passive part is combined with structural damping forces. A set of partially averaged Itô equations for controlled modal energies are derived by applying the stochastic averaging method for quasi-integrable-Hamiltonian systems. The optimal control law is then determined by using the stochastic dynamical programming principle and the cost function is so selected that the optimal control law can be implemented by the MR/ER dampers. The response of semi-active controlled structures is predicted by using the reduced Fokker–Planck–Kolmogorov equation associated with fully averaged Itô equations of the controlled structures. A comparison with clipped linear quadratic Gaussian (LQG) control strategy, for an example, shows that the proposed semi-active control strategy for MR/ER dampers is superior to clipped LQG control strategy.     

Stochastic response analysis of nonlinear systems under Gaussian inputs

In this paper the extension of Itô's rule for the case of vector real valued functions of the response of nonlinear systems excited by zero-mean Gaussian white noise processes is presented. A suitable particularization of the vector function, in order to obtain the statistical moments of every order to the response, is treated, obtaining the differential equations of the response moments in an elegant and compact form. Polynomial expansion and closure schemes are framed in the context outlined here in order to obtain an effective procedure from a computational point of view. An application to a trigonometric nonlinear system, solved in the literature by the stochastic averaging method, is treated here by the moment equation approach using the polynomial expansion of the nonlinear terms in order to evidence the validity of this approach.     

Importance sampling for reliability assessment of dynamic systems under general random process excitation

This paper develops a reliability assessment method for dynamic systems subjected to a general random process excitation. Safety assessment using direct Monte Carlo simulation is computationally expensive, particularly when estimating low probabilities of failure. The Girsanov transformation-based reliability assessment method is a computationally efficient approach intended for dynamic systems driven by Gaussian white noise, and this approach can be extended to random process inputs that can be represented as transformations of Gaussian white noise. In practice, dynamic systems may be subjected to inputs that may be better modeled as non-Gaussian and/or non-stationary random processes, which are not easily transformable to Gaussian white noise. We propose a computationally efficient scheme, based on importance sampling, which can be implemented directly on a general class of random processes — both Gaussian and non-Gaussian, and stationary and non-stationary. We demonstrate that this approach is in fact equivalent to Girsanov transformation when the uncertain inputs are Gaussian white noise processes. The proposed approach is demonstrated on a linear dynamic system driven by Gaussian white noise and Brownian bridge processes, a multi-physics aero-thermo-elastic model of a flexible panel subjected to hypersonic flow, and a nonlinear building frame subjected to non-stationary non-Gaussian random process excitation.     

A physical approach to structural stochastic optimal controls

The generalized density evolution equation proposed in recent years profoundly reveals the intrinsic connection between deterministic systems and stochastic systems by introducing physical relationships into stochastic systems. On this basis, a physical stochastic optimal control scheme of structures is developed in this paper, which extends the classical stochastic optimal control methods, and can govern the evolution details of system performance, while the classical stochastic optimal control schemes, such as the LQG control, essentially hold the system statistics since there is still a lack of efficient methods to solve the response process of the stochastic systems with strong nonlinearities in the context of classical random mechanics. It is practically useful to general nonlinear systems driven by non-stationary and non-Gaussian stochastic processes. The celebrated Pontryagin’s maximum principles is employed to conduct the physical solutions of the state vector and the control force vector of stochastic optimal controls of closed-loop systems by synthesizing deterministic optimal control solutions of a collection of representative excitation driven systems using the generalized density evolution equation. Further, the selection strategy of weighting matrices of stochastic optimal controls is discussed to construct optimal control policies based on a control criterion of system second-order statistics assessment. The stochastic optimal control of an active tension control system is investigated, subjected to the random ground motion represented by a physical stochastic earthquake model. The investigation reveals that the structural seismic performance is significantly improved when the optimal control strategy is applied. A comparative study, meanwhile, between the advocated method and the LQG control is carried out, indicating that the LQG control using nominal Gaussian white noise as the external excitation cannot be used to design a reasonable control system for civil engineering structures, while the advocated method can reach the desirable objective performance. The optimal control strategy is then further employed in the investigation of the stochastic optimal control of an eight-storey shear frame. Numerical examples elucidate the validity and applicability of the developed physical stochastic optimal control methodology.     

Higher order statistics of the response of linear systems excited by polynomials of filtered Poisson pulses

The higher order statistics of the response of linear systems excited by polynomials of filtered Poisson pulses are evaluated by means of knowledge of the first order statistics and without any further integration. This is made possible by a coordinate transformation which replaces the original system by a quasi-linear one with parametric Poisson delta-correlated input; and, for these systems, a simple relationship between first order and higher order statistics is found in which the transition matrix of the dynamical new system, incremented by the correction terms necessary to apply the Itô calculus, appears.     

Nonlinear systems driven by polynomials of filtered Poisson processes

The perturbation method is applied to determine approximately the mean, variance, skewness and kurtosis of the transient and stationary response of nonlinear systems driven by polynomials of filtered Poisson processes. The analysis is based on the classical perturbation method, the Itô differentiation formula, and properties of the response of linear systems subjected to polynomials of filtered Poisson processes. Two examples are presented to demostrate the efficiency and accuracy of this approximate analysis.     

Linear systems driven by martingale noise

A method is developed for calculating second moment properties and moments of order three and higher of the state X of a linear filter driven by martingale noise. The martingale noise is interpreted as the formal derivative of a square integrable martingale with continuous samples. The Gaussian white noise is an example of a martingale noise. It is shown that the differential equations of the mean and correlation functions of the state X developed in the paper resemble the corresponding equations of the classical linear random vibration and coincide with these equations if the input is a Gaussian white noise. The moment equations are derived by (1) the Itô formula for semimartingales and (2) the classical Itô formula applied to a diffusion process whose coordinates include X. An advantage of the second method is use of more familiar concepts. However, this method requires to calculate unnecessary moments and can be applied only for a class of martingale noise processes. Examples are presented to illustrate and evaluate the two methods for calculating moments of X and demonstrate the use of these methods in linear random vibration.     

