Conditions For Mean Square Stability Research Articles

Simple stability criterion for non-linear systems with multiplicative noise

A new sufficient condition for mean square stability of stochastic systems with sector-bounded non-linearity is presented. Verification of modern stability criteria for systems under consideration is a complicated numerical problem. As a rule, in the process of verification of most of these criteria, we repeatedly have to solve non-linear matrix equations. We have used estimates of the solution of this equation and non-traditional matrix calculus. As a result, we obtained a simple, easy-to-check mean-square stability criterion.

Stochastically Excited Linear Nonconservative Systems∗

ABSTRACT Dynamic stability of linear nonconservative systems under stochastic parametric excitation of small intensity is examined. It is assumed that a finite number of modes of a general, multi-degree-of-freedom system undergo flutter instability (nonresonance case). A modified stochastic averaging method is employed to obtain the contribution from the stochastic components of the stable modes to that of the critical modes. The approximate lto equations for the critical modes are then examined. Conditions for mean square stability of dynamic response are obtained. Results are shown to depend only on those values of the excitation spectral density near twice the natural frequencies, the difference and combination frequencies of the system. The results are applied to the problem of a cantilever column subjected to stochastic follower force.

Functional-perturbational analysis of non-linear systems under random external and parametric excitations

A functional-perturbational type approach based on the Volterra-Wiener functional expansion is developed to analyze the non-linear response of dynamic systems to random excitations. The approach is applicable for non-linear systems under parametric excitation provided that the system is also subjected to external excitation. An example of a cantilever beam subjected to axial and lateral excitations is considered. For the linear case the response is determined up to second order approximation. The mean square response is compared with the results of the method of moments. It is found that the response predicted by the functional approach represents an asymptotic approximation to the moment method solution. The condition of mean square stability does not emerge directly from the functional solution. However, one can establish the stability boundary of the system if the expansion is expressed by a closed form analytical function. The non-linear mean square response is determined and also compared with the Gaussian closure scheme solution. The functional solution gives real positive mean square response for all external excitation levels while the Gaussian closure scheme predicts positive mean square response only for a relatively lower spectral density level of external excitation.

Stability and stabilizability of stochastic evolution equations on Hilbert spaces

The stochastic evolution equations in Hilbert space with state- and control-dependent noise are dealt with. For linear stochastic evolution equations with state-dependent noise, the necessary and sufficient conditions for asymptotic mean-square stability are discussed. For linear and Lure-type non-linear stochastic evolution equations with state- and control-dependent noise, the conditions are derived for which there exists a feedback control law such that the closed-loop system is asymptotically stable in a mean-square sense. Attention is also paid to the case in which there is only state-dependent noise or only control-dependent noise and to the case in which the noise intensity is arbitrarily large.

Stability for Stochastic Infinitedimensional Bilinear Systems

Infinite-dimensional discrete-time bilinear models driven by Hilbert space-valued random sequences can be defined as the uniform limit of finite-dimensional bilinear models. In this paper we shall investigate the limiting properties of the state expectation and correlation sequences for infinite-dimensional stochastic discrete bilinear models, where sufficient conditions for mean square stability will be presented

Modal dynamics and stabilizer design for galloping transmission lines

A stochastic model for wind-induced galloping oscillations in an overhead transmission line is presented as a hyperbolic partial differential equation. Energy input to the oscillating system is known to depend on the relative wind velocity and conductor ice accumulation; it varies randomly in space and time and is represented by a distributed Gaussian process. The problem of stabilization of multiple modes of vibration is examined. A finite dimensional approximation of the proposed model covering a desired number of modes of pitching motion is obtained. Sufficient conditions for the exponential mean square stabilization of the oscillations are established based on stochastic calculus. A design procedure for the optimal selection of dampers providing both magnitude and location is presented. This includes both point as well as distributed dampers which cover the current industrial usage. A numerical example illustrates the theoretical results and the design procedure.

