Conditions For Mean Square Stability Research Articles

Stochastic stability and robust control for sampled-data systems with Markovian jump parameters

In this paper, the problems of stochastic stability and robust control for a class of uncertain sampled-data systems are studied. The systems consist of random jumping parameters described by finite-state semi-Markov process. Sufficient conditions for stochastic stability or exponential mean square stability of the systems are presented. The conditions for the existence of a sampled-data feedback control and a multirate sampled-data optimal control for the continuous-time uncertain Markovian jump systems are also obtained. The design procedure for robust multirate sampled-data control is formulated as linear matrix inequalities (LMIs), which can be solved efficiently by available software toolboxes. Finally, a numerical example is given to demonstrate the feasibility and effectiveness of the proposed techniques.

Exponential stability in p-th mean of solutions, and of convergent Euler-type solutions, of stochastic delay differential equations

One concept of the stability of a solution of an evolutionary equation relates to the sensitivity of the solution to perturbations in the initial data; there are other stability concepts, notably those concerned with persistent perturbations. Results are presented on the stability in p-th mean of solutions of stochastic delay differential equations with multiplicative noise, and of stochastic delay difference equations. The difference equations are of a type found in numerical analysis and we employ our results to obtain mean-square stability criteria for the solution of the Euler– Maruyama discretization of stochastic delay differential equations. The analysis proceeds as follows: We show that an inequality of Halanay type (derivable via comparison theory) can be employed to derive conditions for p-th mean stability of a solution. We then produce a discrete analogue of the Halanay-type theory, that permits us to develop a p-th mean stability analysis of analogous stochastic difference equations. The application of the theoretical results is illustrated by deriving mean-square stability conditions for solutions and numerical solutions of a constant-coefficient linear test equation.

Stochastic Stability of Mechanical Systems Under Renewal Jump Process Parametric Excitation

A dynamic system under parametric excitation in the form of a non-Erlang renewal jump process is considered. The excitation is a random train of nonoverlapping rectangular pulses with equal, deterministic heights. The time intervals between two consecutive jumps up (or down), are the sum of two independent, negative exponential distributed variables; hence, the arrival process may be termed as a generalized Erlang renewal process. The excitation process is governed by the stochastic equation driven by two independent Poisson processes, with different parameters. If the response in a single mode is investigated, the problem is governed in the state space by two stochastic equations, because the stochastic equation for the excitation process is autonomic. However, due to the parametric nature of the excitation, the nonlinear term appears at the right-hand sides of the equations. The equations become linear if the state space is augmented by the products of the original state variables and the excitation variable. Asymptotic mean and mean-square stability as well as asymptotic sample (Lyapunov) stability with probability 1 are investigated. The Lyapunov exponents have been evaluated both by the direct simulation of the stochastic equation governing the natural logarithm of the hyperspherical amplitude process and using the modification of the method wherein the time averaging of the pertinent expressions is replaced by ensemble averaging. It is found that the direct simulation is more suitable and that the asymptotic mean-square stability condition is not overly conservative.

A Unified Approach for Stochastic and Mean Square Stability of Continuous-Time Linear Systems with Markovian Jumping Parameters and Additive Disturbances

Necessary and sufficient conditions for stochastic stability (SS) and mean square stability (MSS) of continuous-time linear systems subject to Markovian jumps in the parameters and additive disturbances are established. We consider two scenarios regarding the additive disturbances: one in which the system is driven by a Wiener process, and one characterized by functions in ${L_2^m}{(\Omega,{\cal F}, {\mathbb{P}})}$, which is the usual scenario for the $H_{\infty}$ approach. The Markov process is assumed to take values in an infinite countable set $\mathcal{S}$. It is shown that SS is equivalent to the spectrum of an augmented matrix lying in the open left half plane, to the existence of a solution for a certain Lyapunov equation, and implies (is equivalent for $\mathcal{S}$ finite) asymptotic wide sense stationarity (AWSS). It is also shown that SS is equivalent to the state $x(t)$ belonging to ${L_2^n}{(\Omega,{\cal F}, {\mathbb{P}})}$ whenever the disturbances are in ${L_2^m}{(\Omega,{\cal F}, {\mathbb{P}})}$. For the case in which $\mathcal{S}$ is finite, SS and MSS are equivalent, and the Lyapunov equation can be written down in two equivalent forms with each one providing an easier-to-check sufficient condition.

The Exponential Behaviour of Nonlinear Stochastic Functional Equations of Second Order in Time

Sufficient conditions for exponential mean square stability of solutions to delayed stochastic partial differential equations of second order in time are established. As a consequence of these results, some on the pathwise exponential stability of the system are proved. The stability results derived are applied also to partial differential equations without hereditary characteristics. The results are illustrated with several examples.

Mean square stability of difference equations with a stochastic delay

The paper describes mean-square stability conditions for nonlinear delay difference equations with a stochastic delay. The first part develops a formula for the infinitesimal operator. Using this formula asymptotic mean square stability conditions are derived. A final example is provided.

Stabilization of continuous-time jump linear systems

We investigate almost-sure and moment stabilization of continuous time jump linear systems with a finite-state Markov jump form process. We first clarify the concepts of /spl delta/-moment stabilizability, exponential /spl delta/-moment stabilizability, and stochastic /spl delta/-moment stabilizability. We then present results on the relationships among these concepts. Coupled Riccati equations that provide necessary and sufficient conditions for mean-square stabilization are given in detail, and an algorithm for solving the coupled Riccati equations is proposed. Moreover, we show that individual mode controllability implies almost-sure stabilizability, which is not true for other types of stabilizability. Finally, we present some testable sufficient conditions for /spl delta/-moment stabilizability and almost-sure stabilizability.

