Mean Square Exponential Stability Research Articles

Stability for multi-linked stochastic delayed complex networks with stochastic hybrid impulses by Dupire Itô’s formula

In this paper, the mean-square exponential stability is investigated for multi-linked stochastic delayed complex networks with stochastic hybrid impulses. Distinct from the existing literature, we study the MSDCNs on the basis of the multi-linked stochastic functional differential equations that consider the impact of a certain past interval on the present. Moreover, the stochastic hybrid impulses we discuss possess stochastic impulsive moments and impulsive gain, which make the impulses fit better to the real-world demands for control. Also, a novel concept of average stochastic impulsive gain is proposed to measure the intensity of the stochastic hybrid impulses. By the use of Dupire Itô’s formula, based on Lyapunov method, graph theory and stochastic analysis techniques, two sufficient criteria for the mean-square exponential stability are derived, which are closely related to average stochastic impulsive gain, stochastic disturbance strength as well as the topological structure of the network itself. Finally, an application about neural networks is discussed and corresponding numerical example is presented to demonstrate the feasibility and effectiveness of the theoretical results.

Co-design of transition rates and sliding mode switched controller for Markovian jumping systems under intermittent DoS attacks

This paper focuses on the stabilization problem for a class of Markovian jumping systems (MJSs) subject to intermittent denial-of-service (IDoS) attacks by synthesizing the sliding mode control (SMC) and the transition rate matrix (TRM). The existing conditions for the transition rates are firstly established to ensure the exponential mean-square stability of the unforced uncertain MJSs. And then, a co-design scheme for both the sliding mode controller and TRM is synthesized to achieve the exponential mean-square stability of the closed-loop system under IDoS, in which a switching estimator is utilized to estimate the unmeasurable system state. By introducing a novel Lyapunov function, both the reachability and the stability of sliding mode dynamics are detailedly analyzed, and an iterative optimization algorithm is given for solving the corresponding sufficient conditions. Finally, the proposed co-design SMC strategy is illustrated via the simulation examples.

LMIs-based stability analysis and fuzzy-logic controller design for networked systems with sector nonlinearities: Application in tunnel diode circuit

In this article, the problem of exponential mean-square stability analysis is discussed for uncertain networked control systems expressed by a stochastic T-S fuzzy model. In general, the characteristics of random occurrence for multipath packet dropouts often exist in the signal transmission network. For dealing with this difficult point, a dynamic output feedback strategy combining stochastic Bernoulli theory is employed. Then, delay-dependent stability conditions are derived and closed-loop system is exponentially mean-square stable by designing fuzzy-basis-dependent Lyapunov functional. Furthermore, in terms of linear matrix inequalities (LMIs) technology, sufficient conditions are gained to guarantee the prescribed H-infinity performance. Different from previous literatures, the congruence transformation method is employed to determine controller gain matrices for reducing the computation complexity of solving LMIs. Finally, the proposed method is applied in tunnel diode circuit model to verify the applicability.

Global attracting sets and exponential stability of stochastic functional differential equations driven by the time-changed Brownian motion

In this paper, we study a class of stochastic functional differential equations driven by the time-changed Brownian motion. Some sufficient conditions ensuring the existence of the global attracting sets of considered equations are presented. As applications, some new criteria for the exponential stability in mean square of the considered equations are obtained. Lastly, some examples are investigated to illustrate the theory.

Input-to-State Stability Analysis for Stochastic Mixed Time-Delayed Neural Networks with Hybrid Impulses

This paper investigates the mean-square exponential input-to-state stability (MEISS) of stochastic mixed time-delayed neural networks with hybrid impulses. A generalized comparison principle is introduced and a new inequality about the solution of an impulsive differential equation is established. Moreover, by utilizing the proposed inequality and average impulsive interval approach based on different kinds of impulsive sequences, some novel criteria on MEISS are established. When the external input is removed, several conclusions on mean-square exponential stability (MES) are also derived. Unusually, the hybrid impulses including destabilizing and stabilizing impulses have been taken into account in the presented system. Finally, two simulation examples are provided to demonstrate the validity of our theoretical results.

Prediction-based controller for linear systems with stochastic input delay

We study the stabilization problem of a linear time-invariant system with an unknown stochastic input delay. We propose to robustly compensate for the stochastic delay with a constant time horizon prediction-based controller. We prove the mean-square exponential stabilization of the closed-loop system under a sufficient condition, which requires the range of the delay values to be sufficiently narrow and the constant delay used in the prediction-based controller to be chosen in this range. Numerical simulations illustrate the relevance of this condition and the merits of our control design.

