Conditions For Mean Square Stability Research Articles

Noise and stability in reaction-diffusion equations

<p style='text-indent:20px;'>We study the stability of reaction-diffusion equations in presence of noise. The relationship of stability of solutions between the stochastic ordinary different equations and the corresponding stochastic reaction-diffusion equation is firstly established. Then, by using the Lyapunov method, sufficient conditions for mean square and stochastic stability are given. The results show that the multiplicative noise can make the solution stable, but the additive noise will be not.</p>

Fast Self-Triggered MPC for Constrained Linear Systems With Additive Disturbances

This article proposes a robust self-triggered model predictive control (MPC) algorithm for a class of constrained linear systems subject to bounded additive disturbances, in which the intersampling time is determined by a fast convergence self-triggered mechanism. The main idea of the self-triggered mechanism is to select a sampling interval so that a rapid decrease in the predicted costs associated with optimal predicted control inputs is guaranteed. This allows for a reduction in the required computation without compromising performance. By using a constraint tightening technique and exploring the nature of the open-loop control between sampling instants, a set of minimally conservative constraints is imposed on nominal states to ensure robust constraint satisfaction. A multistep open-loop MPC optimization problem is formulated, which ensures recursive feasibility for all possible realizations of the disturbance. The closed-loop system is guaranteed to satisfy a mean-square stability condition. To further reduce the computational load, when states reach a predetermined neighborhood of the origin, the control law of the robust self-triggered MPC algorithm switches to a self-triggered local controller. A compact set in the state space is shown to be robustly asymptotically stabilized. Numerical comparisons are provided to demonstrate the effectiveness of the proposed strategies.

Phytoplankton-zooplankton interaction under environmental stochasticity: Survival, extinction and stability

Deterministic mathematical models are frequently used to represent the predator-prey interaction for a deeper understanding of population dynamics. This approach, however, always has some limitations as it does not capture the environmental noise or demographic stochasticity which are important features of any natural system. In this paper, we study a minimal deterministic model of phytoplankton-zooplankton (prey and predator, respectively) interaction and compare its dynamics with its stochastic version formulated by two different stochastic perturbation techniques. In the first method, two parameters of the deterministic system are replaced by its average value plus an error term. In the second method, the stochastic perturbation is considered proportional to the distance of state variables from their deterministic equilibrium value. We analyze both the stochastic models and explore their dynamics. In particular, we determine sufficient conditions for extinction probability, stochastic persistence of populations, and the existence of stationary distribution of the first stochastic system. For the other system, we determine sufficient conditions for the asymptotic mean square stability by defining a suitable Lyapunov function. Different analytical results are illustrated numerically and interpreted biologically. The stochastic behaviours of the system are compared with its deterministic counterpart. A case study has also been done considering the 24 months data of phytoplankton-zooplankton interaction in the Lake Trasimeno, Italy.

Quantised output-feedback design for networked control systems using semi-Markov model approach

ABSTRACT This paper focuses on the stability analysis and controller design for a family of discrete-time linear systems with quantised signals in both control input channel and measurement output channel. Quantisation density of each quantizer is considered to be variable, and the variation is governed by a semi-Markov chain. The quantisation density can be adjusted to reduce the amount of transmitted data if the requirement of system performance allows. By constructing a quantisation-error-dependent Lyapunov function, sufficient conditions for σ-error mean-square stability of the considered system are given. With the aid of cone complementarity linearisation algorithm, a set of mode-dependent output-feedback controllers is designed to achieve the required σ-error mean-square stability of the closed-loop system. Finally, a numerical example is provided to illustrate the validity of the proposed control method.

Stability Analysis of Discrete-Time Semi-Markov Jump Linear Systems

This article discusses the stability and stabilization problems for the discrete-time semi-Markov jump linear systems (S-MJLSs) with bounded sojourn times. By using the method of multiple-Lyapunov functions and the concept of discrete-time semi-Markov kernel, we provide the sufficient conditions for σ-error mean square stability of S-MJLSs. In such conditions, the transition probability matrix of embedded Markov chain is considered, which is more general than the scenarios in previous works that only consider the sojourn time of each subsystem mode. Based on such conditions, the stabilization controller design method is also proposed for the resulting closed-loop systems. Two examples are given to illustrate the effectiveness of our results.

