Results obtained by applying the method of stochastic averaging to random vibration problems are discussed. This method is applicable to a variety of problems involving the response of lightly damped systems to broad-band random excitations. Solutions pertaining to both linear and non-linear vibrations are reviewed, and it is shown that the technique enables, in the case of parametric excitation, stability criteria to be established. Some results which have been obtained relating to the first-passage reliability problems are also surveyed. Various applications of the theory to engineering problems are outlined.

Stochastic averaging: An approximate method of solving random vibration problems

In this paper, the generalized Lyapunov equation approach is used to study stochastic stabilization/detectability with state-multiplicative noise. Some practical test criteria for stochastic stabilization and detectability, such as stochastic Popov-Belevitch-Hautus criterion for exact detectability, are obtained. Moreover, useful properties of the generalized Lyapunov equation are derived based on critical stability and exact detectability introduced in this paper. As applications, first, the stochastic linear quadratic regulator as well as the related generalized algebraic Riccati equation are discussed extensively. Second, the infinite horizon stochastic H 2/H infin control with state- and control-dependent noise is also investigated, which extends and improves the recently published results.

Generalized Lyapunov Equation Approach to State-Dependent Stochastic Stabilization/Detectability Criterion

Stochastic bifurcation

An algorithm is derived for solving a large class of Ito random integral equations. The derivation of the algorithm involves approximate discretization of the given Ito equation. The Ito integrals arising out of discretization are expressed as functions of normal random variables. The algorithm gives a sample pathwise solution and is readily implementable on a digital computer.

Numerical Solution of Ito Integral Equations

First-passage probabilities for randomly excited systems: Diffusion methods

The paper is an extension of a previous one (Ariaratnam and Graefe 1965), and presents the analysis of general linear systems governed by n first-Order state equations whose coefficients are subjected to either of two forms of random variation, namely Gaussian white or incremental Brownian type of process. The differences between the two cases are pointed out. The Fokker-Planck equation is set up for both cases and the equations governing the response moments are obtained. An example is given which incorporates both types of random variation of the coefficients.

Linear Systems with Stochastic Coefficients. II

The first-passage time problem for envelope crossing for a linear oscillator subject to stationary wide-band random excitation is investigated. The amplitude of the response is approximated by a one-dimensional Markov process, and differential equations governing the moments of the first-passage time are derived. Analytical solutions are obtained for the mean and the variance

On the first-passage time for envelope crossing for a linear oscillator†

This paper contains three parts. In (a), the Fokker-Planck equation for a linear stochastic central system with additive disturbances is solved. The performance of the optimal system possesses a term for cost duo to the random disturbances at the input. In (b), the optimal control law for a linear system subject to stochastic fluctuations in its parameters and additive disturbances at the input is derived. Stability conditions of the optimal system are considered. In (c), the optimal control of a linear system subject to parametric fluctuations of the truncated white noise type is considered. The corresponding modified Fokker-Planck equation is given, from which the stability conditions of the optimal system are obtained. A second-order example is given to illustrate the methods.

S. T. Ariaratnam

Papers

Linear Systems with Stochastic Coefficients. II

On the first-passage time for envelope crossing for a linear oscillator†

Optimal Control of Linear Stochastic Systems