Probability Of Nonlinear Stochastic Systems Research Articles

On the Partial Stability in Probability of Nonlinear Stochastic Systems

A general class of the nonlinear time-varying systems of Ito stochastic differential equations is considered. Two problems on the partial stability in probability are studied as follows: 1) the stability with respect to a given part of the variables of the trivial equilibrium; 2) the stability with respect to a given part of the variables of the partial equilibrium. The stochastic Lyapunov functions-based conditions of the partial stability in probability are established. In addition to the main Lyapunov function, an auxiliary (generally speaking, vector-valued) function is introduced for correcting the domain in which the main Lyapunov function is constructed. A comparison with the well-known results on the partial stability of the systems of stochastic differential equations is given. An example that well illustrates the peculiarities of the suggested approach is described. Also a possible unified approach to analyze the partial stability of the time-invariant and time-varying systems of stochastic differential equations is discussed.

Global asymptotic stabilisation in probability of nonlinear stochastic systems via passivity

ABSTRACTThe purpose of this paper is to develop a systematic method for global asymptotic stabilisation in probability of nonlinear control stochastic systems with stable in probability unforced dynamics. The method is based on the theory of passivity for nonaffine stochastic differential systems combined with the technique of Lyapunov asymptotic stability in probability for stochastic differential equations. In particular, we prove that a nonlinear stochastic differential system whose unforced dynamics are Lyapunov stable in probability is globally asymptotically stabilisable in probability provided some rank conditions involving the affine part of the system coefficients are satisfied. In this framework, we show that a stabilising smooth state feedback law can be designed explicitly. A dynamic output feedback compensator for a class of nonaffine stochastic systems is constructed as an application of our analysis.

Stabilization in probability of nonlinear stochastic systems with guaranteed region of attraction and target set

We deal with nonlinear dynamical systems, consisting of a linear nominal part perturbed by model uncertainties, nonlinearities and both additive and multiplicative random noise, modeled as a Wiener process. In particular, we study the problem of finding suitable measurement feedback control laws such that the resulting closed-loop system is stable in some probabilistic sense. To this aim, we introduce a new notion of stabilization in probability, which is the natural counterpart of the classical concept of regional stabilization for deterministic nonlinear dynamical systems and stands as an intermediate notion between local and global stabilization in probability. This notion requires that, given a target set, a trajectory, starting from some compact region of the state space containing the target, remains forever inside some larger compact set, eventually enters any given neighborhood of the target in finite time and remains thereinafter, all these events being guaranteed with some probability. We give a Lyapunov-based sufficient condition for achieving stability in probability and a separation result which splits the control design into a state feedback problem and a filtering problem. Finally, we point out constructive procedures for solving the state feedback and filtering problem with arbitrarily large region of attraction and arbitrarily small target for a wide class of nonlinear systems, which at least include feedback linearizable systems. The generality of the result is promising for applications to other classes of stochastic nonlinear systems. In the deterministic case, our results recover classical stabilization results for nonlinear systems.

Stabilization in Probability of Nonlinear Stochastic Systems with Guaranteed Cost

We deal with nonlinear dynamical systems, consisting of a linear nominal part plus model uncertainties, nonlinearities, and both additive and multiplicative random noise, modeled as a Wiener process. In particular, we study the problem of finding suitable measurement feedback control laws such that the resulting closed-loop system is stable in some probabilistic sense and a given cost functional is minimized. We give a Lyapunov-based separation result which splits the control design into a state feedback problem and a filtering problem. Finally, we point out constructive algorithms for solving the state feedback and filtering problems with arbitrarily large region of attraction for a wide class of nonlinear systems, which at least include feedback linearizable systems.

Stability in probability of nonlinear stochastic systems with delay

Stability in probability of nonlinear stochastic systems with delay

First-passage time probability of non-linear stochastic systems by generalized cell mapping method

In this paper, the first-passage time probability of linear and non-linear dynamical systems subjected to white noise excitation is studied by the generalized cell mapping method. A wide range of the damping and non-linear parameters is examined in the study. The results of the first-passage time probability for the systems with large damping and non-linear parameters are difficult to obtain by other methods and they do not seem to have been previously available. The results obtained by the generalized cell mapping method are found to be in good agreement with those obtained by direct simulation.