Time-varying Transition Rates Research Articles

Disturbance-observer-based control for non-homogeneous semi-Markov jump systems with time-varying delays

This paper discusses the issue of disturbance-observer-based control (DOBC) for a kind of continuous-time non-homogeneous semi-Markov jump systems (SMJSs) with time-varying delays. Unlike the existing results, the system examined in this article is about non-homogeneous SMJSs, which indicates that its transition rates (TRs) is time-varying. The versions of TRs matrices are selected by a switching signal, which is independent of the semi-Markov process that decides on the system modes. The time-varying TRs are established to be of two situations: stochastic process and arbitrary switching, respectively. First, we construct a novel class of Lyapunov function that is related to both system mode and switching signal mode, thus it can make more use of information about the system and help reduce conservatism. Second, some conditions are given to guarantee the exponentially mean square stable of the SMJSs and H∞ output tracking performance index. Finally, two examples are provided to demonstrate the efficacy of research methods.

Discrete-time layered-network epidemics model with time-varying transition rates and multiple resources

This paper studies a discrete-time time-varying multi-layer networked SIWS (susceptible-infected-water-susceptible) model with multiple resources under both single-virus and competing multi-virus settings. Besides the human-to-human interaction, we also consider that the disease can diffuse on different types of medium. We use resources to refer to any media through which the pathogen of a virus can spread, and do not restrict the resource only to be water. In the single-virus case, we give a full analysis of the system’s behaviour related to its healthy state and endemic equilibrium. In the multi-virus case, we show analytically that different equilibria appear driven by the competition among all viruses. We also show that some analytical results of the time-invariant system can be expanded into time-varying cases. Finally, we illustrate the results through some simulations.

A novel queue-based stochastic epidemic model with adaptive stabilising control

The main objective of this paper is to propose a novel SEIR stochastic epidemic model. A distinguishing feature of this new model is that it allows us to consider a setup under general latency and infectious period distributions. To some extent, queuing systems with infinitely many servers and a Markov chain with time-varying transition rate comprise the very technical underpinning of the paper. Although more general, the Markov chain is as tractable as previous models for exponentially distributed latency and infection periods. It is also significantly more straightforward and tractable than semi-Markov models with a similar level of generality. Based on stochastic stability, we derive a sufficient condition for a shrinking epidemic regarding the queuing system’s occupation rate that drives the dynamics. Relying on this condition, we propose a class of ad-hoc stabilising mitigation strategies that seek to keep a balanced occupation rate after a prescribed mitigation-free period. We validate the approach in the light of the COVID-19 epidemic in England and in the state of Amazonas, Brazil, and assess the effect of different stabilising strategies in the latter setting. Results suggest that the proposed approach can curb the epidemic with various occupation rate levels if the mitigation is timely.

Multi-state network meta-analysis of progression and survival data.

Multiple randomized controlled trials, each comparing a subset of competing interventions, can be synthesized by means of a network meta-analysis to estimate relative treatment effects between all interventions in the evidence base. Here we focus on estimating relative treatment effects for time-to-event outcomes. Cancer treatment effectiveness is frequently quantified by analyzing overall survival (OS) and progression-free survival (PFS). We introduce a method for the joint network meta-analysis of PFS and OS that is based on a time-inhomogeneous tri-state (stable, progression, and death) Markov model where time-varying transition rates and relative treatment effects are modeled with parametric survival functions or fractional polynomials. The data needed to run these analyses can be extracted directly from published survival curves. We demonstrate use by applying the methodology to a network of trials for the treatment of non-small-cell lung cancer. The proposed approach allows the joint synthesis of OS and PFS, relaxes the proportional hazards assumption, extends to a network of more than two treatments, and simplifies the parameterization of decision and cost-effectiveness analyses.

Extended dissipativity and dynamical output feedback control for interval type-2 singular semi-Markovian jump fuzzy systems

This paper considers the problems of stochastic admissibility and extended dissipativity analysis, as well as dynamical output feedback controller design for interval type-2 singular semi-Markovian jump fuzzy systems. By utilising the partition method and the matrix decomposition technique, an improved approach of handling the time-varying transition rates is proposed. Based on this approach, a sufficient condition with less conservatism is established to guarantee that the closed-loop system is regular, impulse-free, exponentially stable in mean square and extended dissipative. Determining the parametric matrices of the controller via the obtained sufficient condition is difficult, since some parametric matrices factor between two nonsquare matrices. A dynamical output feedback control scheme is proposed by utilising a vital result which is proposed inspired by Projection lemma. It is worth noting that no conservatism is introduced in controller design due to the sufficiency and necessity of this result. Finally, simulation examples are given to show the effectiveness of the proposed scheme.

