Stability Analysis Of Stochastic Systems Research Articles

Output regulation of stochastic sampled-data systems with post-processing internal model

This paper deals with the output regulation problem (ORP) of a linear time-invariant (LTI) system in the presence of sporadically sampled measurement streams with the inter-sampling intervals following a stochastic process. Under such sporadically available measurement streams, a regulator consisting of a hybrid observer, continuous-time post-processing internal model, and stabilizer are proposed, which resets with the arrival of new measurements. The resulting system exhibits a deterministic behavior except for the jumps that occur at random sampling times and therefore the overall closed-loop system can be categorized as a piecewise deterministic Markov process (PDMP). In existing works on ORPs with aperiodic sampling, the requirement of boundedness on inter-sampling intervals precludes extending the solution to the random sampling intervals with possibly unbounded support. Using the Lyapunov-like theorem for the stability analysis of stochastic systems, we offer sufficient conditions to ensure that the overall closed-loop system is mean exponentially stable (MES) and the objectives of the ORP are achieved under stochastic sampling of measurement streams. The resulting LMI conditions lead to a numerically tractable design of the hybrid regulator. Finally, with the help of an illustrative example, the effectiveness of the theoretical results are verified.

Incremental nonlinear stability analysis of stochastic systems perturbed by Lévy noise

AbstractWe present a theoretical framework for characterizing incremental stability of nonlinear stochastic systems perturbed by either compound Poisson shot noise or finite‐measure Lévy noise. For each noise type, we compare trajectories of the perturbed system with distinct noise sample paths against trajectories of the nominal, unperturbed system. We show that for a finite number of jumps arising from the noise process, the mean‐squared error between the trajectories exponentially converge toward a bounded error ball across a finite interval of time under practical boundedness assumptions. The convergence rate for shot noise systems is the same as the exponentially stable nominal system, but with a tradeoff between the parameters of the shot noise process and the size of the error ball. The convergence rate and the error ball for the Lévy noise system are shown to be nearly direct sums of the respective quantities for the shot and white noise systems separately, a result which is analogous to the Lévy–Khintchine theorem. We demonstrate both empirical and analytical computation of the error ball using several numerical examples, and illustrate how varying the parameters of the system affect the tightness of the bound.

A Note on Sufficient Conditions of Almost Sure Exponential Stability for Semi-Markovian Jump Stochastic Systems

In this paper, we are concerned with the stability analysis of stochastic systems switched by the ergodic semi-Markovian process. The sufficient conditions for almost sure exponential stability are derived by using the method of multiple Lyapunov functions and the ergodic property of the semi-Markov process. Particularly, our results generalize and improve some published results in the literature. An example is presented to illustrate the effectiveness of the obtained results.

Robust Control and Stability Analysis for Stochastic Systems with Markov Jump

Stochastic systems with Markov jump is a new type of stochastic system in recent years, which is a new field integrated by information, control and Markov process. This paper introduces the research history and the newest research trends of stochastic systems with Markov jump, and presents many widespread theoretical and application problems. Moreover, some new research topics and directions related to stochastic systems with Markov jump are proposed.

Stochastic Extremum-Seeking Algorithm for One-Dimensional and Multivariate Static Systems

The aim of this paper is to present a stochastic extremum-seeking algorithm for one-dimensional and multivariate optimization of static systems. Extremum-seeking algorithms estimate the optimum value of a function using perturbation signals. The authors propose three schemes (a basic scheme, an annealing parameter scheme, and a high-pass filter scheme) for the one-parameter problem and one scheme (a high-pass filter scheme) for the multivariate problem. These methods employ Wiener processes for the perturbation signals. In this paper, the proposed methods are shown to converge by means of a stability analysis of stochastic systems. The paper presents some numerical examples to demonstrate the effectiveness of the methods.

New delay-dependent stability criterion for stochastic systems with time delays

The problem of concern here is the stability analysis for stochastic systems with a time delay in the state. By employing a novel Lyapunov–Krasovskii functional, a new delay-dependent exponential stability condition is established for such stochastic time-delay systems. Based on the delay fractioning approach, the developed method can greatly reduce conservativeness compared with the existing results. Numerical examples are provided to show the advantage of the proposed techniques.

On the Steady State Behavior of Stochastic Systems with Application to Adaptive Control

A stability analysis of stochastic systems which may be Markovinized under feedback is presented. Two different formulations of stability are introduced along with neces-sary and sufficient conditions for stability. If the closed loop system satisfies a condition known as local stochastic controllability, it is shown that these notions of stability are equivalent.Under the local stochastic controllability assumption a variety of results are presented. It is shown that such systems exhibit very regular asymptotic behavior. For example, the convergence of averages of functions of the state process converge for every initial condition for stable lo-cally stochastically controllable systems. Furthermore, if there is exactly one invariant probability, then the probabilities governing the state process converge to a periodic orbit consisting of weighted averages of restrictions of the invariant probability to cyclical sets.The results are applied to the stability analysis of a random parameter system operating under feedback.Work partially supported by the Natural Sciences and Engineering Research Council of Canada