Conditions For Mean Square Stability Research Articles

A study on generalized balanced split drift stochastic Runge- Kutta methods for stochastic differential equations

To reduce computational complexity, the balanced numerical approximations of the general split drift stochastic Runge-Kutta methods are analyzed. The primary reasons for considering the numerical approximations of these balanced split stochastic Runge-Kutta methods are their improved stability characteristics and lower mean square error compared to other methods. By balancing the drift and diffusion components, the splitting techniques outperform the mean square error over longer time increments. For Ito multi-dimensional stochastic differential equations, we propose a novel family of balanced universal split stochastic Runge-Kutta procedures. The Kronecker product concept is utilized to derive the mean-square stability conditions. We conduct numerical tests to evaluate these methods against an existing weak order 2 split drift method. Ultimately, a specific numerical example validates the theoretical outcomes of the balanced general split stochastic Runge-Kutta procedures.

Sampled-Data H∞ Dynamic Output-Feedback Control for NCSs With Successive Packet Losses and Stochastic Sampling.

The problem of sampled-data H∞ dynamic output-feedback control for networked control systems with successive packet losses (SPLs) and stochastic sampling is investigated in this article. The aim of using sampled-data control techniques is to alleviate network congestion. SPLs that occur in the sensor-to-controller (S-C) and controller-to-actuator (C-A) channels are modeled using a packet loss model. Additionally, it is assumed that stochastic sampling follows a Bernoulli distribution. A model is established to capture the stochastic characteristics of both the SPL model and stochastic sampling. This model is crucial as it allows us to determine the probability distribution of the sampling interval between successive update instants, which is essential for stability analysis. An exponential mean-square stability condition for the constructed equivalent discrete-time stochastic system, which also guarantees the prescribed H∞ performance, is established by incorporating probability theory. The desired controller is designed using a step-by-step synthesis approach, which may offer lower design conservatism compared to some existing methods. Finally, our designed approach using a networked F-404 engine system model is validated and its merits relative to existing results are discussed. The proposed method is finally validated by employing a networked model of the F-404 engine system. Furthermore, the advantages of our method are presented in comparison to previous results.

Stability of equilibrium states for a stochastic difference equation of exponential form

It is well know that many process and problems in different fields of sciences and engineering can be modelled by linear and nonlinear difference equations. Particularly, in case of systems biology, ecology, biochemistry, genetics and physiology dynamics, many population models are governed by exponential difference equations, and lot of papers were published in this matter. The goal of this work is to study a nonlinear second order difference equation of exponential form: Xn+1 = bXn + cXne−σXn−1, n ≥ 0 where the parameters b, c have arbitrary values, σ > 0 and the initial values, X0, X−1 are arbitrary positive numbers. Our equation has an equilibrium points and is exposed to additive stochastic perturbations that are assumed a sequence of independent random variables with zero mean, unit variance and these perturbations are proportional to the deviation of the system state Xn from equilibrium points, and the result is stochastic difference equation. We cannot find the solutions of nonlinear stochastic difference equations of exponential form in all cases. So, one can study the behavior of solutions by asymptotic stability of equilibrium points. Additionally, with the general method of Lyapunov functional constructions for stochastic difference equations with discrete time, we provide the necessary and sufficient conditions for the asymptotic mean square stability of the two equilibrium points (zero and positive) within the linear approximation of the stochastic difference equation. These conditions also serve as sufficient conditions for the stability in probability of the equilibrium points of the initial nonlinear equation. Finally, to demonstrate the validity of the obtained results, some numerical examples and simulations of equations solutions are made and numerous graphical illustrations of stability trajectories of solutions are plotted.

A Decomposition Approach to Multiagent Systems With Bernoulli Packet Loss

In this paper, we extend the decomposable systems framework to multi-agent systems with Bernoulli distributed packet loss with uniform probability. The proposed sufficient analysis conditions for mean-square stability and $H_2$-performance -- which are expressed in the form of linear matrix inequalities -- scale linearly with increased network size and thus allow to analyse even very large-scale multi-agent systems. A numerical example demonstrates the potential of the approach by application to a first-order consensus problem.

Output feedback H∞ control for discrete time singularly perturbed systems with Markov lossy network: The round-robin-like protocol case

In this paper, the output feedback H∞ control is investigated for the networked singularly perturbed systems (SPSs) with Markov lossy network, in which the packet loss probability varies in different modes. A new network mode-dependent round-robin-like protocol (RRLP) is introduced to the networked SPSs, which can dynamically adjust the number of the selected sensor nodes according to the network environment condition. Based on this RRLP and Markov lossy network, a Markov mode and scheduling order co-dependent output feedback controller is constructed to ensure the system control performance. By constructing a novel Lyapunov-Krasovskii functional dependent on the singular perturbation parameter, Markov mode, and scheduling order simultaneously, sufficient conditions of mean square stability for SPSs are obtained, and numerical stiffness is avoided. Finally, a direct current motor-controlled inverted pendulum example is proposed to verify the effectiveness of our main results.