A Switched Vertices Approach to Stability Analysis of Delayed Markov Jump Systems With Time-Varying Transition Rates

In the last five years, the stability of continuous-time Markov jump systems (MJSs) with time-varying transition rates (TRs) has been widely investigated by establishing parameter-dependent Lyapunov-Krasovskii functional (PDLKF), in which the time-varying TRs are considered to be in a polytopic sense. However, in the previous works, the derivative of polytopic parameters in PDLKF is assumed to be in a finite domain, which is limited in actual applications. In this brief, a switched vertices approach is proposed to remove the bounded assumption. On this basis, a new stability criterion is derived to guarantee the MJSs to be stochastically stable, which is more general and practical than the existing results. An example is illustrated to show the effectiveness of the obtained result.

A novel result on H$$_{\infty }$$ performance state estimation for Markovian neural networks with time-varying transition rates

A novel result on H$$_{\infty }$$ performance state estimation for Markovian neural networks with time-varying transition rates

A review of multistate modelling approaches in monitoring disease progression: Bayesian estimation using the Kolmogorov-Chapman forward equations.

There are numerous fields of science in which multistate models are used, including biomedical research and health economics. In biomedical studies, these stochastic continuous-time models are used to describe the time-to-event life history of an individual through a flexible framework for longitudinal data. The multistate framework can describe more than one possible time-to-event outcome for a single individual. The standard estimation quantities in multistate models are transition probabilities and transition rates which can be mapped through the Kolmogorov-Chapman forward equations from the Bayesian estimation perspective. Most multistate models assume the Markov property and time homogeneity; however, if these assumptions are violated, an extension to non-Markovian and time-varying transition rates is possible. This manuscript extends reviews in various types of multistate models, assumptions, methods of estimation and data features compatible with fitting multistate models. We highlight the contrast between the frequentist (maximum likelihood estimation) and the Bayesian estimation approaches in the multistate modeling framework and point out where the latter is advantageous. A partially observed and aggregated dataset from the Zimbabwe national ART program was used to illustrate the use of Kolmogorov-Chapman forward equations. The transition rates from a three-stage reversible multistate model based on viral load measurements in WinBUGS were reported.

Finite-time observer-based control for Markovian jump systems with time-varying generally uncertain transition rates

This paper addresses the finite-time observer-based control for Markovian jump systems with time-varying generally uncertain transition rates. In order to estimate the states, a suitable observer is designed, in which both external disturbance and Brownian motion exist. In order to solve the complex time-varying transition rates, a quantization mechanism is raised to prove the closed-loop system and the observer error system be stable. Sufficient conditions of the existences of both the observer and the observer-based controller are derived in terms of linear matrix inequalities. Eventually, two practical examples are given to testify the correctness of the results.

Observer-based control for singular Markov jump systems with actuator saturation and time-varying transition rates

This paper investigates the issue of observer-based control for singular Markov jump systems (SMJSs) with time-varying delay, actuator saturation and time-varying transition rates. Firstly, the state feedback controller is proposed, which based on the state observer. The time-varying transition rates are quantized by the logarithmic quantizer. In addition, the appropriate Lyapunov function is considered to ensure that the original system and the error system are singular stochastic admissible (SSA) with mixed H∞ and passivity performance. Finally, four examples are provided to expound the consistency of the technique.

Linear quadratic optimal control for a class of continuous-time nonhomogeneous Markovian jump linear systems in infinite time horizon

In this paper, the infinite time horizon linear quadratic optimal control problem is investigated for the continuous-time Itô stochastic Markovian jump linear systems (MJLSs) with time-varying transition rates. It is assumed that the time-varying transition rates of the MJLSs possess piecewise homogeneous time-varying property, which implies that the transition rates are time-varying in the whole time domain but they are time-invariant in some small time intervals. The variations of the transition rates in these small time intervals are considered to be in two cases: arbitrary variation and stochastic variation. With this, the considered system becomes a piecewise homogeneous Itô stochastic MJLS. The main contribution of this paper is that two linear quadratic optimal controllers in infinite time horizon are proposed for the above modeled continuous-time piecewise homogeneous Itô stochastic MJLS in the sense of arbitrary variation and stochastic variation, respectively. Moreover, the sufficient and necessary conditions for the existence of the designed controllers are established based on the existence of the unique positive definite solution of two coupled algebraic Riccati matrix equations. Finally, a simulation example is provided to illustrate the effectiveness of the proposed linear quadratic optimal controllers.