An Input–Output Approach to Structured Stochastic Uncertainty

We consider linear time invariant systems with exogenous stochastic disturbances, and in feedback with structured stochastic uncertainties. This setting encompasses linear systems with both additive and multiplicative noise. Our concern is to characterize second-order properties such as mean-square stability and performance. A purely input-output treatment of these systems is given without recourse to state space models, and thus the results are applicable to certain classes of distributed systems. We derive necessary and sufficient conditions for mean-square stability in terms of the spectral radius of a linear matrix operator whose dimension is that of the number of uncertainties, rather than the dimension of any underlying state space models. Our condition is applicable to the case of correlated uncertainties, and reproduces earlier results for uncorrelated uncertainties. For cases where state space realizations are given, Linear Matrix Inequality (LMI) equivalents of the input-output conditions are given.

Stabilization of Linear Systems Across a Time-Varying AWGN Fading Channel

This technical note investigates the minimum average transmit power required for mean-square stabilization of a discrete-time linear process across a time-varying additive white Gaussian noise (AWGN) fading channel that is presented between the sensor and the controller. We assume channel state information at both the transmitter and the receiver, and allow the transmit power to vary with the channel state to obtain the minimum required average transmit power via optimal power adaptation. We consider both the case of independent and identically distributed fading and fading subject to a Markov chain. Based on the proposed necessary and sufficient conditions for mean-square stabilization, we show that the minimum average transmit power to ensure stabilizability can be obtained by solving a geometric program.

Stability Analysis of Polytopic Markov Jump Linear Systems with Applications to Sequence-Based Control over Networks

This paper deals with sequence-based control over networks with time-varying and generally unknown delay and loss probabilities. We show that the problems of stability analysis and controller synthesis can be addressed using a polytopic Markov jump linear system describing an augmented system. For this kind of systems, we derive a necessary and sufficient condition for mean square stability that extends existing results in literature. Likewise, we provide a sufficient condition for mean square stabilizability in terms of an LMI feasibility test. The results are illustrated in a numerical example.

Sliding mode control of automotive electronic valve system under weighted try-once-discard protocol

In this paper, the sliding mode control problem is addressed for the automotive electronic valve system, which is described by the Markovian model according to the voltage failure. It is supposed that both the system states and the system modes are unavailable to the controller. In order to avoid data collision on the sensor-to-controller transmission, the scheduling among the sensor nodes is ruled by the weighted try-once-discard protocol. A mode detector via a hidden Markovian model is introduced, and an asynchronous token-dependent state observer is proposed. Dependent on the hidden mode information and current token directive, a sliding mode controller is constructed to assure the reachability of a sliding region. Besides, the hidden Markovian model approach is developed to derive mean-square stability conditions for the augmented system. Eventually, simulation studies are provided to demonstrate the validity of the proposed control scheme for the system under consideration.

Control of a group of systems whose communication channels are assigned by a semi-Markov process

ABSTRACTThis technical note is concerned with the problem of medium access constraint for a group of networked systems. The scheduling of each subsystem is defined by a stochastic protocol, which can be modelled by a semi-Makov chain with a time-varying transition probability matrix. The resulting closed-loop nonlinear systems are a semi-Markovian jump system with delay. Sufficient conditions for exponential mean-square stability of the resulting closed-loop systems are derived via a Lyapunov–Krasovskii method. Based on the stability criterion, the controller gain of each subsystem is designed. A simulation example is used to demonstrate the effectiveness of proposed method.

Robust stability of polytopic time-inhomogeneous Markov jump linear systems

The transition probabilities of jumps between operational modes of discrete-time Markov(ian) jump linear systems (dtMJLSs) are generally considered to be time-invariant, certain, and often completely known in the majority of dedicated studies. Still, in most real cases the transition probability matrix (TPM) cannot be computed exactly and is time-varying. In this article, we take into account the uncertainty and time-variance of the jump parameters by considering the underlying Markov chain as polytopic and time-inhomogeneous, i.e., its TPM is varying over time with variations that are arbitrary within a polytopic set of stochastic matrices. We show that the conditions used for time-homogeneous dtMJLSs are not enough to ensure the stability of the time-inhomogeneous system, and that perturbations on values of the TPM can make a stable system unstable. We present necessary and sufficient conditions for mean square stability (MSS) of polytopic time-inhomogeneous dtMJLSs, prove that deciding MSS on such systems is NP-hard and that MSS is equivalent to exponential MSS and to stochastic stability. We also derive necessary and sufficient conditions for robust MSS of dtMJLSs affected by polytopic uncertainties on transition probabilities and bounded disturbances.

