Stochastic Hybrid Systems Research Articles

Uniform-in-time bounds for a stochastic hybrid system with fast periodic sampling and small white-noise

We study the asymptotic behavior, uniform-in-time, of a nonlinear dynamical system under the combined effects of fast periodic sampling with period [Formula: see text] and small white noise of size [Formula: see text]. The dynamics depend on both the current and recent measurements of the state, and as such it is not Markovian. Our main results can be interpreted as Law of Large Numbers (LLN) and Central Limit Theorem (CLT) type results. LLN type result shows that the resulting stochastic process is close to an ordinary differential equation (ODE) uniformly in time as [Formula: see text] Further, in regards to CLT, we provide quantitative and uniform-in-time control of the fluctuations process. The interaction of the small parameters provides an additional drift term in the limiting fluctuations, which captures both the sampling and noise effects. As a consequence, we obtain a first-order perturbation expansion of the stochastic process along with time-independent estimates on the remainder. The zeroth- and first-order terms in the expansion are given by an ODE and SDE, respectively. Simulation studies that illustrate and supplement the theoretical results are also provided.

Partial Stability of Linear Hybrid Discrete–Continuous Itô Systems with Aftereffect

This paper offers several new sufficient conditions of the partial moment stability of linear hybrid stochastic systems with delay. Despite its potential applications in economics, biology and physics, this problem seems to have not been addressed before. A number of general theorems on the partial moment stability of stochastic hybrid systems are proven herein by applying a specially designed regularization method, based on the connections between Lyapunov stability and input-to-state stability, which are well known in control theory. Based on the results obtained for stochastic hybrid systems, some new conditions of the partial stability of deterministic hybrid systems are derived as well. All stability conditions are conveniently formulated in terms of the coefficients of the systems. A numerical example illustrates the feasibility of the suggested framework.

THE EXISTENCE AND UNIQUENESS THEOREMS FOR STOCHASTIC DIFFERENTIAL-DIFFERENCE HYBRID SYSTEMS

A stochastic differential-difference hybrid system is a system of interacting variables whose dynamics are described by stochastic differential equations for some of them and difference equations for others. Systems with two types of difference equations are examined: first, a difference equation in the form of a process involving the multiplicative Wiener process, and second, a difference equation with delay. The existence and uniqueness theorems for both systems have been proven. The basic conditions on the system’s parameters are local Lipschitz conditions and linear growth order.

Environmental management and restoration under unified risk and uncertainty using robustified dynamic Orlicz risk

Environmental management and restoration under unified risk and uncertainty using robustified dynamic Orlicz risk

AoI Analysis of Satellite–UAV Synergy Real-Time Remote Sensing System

With the rapid development of space–air–ground integrated networks (SAGIN), the synergy between the satellite and unmanned aerial vehicles (UAVs) in sensing environmental status information reveals substantial potential. In SAGIN, applications such as disaster response and military operations require fresh status information to respond effectively. The freshness of information, quantified by the age of information (AoI) metric, is crucial for an effective response. Therefore, it is urgent to investigate the AoI in real-time remote sensing systems leveraging satellite–UAV synergy. To this end, we first establish a comprehensive system model, corresponding to the satellite–UAV “multiscale explanation” synergy remote sensing system in SAGIN, in which we focus on the typical information transmission and fusion strategies of the system, the analysis framework of AoI, and the temporal evolution of AoI. Subsequently, the time-varying process of the system model is transformed into a corresponding finite-states continuous-time Markov chain, enabling a precise analysis of its stochastic behavior. By employing the stochastic hybrid system (SHS) approach, the moment generating functions (MGFs) and mean AoI, offering quantitative insights into the freshness of status information, are derived. Following this, a comparative analysis of AoI under different queuing disciplines, highlighting their respective performance characteristics, is conducted. Furthermore, considering transmit power and bandwidth constraints of the system, the AoI performances under full frequency reuse (FFR), and frequency division multiple access (FDMA) strategies are analyzed. The energy advantage and spectrum advantage associated with AoI are also examined to explore the superior AoI-related performance of the FFR strategy in SAGIN.

Stability of highly nonlinear stochastic systems with non-random switching signals and multiple delays

The situation of multiple delays often occur in hybrid stochastic systems. In this paper, multiple time-varying delays are considered for highly nonlinear switching stochastic systems. The function of time-varying delays is differentiable, and its derivative is a positive constant less than 1. By the Lyapunov functional method and the mode-dependent average dwell time approach, existence and uniqueness of solution, asymptotic stability and pth moment exponential stability for highly nonlinear switching stochastic multiple delays systems are obtained. Finally, an example is afforded to explicate the effectiveness our theoretical analysis results.