Observer-based dynamic event-triggered [formula omitted] LFC for power systems under actuator saturation and deception attack

This paper focuses on the observer-based H∞ load frequency control (LFC) of multi-area power systems with actuator saturation and deception attack. Firstly, a state observer is constructed to estimate the unmeasurable states and a time-delay model with the area control error is established. Secondly, a dynamic event triggering mechanism is developed, where dynamic parameter and a novel error-related exponential term are introduced to reduce the communication burden further in the network. Thirdly, considering time delays, data-packet losses and stochastic deception attack, a resilient LFC controller of power systems based on actuator saturation is designed and the security of the system is guaranteed with the condition of H∞ mean-square exponential stability. Sufficient conditions to co-design the event-triggered scheme, observer and desired controller gain are derived by Lyapunov functional analysis approach. Finally, simulation example is given to demonstrate the efficiency of the developed method.

Exponential stability in mean square of neutral stochastic pantograph integro-differential equations

In this paper, we show two new results on the existence and uniqueness of the solution of Neutral Stochastic Pantograph Integro-Differential Equations (NSPIDE) and the exponential stability in mean square using the one-sided Growth Condition. One example is exhibited to show the interest of our results.

New criteria for exponential stability in mean square of stochastic functional differential equations with infinite delay

<p style='text-indent:20px;'>Stochastic functional differential equations with infinite delay are considered. A novel approach to exponential stability of such equations is proposed. New criteria for the mean square exponential stability of general stochastic functional differential equations with infinite delay are presented. Illustrative examples are given.</p>

Sparse Control Synthesis for Uncertain Responsive Loads With Stochastic Stability Guarantees

Recent studies have demonstrated the potential of flexible loads in providing frequency response services. However, uncertainty and variability in various weather-related and end-use behavioral factors often affect the demand-side control performance. This work addresses this problem with the design of a demand-side control to achieve frequency response under load uncertainties. Our approach involves modeling the load uncertainties via stochastic processes that appear as both multiplicative and additive to the system states in closed-loop power system dynamics. Extending the recently developed mean square exponential stability (MSES) results for stochastic systems, we formulate multi-objective linear matrix inequality (LMI)-based optimal control synthesis problems to not only guarantee stochastic stability, but also promote sparsity, enhance closed-loop transient performance, and maximize allowable uncertainties. The fundamental trade-off between the maximum allowable ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">critical</i> ) uncertainty levels and the optimal stochastic stabilizing control efforts is established. Moreover, the sparse control synthesis problem is generalized to the realistic power systems scenario in which only partial-state measurements are available. Detailed numerical studies are carried out on IEEE 39-bus system to demonstrate the closed-loop stochastic stabilizing performance of the sparse controllers in enhancing frequency response under load uncertainties; as well as illustrate the fundamental trade-off between the allowable uncertainties and optimal control efforts.

Sign Stability of Dual Switching Linear Continuous-Time Positive Systems

This paper investigates 1-moment exponential stability and exponential mean-square stability (EMS stability) under average dwell time (ADT) and the preset deterministic switching mechanism of dual switching linear continuous-time positive systems when a numerical realization does not exist. The signs of subsystem matrices, but not their structures of magnitude, are key information that causes a qualitative concept of stability called sign stability. Both 1-moment exponential stability and EMS stability, which are the traditional stability concepts, are generalized intrinsically. Hence, both 1-moment exponential sign stability and EMS sign stability are introduced and are proven based on sign equivalency. It is shown that they are symmetrically and qualitatively stable. Notably, the notion of stability can be checked quantitatively using some examples.

Exact detectability: Application to generalized Lyapunov and Riccati equations

This paper is devoted to the application of the recently developed results about exact detectability of a large class of time-varying hybrid stochastic systems (Drăgan et al., 2020). More specifically, we will show from one side the connection between this new detectability notion and the derivation of a criterion for exponential stability in mean square for this class of systems. In the second part, we will show the central role played by exact detectability in Riccati equations arising in linear quadratic (LQ) regulator problems for such stochastic systems: it is a necessary and sufficient condition for the Riccati equation to have a unique positive semi-definite and stabilizing solution.