Stochastic stability of jump linear systems

In this note, some testable conditions for mean square (i.e., second moment) stability for discrete-time jump linear systems with time-homogenous and time-inhomogenous finite state Markov chain form processes are presented.

LMI-based robust H∞ control of uncertain linear jump systems with time-delays

This paper examines the robust mean square stabilization and robust H ∞ control problem for a class of linear systems. The systems under consideration involve time delays, parameter uncertainties, and jumping parameters which may be seen as, for example, failures that occur in the systems at certain probabilities. The uncertain parameters are assumed to be time-varying and satisfy boundness conditions. A systematic approach via linear matrix inequalities (LMI) is proposed to solve the problem of concern. Sufficient conditions for robust mean square stabilization and robust H ∞ control are given in the form of LMI and the control law is computed using standard LMI techniques.

MEAN-SQUARE STABILITY OF NONLINEAR SYSTEMS WITH TIME-VARYING, RANDOM DELAY

A class of nonlinear systems with a time-varying delay is considered.The delay is modeled by a continuous-time Markov process with a finite number of states. Systems of this type may arise in real-time control applications. Employing a “delay-averaging” approach we demonstrate how certain mean-square stochastic stability conditions can be derived in terms of transition functions of the Markov process and stability properties of a system with a constant delay.

State feedback control of linear systems in the presence of devices with finite signal-to-noise ratio

This paper provides necessary and sufficient conditions for mean-square state-feedback stabilization of linear systems whose white noise sources have intensities affinely related to the variance of the signal they corrupt. Systems with such noise sources have been called FSN (finite signal-to-noise) models, and the stability results provided in prior work were only sufficient conditions. Upper bounded ℒ2 performance is also guaranteed herein by solving a control problem which is non-convex only due to a certain scaling parameter. By fixing this parameter convex programming algorithms provide controllers.

Mean-square stability of linear stochastic dynamical systems under parametric wide-band excitations

A new simplified condition is developed for determining the exponential meansquare stability margins of linear stochastic dynamical systems. It is well-known that under parametric wide-band noise disturbances, the governing equations of motion of such a system can be approximated by linear Itô stochastic differential equations (SDE). A necessary and sufficient condition for exponential mean square stability of the resulting ltô SDE is that the real parts of all the eigenvalues of the matrix describing the system of second-order moments are negative. Equivalently, the Routh-Hurwitz procedure provides conditions for stability in the form of several inequalities. In this study, it is shown that a necessary condition for the system configuration to correspond to a point on the stability boundary is that the determinant of the matrix describing the system of secondorder moments be zero. This condition is a single algebraic expression allowing for the straightforward calculation of all candidate stability boundaries. In addition, the topological properties of the stability domain are presented and shown to be useful in identifying stability boundaries and stability domains from the developed single stability boundary condition. This simplified condition provides significant advantages in the analytical and numerical estimation of the stability border and stability region of dynamical systems. The usefulness and superiority of the new condition is demonstrated by applications to example dynamical systems, including a long-span bridge model subjected to turbulent wind.

Necessary and sufficient conditions of asymptotic mean square stability for stochastic linear difference equations

Many processes in automatic regulation, physics, mechanics, biology, economy, ecology, etc. can be modelled by hereditary systems (see, e.g., [1–4]). One of the main problems for the theory of such systems and their applications is connected with stability (see, e.g., [2–4]). Many stability results were obtained by the construction of appropriate Lyapunov functionals. At present, the method is proposed which allows us, in some sense, to formalize the procedure of the corresponding Lyapunov functionals construction [5–10]. In this work, by virtue of the proposed procedure, the necessary and sufficient conditions of asymptotic mean square stability for stochastic linear difference equations are obtained.

Noise and stability in differential delay equations

We study the stability of linear stochastic differential delay equations in the presence of additive or multiplicative white and colored noise. Using a stochastic analog of the second Liapunov method, sufficient conditions for mean square and stochastic stability are derived.

Stability Results for Discrete-Time Linear Systems with Markovian Jumping Parameters

Stability results for discrete-time linear systems subject to random jumps in the parameters are obtained. First, necessary and sufficient conditions for mean square stability (MSS), including the case in which the system is driven by an independent wide-sense stationary random sequence, are derived. It is shown that MSS is equivalent to the spectral radius of an augmented matrix being less than one or to the existence of a solution of a certain Lyapunov equation. In addition it is shown that the Lyapunov equation can be derived in four equivalent forms and, as a by-product, each one gives risc to an easier-to-check sufficient condition. These results give, inter alia, a unified and rather complete picture for MSS of Markovian jump linear systems. Next we derive sufficient conditions for almost sure stability of the noiseless ease. Finally, we conclude by presenting an illustrative application to adaptive filtering where a mild condition for almost sure convergence is provided.

On the mean-square stability for a harmonic oscillator with random parameter

Sufficient mean-square stability conditions of a harmonic oscillator whose random parameter is an Ornstein-Uhlenbeck process are obtained.

Stochasic stability of viscoewtic bars

This paper is devoted to the investigation of stability of viscoelsstic bars compressed by stochastic forces at infinite time interval, The problem of the bar buckling is considered in dynamic statement. Some sufficient conditions of mean square stability of viscoelastic bars are derived for arbitrary relaxation measure and different types of the end fixing

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.