Robust multitask diffusion normalized M-estimate subband adaptive filter algorithm over adaptive networks

In recent years, the multitask diffusion least mean square (MD-LMS) algorithm has been extensively applied in distributed parameter estimation and target tracking of multitask network. However, its performance degrades under colored input signals or impulsive noises. To overcome these two drawbacks, this paper introduces the subband adaptive filter (SAF) into a multitask network for the first time, and a robust multitask diffusion normalized M-estimate subband adaptive filtering (MD-NMSAF) algorithm is proposed, which solves the global network optimization problem based on the modified Huber (MH) function in a distributed manner, achieves robustness to impulsive noise, and significantly improves the convergence performance of the MD-LMS algorithm. Compared with existing robust multitask diffusion affine projection algorithms (MD-APAs), the computational complexity of the proposed MD-NMSAF algorithm is greatly reduced. In addition, we analyze the mean and mean-square stability conditions of MD-NMSAF, provide theoretical models characterizing the network mean square deviation (MSD) behaviors of transient and steady-state, and further verify their correctness by computer simulations. Simulation results under different network topologies, input signals, filter lengths and impulsive noise types fully demonstrate the performance advantages of the proposed MD-NMSAF algorithm over its competitors in terms of steady-state estimation accuracy and convergence speed.

Regulation of Uncertain Markov Jump Linear Systems With Application on Automotive Powertrain Control

Modeling and control of dynamic systems that experience abrupt changes are challenging and fundamental tasks in applications of different areas of engineering. Among these, control synthesis for autonomous heavy-duty vehicles has the potential to highly benefit transportation systems. That said, we propose a robust recursive regulator for discrete-time Markov jump linear systems with polytopic uncertain data. We assume the uncertainties affect the state-space parameters as well as the transition probability matrix. We formulate an optimization problem using the penalty function method and design a recursive solution, whereby we attain the robust state feedback gains. We impose no restrictions on how fast the uncertainties can change between two consecutive iterations. Conditions for convergence and mean-square stability are well defined in terms of Riccati recursive equations. We also set up a procedure for powertrain polytopic model identification based on CAN bus data of an autonomous heavy-duty ground vehicle. We provide two numerical examples to verify the effectiveness of the robust recursive regulator. Moreover, in a third example, we validate the obtained powertrain model on a trajectory tracking problem and compute acceleration and braking inputs using the robust regulator.

Asymptotic mean square stability of Predictor-Corrector methods for stochastic delay ordinary and partial differential equations

In this paper, we focus on the asymptotic mean square stability of stochastic delay ordinary and partial differential equations. By virtue of root locus technique, the sufficient and necessary conditions of asymptotic mean square stability for Predictor-Corrector methods are obtained, which are suitable for solving above two types of problems. Furthermore, by regulating the value of parameter θ in drift term of the schemes, we could investigate a series of schemes to explore different size of stable regions, this amounts to provide a road to obtain the optimal stable regions. Several theorems prove that the Predictor-Corrector schemes could inherit the asymptotic mean square stability for above two original problems. Numerical experiments verify the stability theorems, precision and the parallelism.

Classical Control System Analysis Over Lossy Networks: An Easy-to-Use Nyquist Plot Approach

This article presents an approach based on the Nyquist plot to analyze the mean-square stability of a networked control system in which data transmission is subject to the occurrence of packet losses. This result is obtained by translating well known LMI conditions for mean-square stability to conditions that take the form of upper bounds on the <inline-formula><tex-math notation="LaTeX">$\mathcal {H}_{\infty }$</tex-math></inline-formula> norms of the sensitivity and complementary sensitivity transfer matrices. Though the results obtained provide only sufficient conditions, the proposed approach allows to inspect if the stochastic feedback system is mean-square stable by conveniently using classical Nyquist plots. A numerical example is given to illustrate the ease of use of the proposed tool.

Stability analysis and synthesis of discrete-time semi-Markov jump singular systems

This paper investigates the σ-error pth 0) $ ]]> ( p > 0 ) -moment stability and synthesis problems for a class of discrete-time semi-Markov jump singular systems (DTSMJSSs). In view of the Lyapunov functions approach, we first derive the σ-error pth 0) $ ]]> ( p > 0 ) -moment stable behaviour of interested systems. Based on semi-Markov kernel approach and the techniques of eliminating power of matrices (EPMs), we then obtain several LMI conditions for σ-error mean square stability. Subsequently, due to the time-varying singular E-matrix in DTSMJSSs, two different forms of controller gain are exploited in order to design stabilising state-feedback strategies. In particular, computational analysis and comparison of the proposed control schemes are provided, which shows the numerical complexity of the proposed control schemes is vastly lower than that of the existing literature. Finally, an example has been performed to verify the obtained analytical results.