Correction to: Exponential mean-square stability properties of stochastic linear multistep methods

A Correction to this paper has been published: https://doi.org/10.1007/s10444-021-09901-7

Split-step balanced [formula omitted]-method for SDEs with non-globally Lipschitz continuous coefficients

In this paper, a split-step balanced θ-method (SSBT) has been presented for solving stochastic differential equations (SDEs) under non-global Lipschitz conditions, where θ∈[0,1] is a parameter of the scheme. The moment boundedness and strong convergence of the numerical solution have been studied, and the convergence rate is 0.5. Moreover, under some conditions it is proved that the SSBT scheme can preserve the exponential mean-square stability of the exact solution when θ∈(1/2,1] for every step size h>0. Numerical examples verify the theoretical findings.

Stability analysis for semi-linear stochastic differential equation with piecewise continuous arguments

In this paper, the exponential stability in mean square of the exponential Euler method for semi-linear stochastic differential equation with piecewise continuous arguments is obtained. Firstly, the exponential Euler scheme of the equation is given. Secondly, under Lipschitz condition, sufficient conditions of exponential stability to the exact solution are achieved in mean square. Furthermore, it is proved that exponential Euler method preserves exponential stability of the exact solution without any restriction on the step-size. Finally, an example is provided to illustrate our theories.

Stability analysis of theθ-method for hybrid neutral stochastic functional differential equations with jumps

• Few results on the stability of numerical methods for the neutral stochastic functional differential equations with Markovian switching and jumps, results obtained in this manuscript can enrich this area. • Illustrate the almost sure exponential stability and the mean-square exponential stability of the trivial solution. • Show the θ -method can preserve the almost sure exponential stability and the mean-square exponential stability of the trivial solution under some appropriate condition. • The almost sure exponential stability of the θ -method is obtained by the discrete semi-martingale convergence theorem without resorting to using the Borel-Cantelli lemma and the Chebyshev inequality. Few results seems to be known about the stability of numerical methods for the hybrid neutral stochastic functional differential equations with jumps (also known as the neutral stochastic functional differential equations with Markovian switching and jumps (NSFDEwMJs)). This paper mainly investigates the exponential stability (both the almost sure and the mean-square exponential stability) of the θ -method for NSFDEwMJs. Precisely, it is first illustrated that the trivial solution of the NSFDEwMJ is almost surely and mean-square exponentially stable. It is then shown that the θ -method can preserve the same conclusions of the trivial solution. Numerical examples are demonstrated to illustrate the obtained results.

Sampled-data non-fragile state estimation for delayed genetic regulatory networks under stochastically switching sampling periods

This paper is concerned with the non-fragile state estimation (NFSE) problem for a class of delayed genetic regulatory networks (GRNs) under stochastic sampling mechanisms. The measurement output is sampled before being transmitted to the estimator where the sampling period switches between two different values in a random way. By transforming the sampling periods into bounded time-delays, the issue of designing a non-fragile state estimator based on stochastically sampled measurements is converted into the NFSE problem for GRNs with multiple probabilistic interval delays. Then, by constructing a Lyapunov–Krasovskii functional and employing the Gronwall’s inequality together with Jensen’s inequality, a sufficient criterion is obtained to ensure the exponential mean-square stability of the corresponding estimation error dynamics. Furthermore, the desired non-fragile estimator gain is obtained via solving a convex optimization problem. Finally, the validity of the developed stochastic sampled-data NFSE scheme is verified via a numerical example.

On exponential stability in mean square of neutral stochastic functional differential equations

General neutral stochastic functional differential equations are considered. A novel approach to exponential stability of such equations is proposed. New criteria for the mean square exponential stability of neutral stochastic functional differential equations are given. A discussion and illustrative examples are presented.

Exponential Stability in Mean Square of Stochastic Functional Differential Equations with Infinite Delay

Exponential Stability in Mean Square of Stochastic Functional Differential Equations with Infinite Delay

Outlier-resistant observer-based [formula omitted] PID control under stochastic communication protocol

This paper addresses the H∞ proportional-integral-derivative control problem based on the outlier-resistant observer for discrete-time systems under the schedule of stochastic communication protocol (SCP). A novel observer with saturation constraint is structured in order to reduce the negative influence of abnormal measurement outputs, where the saturation limit is dynamically adjusted in the light of the observation errors. For the sake of avoiding the phenomenon of data congestion, the SCP is deployed to orchestrate the order of data transmission in the sensor-to-observer channel. In view of the Lyapunov stability theory and the orthogonal decomposition method, sufficient conditions are obtained to make sure that the exponential mean-square stability is met and the H∞ performance index is satisfied. Eventually, a simulation example is provided to confirm the effectiveness and correctness of the presented controller design algorithm.