Stochastically Perturbed Linear Gyroscopic Systems

ABSTRACT Dynamic stability of linear conservative gyroscopic systems under stochastic parametric excitations of small intensity is examined. Conditions for mean square stability of dynamic response are obtained. Results are shown to depend only on those values of the excitation spectral density near twice the natural frequencies and the combination frequencies of the system. These results are applied to the problem of flow induced vibration in a supported pipe conveying fluid with pulsating velocity. The effects of mean flow velocity and virtual mass on the extent of parametric instability regions are then discussed.

Stochastic modal interaction in linear and nonlinear aeroelastic structures

The linear and autoparametric modal interactions in a three defree-of-freedom structure under wide band random excitation are examined. For a structure with constant parameters the linear response is obtained in a closed form. When the structure stiffness matrix involves random fluctuations, the governing equations of motion, in terms of the normal coordinates, are found to be coupled through parametric terms. The structural response is mainly governed by the condition of mean square stability. The boundary of stable-unstable responses is obtained as a function of the internal detuning parameter. The results of the linear system with constant parameters are used as a reference to measure the deviation of the system response when the nonlinear inertia coupling is included. In the neighbourhood of combination internal resonance the system random response is determined by using the Fokker Planck equation approach together with the Gaussian closure scheme. This approach results in 27 coupled first order differential equations in the first and second response moments. These equations are solved numerically. The response is found to deviate significantly from the linear solution when the system internal detuning is close to the exact internal resonance. The autoparametric interaction is found to depend significantly on the system damping ratios and a nonlinear coupling parameter. In the vicinity of combination internal resonance, the second normal mode mean square exhibits an increase associated with a corresponding decrease in the first and third normal modes.

Mean square stability conditions for discrete stochastic bilinear systems

Necessary and sufficient conditions for mean square stability are proved for the following class of nonlinear dynamical systems: finite-dimensional bilinear models, evolving in discrete-time, and driven by random sequences. The stochastic environment under consideration is characterized only by independence, wide sense stationarity, and second-order properties. Thus, we do not assume random sequences to be Gaussian, zero-mean, or ergodic. The probability distributions involved are allowed to be arbitrary and unknown. Limiting state moments are given in terms of the model parameters and disturbances moments.

Algebraic criterion for the stochastic stability of linear systems with the parametric action of correlated white noise: PMM vol. 36, n≗3, 1972, pp. 546–551

We give the necessary and sufficient conditions for the exponential mean square stability of linear systems with constant coefficients subjected to the action of correlated white noise. These conditions are expressed in terms of the transfer functions. We present example. Papers [1– 9] were devoted to the stability problem for stochastic systems. Exponential mean square stability of linear systems with white noise was examined in detail in [1,4,6–9]. The stability criterion proposed in [3, 4] requires the computations of determinants of upto order ν (ν + 1) 2 , where ν is the system's order. A criterion was established in [9] for a special class of systems, requiring a knowledge only of the system's transfer matrix from noise to outputs, moreover, taking count in the systems of the number of perturbing noise sometimes makes it possible to avoid the laborious calculations of higher-order determinants. In this paper the investigative method in [9] is extended to a wider class of linear systems which are under the parametric action of dependent noise. In many cases the criterion proposed here permits us to restrict ourselves to the computation of determinants of orders less than both the number of noise perturbing the system as well as the quantity ν (ν + 1) 2 . Just as in [9] the criterion is applicable to systems given by a transfer matrix from noise to outputs. Everywhere below, by the stability of a system with noise we mean the exponential mean square stability.

Stochastic Differential Equations with Applications to Random Harmonic Oscillators and Wave Propagation in Random Media

The two-time method is used to obtain an expansion, valid for $\varepsilon $ small and t large, of the vector solution $u( {t,\varepsilon } )$ of an abstract ordinary differential equation involving $\varepsilon $. The same method is used to get expansions of functions of u. The results are shown to apply to the solutions of stochastic equations. They are used to find the first two moments and the transition probability of the displacement of a harmonic oscillator with spring constant a random function of t. The result contains the condition for mean square stability due to Stratonovich. The results are also applied to one-dimensional wave propagation through a layer with refractive index a random function of position. They are used to find the mean square amplitude reflection and transmission coefficients, which are just the mean power reflection and transmission coefficients. A graph of the mean square transmission coefficient as a function of layer thickness is presented. The results are also compared with those obtainable by other methods.