Stability analysis and controller synthesis of continuous-time nonhomogeneous Markovian jump systems with state and input delays

This paper focuses on the problems of stability analysis and controller synthesis for a class of nonhomogeneous Markovian jump systems with both state and input delays. The time-varying transition rate matrix is described as a polytope set. By constructing a Lyapnouv–Krasovskii functional and introducing some appropriately slack matrices, a stochastic stability condition is proposed in term of infinite matrix inequalities. Based on this, a new stochastic stability condition is further derived in form of a finite set of LMIs. Then, in terms of linear matrix inequalities (LMIs) techniques, the sufficient condition on the existence of state-feedback controller is presented and proved. Finally, three numerical examples are provided to illustrate the effectiveness and usefulness of the obtained results.

Robust exponential synchronization of a Markovian jump complex dynamical network with piecewise homogeneous Markovian parameters

Abstract This paper establishes a stochastic synchronization method for a Markovian jump complex dynamical network (MJCDN) with time-delay and uncertainties. The considered Markovian structure is piecewise-homogeneous with piecewise-constant time-varying transition rates (TRs). Two Markovian signals are utilized to construct the piecewise-homogeneous Markovian structure. A low-level Markovian signal with time-varying TRs governs the switching between the system dynamics while it is managed by a high-level Markovian signal. Due to the effect of imperfections induced by modeling errors in the system dynamics, some parametric norm-bounded uncertainties are considered. In addition, uncertain TR matrix is considered which means that inaccurate or uncertain information for each element of the TR matrix is allowable. This modelling makes the MJCDN to be more general and applicable than the existing ones. Synchronization conditions are obtained and reported in the form of linear matrix inequalities by the help of Lyapunov–Krasovskii theory, Wirtinger-based integral inequality approach and reciprocally convex technique. Finally, a numerical example is presented to verify the effectiveness of the proposed method.

Fault Detection Filter Design of Polytopic Uncertain Continuous-Time Singular Markovian Jump Systems with Time-Varying Delays

This paper investigates the problem of full-order and reduced-order fault detection filter (FDF) design under unified linear matrix inequality (LMI) conditions for a class of continuous-time singular Markovian jump systems (CTSMJSs) with time-varying delays and polytopic uncertain transition rates. By constructing a new Lyapunov function, sufficient conditions are firstly provided for the singular model error augmented system such that the system is stochastically admissible with an H∞ performance level γ. And then, by applying a novel convex polyhedron technique to decoupled linear matrix inequalities, the full-order and reduced-order fault detection filter parameters can be obtained within a convex optimization frame. The reduced-order fault detection filter (FDF) can not only meet the fault detection accuracy requirements of complex systems but also improve the fault detection efficiency. Finally, a DC motor and an illustrative simulation example are given to verify the feasibility and effectiveness of the proposed algorithms.

Exponential Synchronization of Markovian Jump Complex Dynamical Networks with Uncertain Transition Rates and Mode-Dependent Coupling Delay

This paper aimed at investigating the exponential synchronization problem for a Markovian jump complex dynamical network through designing a state feedback controller. In this paper, it is supposed that both time delay and coefficient matrices switch between finite modes governed by a time-varying Markov process. The transition rate (TR) matrix of the Markov process is supposed to vary with time, and to be piecewise-constant. The time-varying transition rates are investigated under two cases: completely known TRs and partly unknown TRs, respectively. The synchronization problem of the proposed model is inspected by developing Lyapunov–Krasovski function with Markov-dependent Lyapunov matrices. The controller gain matrix for guaranteeing the synchronization problem is derived by using linear matrix inequalities. The resulted criteria depend on both delay size and the probability of the delay-taking value. Finally, a numerical example is provided to demonstrate the effectiveness of the theoretical results.

