Stochastic Mean Square Research Articles

Observer‐based sampled‐data control for networked systems with consecutive packet dropouts

AbstractThis paper investigates the observer‐based sampled‐data control problem for networked systems with consecutive packet dropouts. A new consecutive packet dropout model is established to describe the phenomenon of packet dropout in both the sensor‐controller (S‐C) and controller‐actuator (C‐A) channels. First, the considered system with observer‐based control scheme is studied in the discrete‐time domain by using equivalent discretization transformation. In order to simplify subsequent analysis, matrix exponential computation is employed to transform the discrete‐time system into a more tractable form. Then, the proposed new consecutive packet dropout model is taken into consideration to establish the stochastic mean square exponential stability condition. Furthermore, a novel two‐step synthesis approach is used to design the observer‐based controller. Finally, the effectiveness of the proposed observer‐based sampled‐data control approach is verified using a numerical example. In addition, the advantages of the proposed approach are validated through comparisons.

Nonstationary quantized control for discrete-time Markov jump singularly perturbed systems against deception attacks

This paper addresses the nonstationary quantized control problem for the discrete-time Markov jump singularly perturbed systems (MJSPSs) subject to deception attacks (DAs). The control inputs are characterized by randomly occurring DAs and nonstationary quantization simultaneously, where the DAs are depicted by means of a Bernoulli distributed sequence. By applying a multi-layer structure methodology (MLSM), the nonstationary controllers are devised for MJSPSs. Meanwhile, the correlation among system mode, controller mode, and quantizer mode are portrayed via the nonstationary Markov process. Based on a mode-dependent Lyapunov functional, sufficient criteria are established such that the resulting closed-loop system (CLS) is stochastic mean square exponential ultimately bounded (SMSEUB), and the desired controller is designed. Ultimately, two simulation examples are offered to elaborate on the validity and superiority of the proposed theoretical results.

The TVICMs method for weak signal detection based on a nonlinear stochastic delay differential system

A nonlinear stochastic delay differential system with symmetry SD oscillator is investigated to detect weak signal. Firstly, the stochastic Melnikov function and the chaos threshold are obtained by using the stochastic mean square criterion. Secondly, the stochastic center manifold and the stochastic averaging method are used to obtain the one-dimensional Itoˆ equation and the corresponding FPK equation. The stationary probability density of the FPK equation is studied and the corresponding nonlinear stochastic dynamical behaviors are analyzed. Finally, the highlight of this paper is to detect the weak signal with ‘transient vacancy in chaotic motions’ (TVICMs) method successfully and extract the features information such as frequency and amplitude on the premise of unknown features information. Astonishingly, numerical simulation shows that the time delay can enhance the detection capability of the system and the effectiveness of the new method.

Efficient calculation of the response statistics of two-dimensional fractional diffusive systems

Various natural phenomena are described by anomalous diffusion processes. Notable examples relate to the study of diffusion of tracers in particles in turbulent flows, or in the propagation of acoustic waves. In this context, the governing equations involve a fractional Laplacian operator, which replaces the classical Laplacian, and may involve nonlinear terms. This leads to problems described by fractional nonlinear diffusion equations. In general, no analytical solutions are available for determining the response of these systems. Thus, the development of approximate approaches circumventing the use of computationally demanding numerical techniques is desirable. This paper proposes a statistical linearization based approach, which allows calculating approximately, albeit iteratively, the response statistics. The method is developed using a recently proposed representation of the fractional Laplacian in conjunction with a mode expansion of the system response. It is implemented by introducing non-orthogonal eigenfunctions of the fractional Laplacian of the response, which are obtained from the linear modes of the classical diffusion equation. Such a representation allows deriving a system of nonlinear ordinary differential equations, which is linearized in a stochastic mean square sense. Then, the response statistics and power spectral density are determined by an iterative procedure. Numerical results pertaining to a system with white noise excitation demonstrate the efficiency of the proposed approximate approach. Further, comparisons with data from relevant Monte Carlo results assess the reliability of the estimated response.

Stochastic mean square exponential synchronisation of time-varying complex dynamical networks via pinning control

ABSTRACTThis paper investigates the problem of stochastic mean square exponential synchronisation of complex dynamical networks with time-varying delay via pinning control. By applying the Lyapunov method and stochastic analysis, criteria on mean square exponential synchronisation are established under linear feedback pinning control and adaptive feedback pinning control, which depend on the time-varying delay and stochastic perturbation. These results complement and improve the previously known results. Two numerical examples are given to illustrate the effectiveness and correctness of the derived theoretical results.