Control of NCSs with energy efficient channel assignment and power allocation

Abstract This paper investigates the simultaneous stabilization of a collection of battery-powered networked control systems (NCSs) with medium access constraint and communication energy limitation. Each time the channels can only accommodate a limited number of systems for communication and the successful packet arrival rate at the controller depends on the power used by the sensor. Then in order to satisfy the medium access constraint, reduce the communication energy cost and stabilize the collection of NCSs, one needs to decide when to schedule which plant to communicate by which channel and by which power level. Based on the average dwell time technique, sufficient conditions for the exponential mean square stability of a single plant are given at first. Then, based on the stability conditions, a period scheduling policy is proposed to assign the channel accessing sequence and power level such that all the plants are simultaneously stabilized. Finally, a new framework for channel assignment, power allocation and controller design is constructed, which can guarantee a desired decay rate for each system and a minimal energy consumption for the group of systems. The effectiveness of the methodology is demonstrated with numerical simulations.

Quantized stabilization of stochastic systems with multiplicative noise under Markovian switching

A problem of quantized state feedback quadratic mean-square stabilization of discrete-time stochastic processes under Markovian switching and multiplicative noise is considered. A static quantizer is used in the feedback channel and the jump Markovian switching is modeled by a discrete-time Markov chain. The control input is simultaneously applied to both the rate vector and the diffusion term. It is shown that the coarsest quantization density that permits quadratic mean-square stabilization of this system is achieved with the use of a logarithmic quantizer, and the coarsest quantization density is determined by an algebraic Riccati equation, which is also the solution to a special linear stochastic Markovian switching control system. Also, sufficient conditions for exponential mean-square stabilization of such systems are also explored. An example is given to demonstrate the obtained results.

Stabilization for switched stochastic systems with semi-Markovian switching signals and actuator saturation

In this paper, the asymptotic stabilization problem is investigated for a class of switched stochastic systems with semi-Markovian switching signals and actuator saturation. By using the stochastic analysis theory and multiple Lyapunov function method, sufficient conditions for the local asymptotic mean square stability of the related system are established based on the stationary distribution of the embedded Markov chain. Moreover, the mode-dependent state feedback controller and estimation of domain of attraction in mean square sense are proposed in terms of a family of decoupled linear matrix inequalities (LMIs). Finally, two examples are provided to illustrate the effectiveness of the proposed results.

Stability and stabilizability of discrete-time dual switching systems with application to sampled-data systems

In this paper, stability and stabilizability of discrete-time dual switching linear systems is investigated. The switched systems under consideration have two switching variables. One of them is stochastic, described by an underlying Markov chain; the other one can be regarded either as a deterministic disturbance or as a control input, leading to stability or stabilizability problems, respectively. For the considered class of systems, sufficient conditions for mean square stability (with or without control gain synthesis) and mean square stabilizability are provided in terms of matrix inequalities. When the stochastic switching is driven by an independent identically distributed sequence, we establish simpler conditions without additional conservatism. Then, it is shown how the proposed framework can be used to study aperiodic sampled-data systems with stochastic computation times. The results are illustrated on examples borrowed from the literature.

Mean-Square Stabilization Over Gaussian Finite-State Markov Channels

This paper studies the mean-square stabilization problem of discrete-time linear time-invariant systems over Gaussian finite-state Markov channels, which suffer from both signal-to-noise ratio constraint and correlated channel fading modeled by a Markov process. The existence of a fundamental limitation for mean-square stabilization is first established. Then, sufficient stabilization conditions under a time-division multiple access (TDMA) communication scheme are derived in terms of the stability of a Markov jump linear system. Moreover, we present a necessary and sufficient condition for mean-square stabilization of 2-D systems controlled over power-constrained Markov lossy channels. Furthermore, improved sufficient stabilization conditions are derived based on an adaptive TDMA communication scheme for general high-dimensional systems, which achieves a larger stabilization region than the TDMA communication scheme.