Mean-square stability of discrete-time switched systems under modeled random switching

This paper studies the mean-square stability of discrete-time linear switched systems with random switching, where the switching signal is the output of a logic dynamical switching model driven by an independent and identically distributed (i.i.d.) process. This class of switching is referred to as the modeled random switching. By regarding the switching model as a part of the system, a combined switched system with i.i.d. switching is obtained, which is of a hybrid nature, that is, the augmented state vector has both logic and continuous components. The semi-tensor product of matrices and the vector representation of logic are applied to merge the logic and the continuous components of the state. The equivalence between the mean-square stability of the original switched system and that of the merged switched system under i.i.d. switching is proved. By deriving the dynamics of the column-stacking form of the covariance matrix, necessary and sufficient conditions of mean-square stability for discrete-time linear switched systems with modeled random switching are obtained. An illustrative example is provided to demonstrate the usage of the proposed results.

Multiple leaders formation tracking with switching topology and aperiodic sampled data under intermittent control scheme

AbstractThis article studies the problem of second‐order formation tracking with multiple leaders under intermittent control scheme in stochastic multi‐agent networked systems. Based on the aperiodic sampled data, the formation tracking control protocol is proposed by considering switching topology and the time‐varying transmission delay of information interaction between each agent. Using the algebraic graph theory, stochastic systems theory, Lyapunov theory, and Halanay inequality, the sufficient conditions of mean square stability for formation tracking with multiple leaders are obtained. To further alleviate the computation resources of the controller, the intermittent control strategy is proposed. Meanwhile, a novel stability analysis method for intermittent control is introduced based on an auxiliary system and comparison principle. Furthermore, the lower bound of the duty cycle for a certain control period is obtained and the desired formation mode is also reached under the designed intermittent control scheme. Finally, numerical examples are given to verify the correctness and effectiveness of the proposed methods.

Robust Affine Projection Tanh Algorithm and Its Performance Analysis

The affine projection algorithm (APA) suffers from performance degradation when dealing with non-Gaussian noises. To address this issue, a novel robust affine projection tanh algorithm (APTA) has been proposed according to a new constructed constraint optimization model with the smoothed and bounded tanh function. Generally, it is difficult to analyze the performance of APTA, since the error vector constrained by the nonlinear function cannot be extracted directly. Therefore, a vector-formed Taylor series expansion is first constructed to decompose the tanh error vector. Then, the steady-state performance, mean and mean-square stability conditions, and transient performance of APTA are carried out for performance analyses. Finally, simulation results verify the effectiveness of theoretical analyses and the superiority of APTA from the aspects of filtering accuracy and robustness.

Multiobjective Control Design for Human–Machine Systems With Safety Performance Constraints

This article studies the problem of multiobjective control design for a class of human–machine systems (HMSs) with safety performance constraints containing state constraints and input constraints. The HMSs under consideration not only monitor the human but also need to take proper actions to both the human and machine. A model of controlled hidden Markov jump system (CHMJS) is applied to represent the HMSs. Based on the CHMJS model, a sufficient condition for the practical mean square stability of the unconstrained HMSs is first derived using a stochastic Lyapunov functional. A sufficient condition for ensuring the safety performance constraints of the HMSs is also deduced by employing reachability analysis and set invariance theory. Subsequently, a bilinear matrix inequality-based control design is presented to guarantee both the practical mean square stability and safety performance constraints of the HMSs. A multiobjective optimization problem (MOP) is then formulated to determine a feedback controller for the human and a human-assistance controller for the machine such that both the practical mean square stability and safety performance constraints as well as the less human intervention can be satisfied. An algorithm that mixes the multiobjective particle swarm optimization and linear matrix inequality technique is developed to solve this MOP. Finally, a lane departure example is given to illustrate its effectiveness.

Input-output analysis of stochastic base flow uncertainty

We adopt an input-output approach to analyze the effect of persistent white-in-time structured stochastic base flow perturbations on the mean-square properties of the linearized Navier-Stokes equations. Such base flow variations enter the linearized dynamics as multiplicative sources of uncertainty that can alter the stability of the linearized dynamics and their receptivity to exogenous excitations. Our approach does not rely on costly stochastic simulations or adjoint-based sensitivity analysis. We provide verifiable conditions for mean-square stability and study the frequency response of the flow subject to additive and multiplicative sources of uncertainty using the solution to the generalized Lyapunov equation. For small-amplitude base flow perturbations, we bypass the need to solve large generalized Lyapunov equations by adopting a perturbation analysis. We use our framework to study the destabilizing effects of stochastic base flow variations in transitional parallel flows, and the reliability of numerically estimated mean velocity profiles in turbulent channel flows. We uncover the Reynolds number scaling of critically destabilizing perturbation variances and demonstrate how the wall-normal shape of base flow modulations can influence the amplification of various length scales. Furthermore, we explain the robust amplification of streamwise streaks in the presence of streamwise base flow variations by analyzing the dynamical structure of the governing equations as well as the Reynolds number dependence of the energy spectrum.