Consensus stabilization of stochastic multi-agent system with Markovian switching topologies and stochastic communication noise

In this paper, we study stochastic consensus stabilization problems for a stochastic high-order multi-agent system with Markovian switching topologies and stochastic communication noise. By using the generalized Ito^ formula for the Markovian switching multi-agent system with stochastic communication noise, a state-feedback controller is constructed to ensure that the stochastic high-order multi-agent system reaches consensus in mean square sense when each agent's dynamics has unstable open-loop poles. A necessary and sufficient condition for the stochastic mean square consensus stabilization of the stochastic multi-agent system subject to Markovian switching topologies and additive disturbance is established under a distributed control protocol, i.e., the digraph is balanced and the union of the communication topology set contains a spanning tree. Numerical simulation is presented to demonstrate the theoretical analysis.

Nonlinear Random Vibrations of Beams with Fractional Derivative Elements

AbstractThis paper deals with nonlinear random vibrations of a beam comprising a fractional derivative element; the nonlinear term arises from the assumption of moderately large beam displacements. It is shown that the beam response can be determined reliably via an optimal statistical linearization procedure. Specifically, the solution is obtained by utilizing an appropriate iterative representation of the stochastic response spectrum, which involves the linear modes of vibration of the beam. Such a representation allows retaining the nonlinearity in the time-dependent part of the response, which, in turn, is linearized in a stochastic mean square sense. The reliability of the proposed approximate solution is assessed in relation to the results of relevant Monte Carlo simulations. In this regard, a boundary integral method (BIM)–based algorithm is employed, in conjunction with a Newmark integration scheme, for estimating the beam response from spectrum-compatible realizations of the excitation, while acc...

RobustH∞Control for Linear Stochastic Partial Differential Systems with Time Delay

This paper investigates the problems of robust stochastic mean square exponential stabilization and robustH∞for stochastic partial differential time delay systems. Sufficient conditions for the existence of state feedback controllers are proposed, which ensure mean square exponential stability of the resulting closed-loop system and reduce the effect of the disturbance input on the controlled output to a prescribed level ofH∞performance. A linear matrix inequality approach is employed to design the desired state feedback controllers. An illustrative example is provided to show the usefulness of the proposed technique.

Mean square BIBO stability of discrete-time stochastic control systems with delays and nonlinear perturbations

The problem of mean square bounded-input bounded-output(BIBO) stability is investigated for a class of discrete-time stochastic control systems with time delays and non-linear perturbations. In this paper, a special point δ in the time delay's variation interval is introduced, and the variation interval is divided into two subintervals. Then, by defining a special Lyapunov-Krasovskii functional and checking its variation in the two subintervals, respectively, some novel delay-dependent stability criteria for the discrete-time stochastic control systems are derived. These conditions are expressed in the forms of linear matrix inequalities(LMIs), whose feasibility can be easily checked by using Matlab LMI Toolbox. Meanwhile, this paper provides a new method for studying discrete-time stochastic mean square BIBO stability. Finally, a numerical example is given to illustrate the validity of the main results.

A Lax equivalence theorem for stochastic differential equations

In this paper, a stochastic mean square version of Lax’s equivalence theorem for Hilbert space valued stochastic differential equations with additive and multiplicative noise is proved. Definitions for consistency, stability, and convergence in mean square of an approximation of a stochastic differential equation are given and it is shown that these notions imply similar results as those known for approximations of deterministic partial differential equations. Examples show that the assumptions made are met by standard approximations.

Input-Output Stability Criteria for a Broad Class of Stochastic Nonlinear Distributed Systems Defined by Green’s Function

This paper deals with the input-output stability of a class of nonlinear distributed systems defined by their Laplace-Green’s functions, or similarly from a practical point of view, by their distributed transfer functions. The time dependent nonlinear feedback element is distributed and bounded by two limiting gains which depend explicitly upon the distributed parameter. These systems are disturbed by a state and space dependent Gaussian noise which is added to the input of their linear components. This noise depends explicitly upon the output of the system via its own nonlinear feedback gain. Some input-output stability criteria are stated, which can be considered as being stochastic distributed versions of the circle criterion available for deterministic lumped parameter systems. They involve the stochastic mean square norm and they are expressed in term of the relative positions, in the complex plane, of a circle which depends upon the nonlinearities and the variance of the noise on the one hand; and a locus which may be interpreted as being the Nyquist locus of the linear part on the other hand.