Deterministic and Stochastic Models of Arabidopsis thaliana Flowering

Experimental studies of the flowering of Arabidopsis thaliana have shown that a large complex gene regulatory network (GRN) is responsible for its regulation. This process has been mathematically modelled with deterministic differential equations by considering the interactions between gene activators and inhibitors (Valentim et al. in PLoS ONE 10(2):e0116973, 2015; van Mourik et al. in BMC Syst Biol 4(1):1, 2010). However, due to complexity of the model, the properties of the network and the roles of the individual genes cannot be deducted from the numerical solution the published work offers. Here, we propose simplifications of the model, based on decoupling of the original GRN to motifs, described with three and two differential equations. A stable solution of the original model is sought by linearisation of the original model which contributes to further investigation of the role of the individual genes to the flowering. Furthermore, we study the role of noise by introducing and investigating two types of stochastic elements into the model. The deterministic and stochastic nonlinear dynamic models of Arabidopsis flowering time are considered by following the deterministic delayed model introduced in Valentim et al. (2015). Steady-state regimes and stability of the deterministic original model are investigated analytically and numerically. By decoupling some concentrations, the system was reduced to emphasise the role played by the transcription factor Suppressor of Overexpression of Constants1 (textit{SOC}1) and the important floral meristem identity genes, Leafy (textit{LFY}) and Apetala1 (textit{AP}1). Two-dimensional motifs, based on the dynamics of textit{LFY} and textit{AP}1, are obtained from the reduced network and parameter ranges ensuring flowering are determined. Their stability analysis shows that textit{LFY} and textit{AP}1 are regulating each other for flowering, matching experimental findings. New sufficient conditions of mean square stability in the stochastic model are obtained using a stochastic Lyapunov approach. Our numerical simulations demonstrate that the reduced models of Arabidopsis flowering time, describing specific motifs of the GRN, can capture the essential behaviour of the full system and also introduce the conditions of flowering initiation. Additionally, they show that stochastic effects can change the behaviour of the stability region through a stability switch. This study thus contributes to a better understanding of the role of textit{LFY} and textit{AP}1 in Arabidopsis flowering.

Discretization-based stabilization for a class of switched linear systems with communication delays

The stabilization problem for a class of switched linear systems is investigated in the network environment. Both the synchronous and asynchronous cases are considered according to the availability of the current activated system mode to the actuator. The random communication delay is assumed to be Markovian, resulting in a sampled-data synchronous or asynchronous switched system with Markovian delay as the closed-loop system. We extend the discretization approach to deal with such sampled-data system through exploring the stability conditions of the corresponding discrete-time system. For the asynchronous case, we formulate the closed-loop system as a hybrid system with the switching between its subsystems governed by a switching signal and a Markov chain. By studying the switching number and one-step reachable mode set of the constructed vector-valued switching signal, the exponential mean-square stability (EMSS) conditions and the corresponding mode-dependent controller are obtained with a more general constraint on the designed switching signal. These results are finally verified by two illustrated numerical examples.

Generalized two-step Maruyama methods for stochastic differential equations

In this paper, we propose generalized two-step Maruyama methods for solving Itô stochastic differential equations. Numerical analysis concerning consistency, convergence and numerical stability in the mean-square sense is presented. We derive sufficient and necessary conditions for linear mean-square stability of the generalized two-step Maruyama methods. We compare the stability region of the generalized two-step Maruyama methods of Adams type with that of the corresponding two-step Maruyama methods of Adams type and show that our proposed methods have better linear mean-square stability. A numerical example is given to confirm our theoretical results.

Steady-state and stability analyses of diffusion sign-error LMS algorithm

Diffusion sign-error LMS is a useful algorithm that allows agents in an adaptive network to perform in a robust manner under impulsive noise. In the literature, the mean stability condition and mean-square transient performance of this algorithm were analyzed with a reasonable impulsive noise model. However, it is important to analyze its mean-square stability condition and second-order steady-state performance for both theoretical understanding and practical implementation of the algorithm. This paper complements previous works by achieving this analysis with the same noise model. Simulations are performed to validate the theoretical findings.