Remote State Estimation With Smart Sensors Over Markov Fading Channels

We consider a fundamental remote state estimation problem of discrete-time linear time-invariant (LTI) systems. A smart sensor forwards its local state estimate to a remote estimator over a time-correlated multistate Markov fading channel, where the packet drop probability is time-varying and depends on the current fading channel state. We establish a necessary and sufficient condition for mean-square stability of the remote estimation error covariance in terms of the state transition matrix of the LTI system, the packet drop probabilities in different channel states, and the transition probability matrix of the Markov channel states. To derive this result, we propose a novel estimation-cycle based approach and provide new elementwise bounds of matrix powers. The stability condition is verified by numerical results and is shown more effective than existing sufficient conditions in the literature. We observe that the stability region in terms of the packet drop probabilities in different channel states can either be convex or nonconvex depending on the transition probability matrix of the Markov channel states. Our numerical results suggest that the stability conditions for remote estimation may coincide for setups with a smart sensor and with a conventional one (which sends raw measurements to the remote estimator) though the smart sensor setup achieves a better estimation performance.

Mean-Square Strong Stability and Stabilization of Discrete-Time Markov Jump Systems with Multiplicative Noises

In this paper, the mean-square strong stability and stabilization of discrete-time Markov jump systems are studied. Firstly, the definition of mean-square strong stability is given, and the necessary and sufficient conditions for mean-square strong stability are derived. Secondly, several necessary and sufficient conditions for mean-square strong stabilization via a state feedback controller and an output feedback controller are obtained. Furthermore, explicit expressions for the state feedback controller and static output feedback controller are obtained. Finally, two examples are given to illustrate the validity of the above results.

Stability of Cyber-Physical Systems of Numerical Methods for Stochastic Differential Equations: Integrating the Cyber and the Physical of Stochastic Systems

This paper presents the cyber-physical system (CPS) of a numerical method (the widely-used Euler- Maruyama method) and establishes a foundational theory of the CPSs of numerical methods for stochastic differential equations (SDEs), which transforms the way we understand the relationship between the numerical method and the underlying dynamical system. Unlike in the literature where they are treated as separate systems linked by inequalities, the CPS is a seamless integration of the SDE and the numerical method and we construct a new and general class of stochastic impulsive differential equations (SiDEs) that can serve as a canonic form of the CPSs of numerical methods. By the CPS approach, we show the equivalence and intrinsic relationship between the stability of the SDE and the stability of the numerical method using the Lyapunov stability theory we develop for our class of SiDEs. Applying our established theory, we present the CPS Lyapunov inequality that is the necessary and sufficient condition for meansquare stability of the CPS of the Euler-Maruyama method for linear SDEs. The proposed CPS and theory initiate the study of systems numerics and provoke many open and interesting problems for future work.

Stability analysis of stochastic and random systems in the Lyapunov sense

Lyapunov approach plays a vital role in addressing the issue of stability in deterministic and random (stochastic) systems. This paper is involved in the study of stability of linear and nonlinear systems perturbed by a standard Brownian motion, random coefficients or coefficient functions acting as stochastic processes. We mainly focus on the stability in mean-square and stochastic stability of these systems. Some suitable Lyapunov functions imply the necessary conditions of mean-square stability and stability in probability. Stability of systems with random variable coefficients are firstly discussed in this paper. The theoretical findings are supported by some examples with stability regions and some numerical simulations.

Optimal Control and Stabilization for Networked Systems With Input Delay and Markovian Packet Losses

The linear quadratic regulation (LQR) problem for discrete-time networked control systems (NCSs) is investigated in this article. The difference from most previous works is that input delay and packet losses occur simultaneously in the communication channel connecting the controller to the actuator. Moreover, the data packet dropout is modeled as a time-homogeneous Markov process which poses challenges due to the temporal correlation. The contributions of this article are twofold. First, by applying the maximum principle involving Markov jumps and delay, the linear quadratic optimal control problem in finite horizon is solved and the solution is given in terms of a forward and a backward stochastic difference equations (FBSDEs-M). Second, under a basic assumption, the infinite horizon optimal control problem is solved and the necessary and sufficient condition for mean square stabilization is given in terms of the solutions to coupled algebraic Riccati-type equations with Markov jumps. The presented results are new to the best of our knowledge since there is no existing work that tackles delay and Markovian packet dropouts simultaneously. It is a generalization of the previous work in which the packet dropouts is modeled as an independent identically distributed Bernoulli process.