Sliding Mode Control for Nonlinear Stochastic Semi-Markov Switching Systems With Application to SRMM

In this paper, the sliding mode control (SMC) design for nonlinear stochastic semi-Markov switching systems (S-MSSs) is studied via the bound of time-varying transition rate matrix, in which semi-Markov switching parameters, stochastic disturbance, uncertainty, and nonlinearity are all considered in a unified framework. The system under consideration is more general, which covers the Markov switching system with sojourn-time-independent transition rate matrix as a special case. Many practical systems subject to unpredictable structural variations can be characterized by nonlinear stochastic S-MSSs with sojourn-time-dependent transition rate matrix. The specific information about the bound of time-varying transition rate matrix is known for the sliding mode controller design. First, by using the stochastic semi-Markov Lyapunov function, sojourn-time-dependent sufficient conditions are developed to guarantee the closed-loop sliding mode dynamics stochastically stable. Then, the SMC law is constructed to ensure the reachability of the sliding mode dynamics in a finite-time level. Finally, one joint of space robot manipulator model is described as nonlinear stochastic S-MSSs to illustrate the validity of the proposed SMC design method.

Input–output finite-time mean square stabilization of nonlinear semi-Markovian jump systems

This paper is concerned with the input–output finite-time mean square stabilization problem for a class of nonlinear semi-Markovian jump systems (SMJSs) with time-varying delay. Both time-varying factors of transition rates and states delay are considered to be bounded. Our main purpose is to derive some parameter-dependent sufficient criteria by designing a state feedback controller, to guarantee the input–output finite-time mean square stabilization for the underlying systems where the nonlinear term satisfies the conic sector constraint. Then, the stabilization conditions are obtained by employing a novel Lyapunov–Krasovskii functional, differential form Gronwall’s inequality, and an extra slack variable which is composed of the time-varying transition rates. Finally, a simulation example is provided to illustrate the effectiveness and merits of the proposed method.

Relaxed Robust Stabilization Conditions for Nonhomogeneous Markovian Jump Systems with Actuator Saturation and General Switching Policies

This paper investigates the robust control synthesis problem of nonhomogeneous Markovian jump systems with actuator saturation and uncertainties. First of all, to make a comprehensive mode-transition description, three separate mode sets are established according to the property of transition rates. And then, to improve the numerical solvability of inequality conditions dependent on time-varying transition rates, two useful relaxation methods are developed on the basis of the mode sets. Especially, by taking advantage of a new zero-sum constraint, the relaxation process is further evolved such that the conservatism arising from incomplete knowledge of transition rates can be reduced. Lastly, this paper provides the LMI-based conditions capable of estimating the attraction domain as well as of designing a mode-dependent robust control.

Reduced-Order Observer-Based Sliding Mode Control for Singular Markovian Jump System With Time-Varying Transition Rate

This paper presents a stabilisation problem for singular Markovian jump system with time-varying transition rate based on a reduced-order observer. To design the reduced-order observer, a canonical equivalent form for singular Markovian jump system is given. By the output variables of the equivalent form of singular Markovian jump system and the state of the reduced-order observer, an integral sliding surface is designed. Then, the sufficient condition for the stochastic admissibility of sliding motion is given, in which the time-varying transition rate is handled by the quantized method. Furthermore, a sliding mode controller is developed such that the system can be driven to the sliding surface in finite time. Finally, the effectiveness and superiority of the obtained results are illustrated by two examples, a numerical example and a practical example of DC motor model respectively.

Bilateral Teleoperation of Single-Master Multislave Systems With Semi-Markovian Jump Stochastic Interval Time-Varying Delayed Communication Channels.

Communication time delays in a bilateral teleoperation system often carries a stochastic nature, particularly when we have multiple masters or slaves. In this paper, we tackle the problem for a single-master multislave (SMMS) teleoperation system by assuming an asymmetric and semi-Markovian jump protocol for communication of the slaves with the master under time-varying transition rates. A nonlinear robust controller is designed for the system that guarantees its global robust H∞ stochastic stability in the sense of the Lyapunov theory. Employing the nonlinear feedback linearization technique, the dynamics of the closed-loop teleoperator is decoupled into two interconnected subsystems: 1) master-slave tracking dynamics (coordination) and 2) multislave synchronization dynamics. Employing an improved reciprocally convex combination technique, the stability analysis of the closed-loop teleoperator is conducted using the Lyapunov-Krasovskii methodology, and the stability conditions are expressed in the form of linear matrix inequalities that can be solved efficiently using numerical algorithms. Numerical studies and simulation results validate the effectiveness of the proposed controller design algorithm in both tracking and synchronization performance of the SMMS system, and robustly handling the stochastic and nondifferentiable nature of communication delays.