Identification of a non-linear spring through the Fokker–Planck equation

Standard methods for the dynamic identification of civil engineering structures and mechanical systems generally rely on the determination of eigenfrequencies/eigenvectors and damping. A basic assumption for the use of these methods is the linear elastic behaviour of the tested structures. Nevertheless many structures exhibit a non-linear behaviour even at low levels of external excitation. For instance, reinforced concrete structures cracked by shrinkage or by the overcoming of the concrete tensile strength, exhibit different values of the flexural stiffness depending on the opening of the cracks. In the present paper a new methodology is presented to identify the characteristics of a non-linear mechanical system using the Fokker–Plank Equation (FPE) that allows evaluating the probability density function of the response of structural systems loaded with Gaussian white noise.     

Efficient solution of the first passage problem by Path Integration for normal and Poissonian white noise

In this paper the first passage problem is examined for linear and nonlinear systems driven by Poissonian and normal white noise input. The problem is handled step-by-step accounting for the Markov properties of the response process and then by Chapman–Kolmogorov equation. The final formulation consists just of a sequence of matrix–vector multiplications giving the reliability density function at any time instant. Comparison with Monte Carlo simulation reveals the excellent accuracy of the proposed method.     

Probabilistic response of nonlinear systems under combined normal and Poisson white noise via path integral method

In this paper the response in terms of probability density function of nonlinear systems under combined normal and Poisson white noise is considered. The problem is handled via a Path Integral Solution (PIS) that may be considered as a step-by-step solution technique in terms of probability density function. A nonlinear system under normal white noise, Poissonian white noise and under the superposition of normal and Poisson white noise is performed through PIS. The spectral counterpart of the PIS, ruling the evolution of the characteristic functions is also derived. It is shown that at the limit when the time step becomes an infinitesimal quantity an equation ruling the evolution of the probability density function of the response process of the nonlinear system in the presence of both normal and Poisson White Noise is provided.     

Higher order statistics of the response of MDOF linear systems under polynomials of filtered normal white noises

This paper exploits the work presented in the companion paper in order to evaluate the higher order statistics of the response of linear systems excited by polynomials of filtered normal processes. In fact, by means of a variable transformation, the original system is replaced by a linear one excited by external and linearly parametric white noise excitations. The transition matrix of the new enlarged system is obtained simply once the transition matrices of the original system and of the filter are evaluated. The method is then applied in order to evaluate the higher order statistics of the approximate response of nonlinear systems to which the pseudo-force method is applied.     

On the dynamic stochastic response of FE models

A numerical procedure to compute the mean and covariance matrix of the response of nonlinear structures modeled by large FE models is presented. Non-white, non-zero mean, non-stationary Gaussian distributed excitation is represented by the well known Karhunen–Loéve expansion, which allows to describe any type of non-white Gaussian excitation in contrast to filtered white noise which might not be easily adjusted to available statistical data. The solution procedure differs considerably from standard methodologies using a state space representation. In the proposed approach, step-by-step integration procedures developed for deterministic FE analysis are applied to compute the first two moments of the stochastic response.     

New partial differential equations governing the joint, response–excitation, probability distributions of nonlinear systems, under general stochastic excitation

In the present work the problem of determining the probabilistic structure of the dynamical response of nonlinear systems subjected to general, external, stochastic excitation is considered. The starting point of our approach is a Hopf-type equation, governing the evolution of the joint, response–excitation, characteristic functional. Exploiting this equation, we derive new linear partial differential equations governing the joint, response–excitation, characteristic (or probability density) function, which can be considered as an extension of the well-known Fokker–Planck–Kolmogorov equation to the case of a general, correlated excitation and, thus, non-Markovian response character. These new equations are supplemented by initial conditions and a marginal compatibility condition (with respect to the known probability distribution of the excitation), which is of non-local character. The validity of this new equation is also checked by showing its equivalence with the infinite system of moment equations. The method is applicable to any differential system, in state-space form, exhibiting polynomial nonlinearities. In this paper the method is illustrated through a detailed analysis of a simple, first-order, scalar equation, with a cubic nonlinearity. It is also shown that various versions of Fokker–Planck–Kolmogorov equation, corresponding to the case of independent-increment excitations, can be derived by using the same approach.

A numerical method for the solution of these new equations is introduced and illustrated through its application to the simple model problem. It is based on the representation of the joint probability density (or characteristic) function by means of a convex superposition of kernel functions, which permits us to satisfy a priori the non-local marginal compatibility condition. On the basis of this representation, the partial differential equation is eventually transformed to a system of ordinary differential equations for the kernel parameters. Extension to general, multidimensional, dynamical systems exhibiting any polynomial nonlinearity will be presented in a forthcoming paper.     

The Itô and Stratonovich integrals for stochastic differential equations with poisson white noise

The relationship between the Itô and the Stratonovich integrals used for solving stochastic differential equations with Gaussian white noise is well known. However, this relationship seems to be less clear when dealing with stochastic differential equations driven by Poisson white noise. It is shown that there is no difference between the Itô and the Stratonovich integrals used to define the solution of stochastic differential equations with Poisson white noise. This result is in disagreement with findings of some previous publications but in agreement with the classical definition of the Itô and Stratonovich integrals. Intuitive considerations, arguments based on the theory of stochastic integrals with semimartingales, and examples are used to prove and demonstrate the claimed equality of the Itô and Stratonovich integrals.