Contingency detection in modern power systems: A stochastic hybrid system method

Contingency detection in modern power systems: A stochastic hybrid system method

Existence and Uniqueness Theorems for Stochastic Differential-Difference Hybrid Systems

Existence and Uniqueness Theorems for Stochastic Differential-Difference Hybrid Systems

“PEOPLE” MEET “MARKOVIANS” — INDIVIDUAL-BASED MODELING WITH HYBRID STOCHASTIC SYSTEMS

Individual-based models (IBMs) enable modelers to avoid far-reaching abstractions and strong simplifications by allowing for a state-based representation of individuals. The fact that IBMs are not represented using a standardized mathematical framework such as differential equations makes it harder to reproduce IBMs and introduces difficulties in the analysis of IBMs. We propose a model architecture based on representing individuals via Markov models. Individuals are coupled to populations — for which individuals are not explicitly represented — that are modeled by differential equations. The resulting models consisting of continuous-time finite-state Markov models coupled to systems of differential equations are examples of piecewise-deterministic Markov processes (PDMPs). We will demonstrate that PDMPs, also known as hybrid stochastic systems, allow us to design detailed state-based representations of individuals which, at the same time, can be systematically analyzed by taking advantage of the theory of PDMPs. We will illustrate design and analysis of IBMs using PDMPs via the example of a predator that intermittently feeds on a logistically growing prey by stochastically switching between a resting and a feeding state. This simple model shows a surprisingly rich dynamics which, nevertheless, can be comprehensively analyzed using the theory of PDMPs.

Analyzing the age of information in prioritized status update systems under an interruption-based hybrid discipline

Analyzing the age of information in prioritized status update systems under an interruption-based hybrid discipline

Stability analysis for nonautonomous impulsive hybrid stochastic delay systems

Stability analysis for nonautonomous impulsive hybrid stochastic delay systems

Enhancing information freshness in multi-class mobile edge computing systems using a hybrid discipline

Timely status updating in mobile edge computing (MEC) systems has recently gained the utmost interest in internet of things (IoT) networks, where status updates may need higher computations to be interpreted. Moreover, in real-life situations, the status update streams may also be of different priority classes according to their importance and timeliness constraints. The classical disciplines used for priority service differentiation, preemptive and non-preemptive disciplines, pose a dilemma of information freshness dissatisfaction for the whole priority network. This work proposes a hybrid preemptive/non-preemptive discipline under an M/M/1/2 priority queueing model to regulate the priority-based contention of the status update streams in MEC systems. For this hybrid discipline, a probabilistic discretionary rule for preemption is deployed to govern the server and buffer access independently, introducing distinct probability parameters to control the system performance. The stochastic hybrid system approach is utilized to analyze the average age of information (AoI) along with its higher moments for any number of classes. Then, a numerical study on a three-class network is conducted by evaluating the average AoI performance and the corresponding dispersion. The numerical observations underpin the significance of the hybrid-discipline parameters in ensuring the reliability of the whole priority network. Hence, four different approaches are introduced to demonstrate the setting of these parameters. Under these approaches, some outstanding features are manifested: exploiting the buffering resources efficiently, conserving the aggregate sensing power, and optimizing the whole network satisfaction. For this last feature, a near-optimal low-complex heuristic method is proposed.

Stability in Distribution of Highly Nonlinear Hybrid Stochastic Delay Systems by Delay Feedback Control

Stability in Distribution of Highly Nonlinear Hybrid Stochastic Delay Systems by Delay Feedback Control

Truncated stochastically switching processes.

There are a large variety of hybrid stochastic systems that couple a continuous process with some form of stochastic switching mechanism. In many cases the system switches between different discrete internal states according to a finite-state Markov chain, and the continuous dynamics depends on the current internal state. The resulting hybrid stochastic differential equation(hSDE) could describe the evolution of a neuron's membrane potential, the concentration of proteins synthesized by a gene network, or the position of an active particle. Another major class of switching system is a search process with stochastic resetting, where the position of a diffusing or active particle is reset to a fixed position at a random sequence of times. In this case the system switches between a search phase and a reset phase, where the latter may be instantaneous. In this paper, we investigate how the behavior of a stochastically switching system is modified when the maximum number of switching (or reset) events in a given time interval is fixed. This is motivated by the idea that each time the system switches there is an additive energy cost. We first show that in the case of an hSDE, restricting the number of switching events is equivalent to truncating a Volterra series expansion of the particle propagator. Such a truncation significantly modifies the moments of the resulting renormalized propagator. We then investigate how restricting the number of reset events affects the diffusive search for an absorbing target. In particular, truncating a Volterra series expansion of the survival probability, we calculate the splitting probabilities and conditional MFPTs for the particle to be absorbed by the target or exceed a given number of resets, respectively.

A Stochastic Hybrid Framework for Event-Triggered Nonlinear Control Systems With Stochastic Denial of Service Attacks

In this paper, the stability problem for event-triggered nonlinear control systems is addressed in the presence of stochastic denial of service (DoS) attacks on the communication network. Dynamic event-triggered control (DETC) and periodic event-triggered control (PETC) are considered under the stochastic attacks. The stochastic attacks are modeled by a stochastic hybrid timer and a logic variable which is used to identify whether the network is attacked or not. Then the entire closed-loop system can be modeled into the framework of stochastic hybrid systems. Uniform global asymptotic stability in probability is concluded for the resulting stochastic system by proving that the equilibrium set is uniformly globally stable in probability and each neighborhood of the equilibrium set is uniformly globally recurrent. A nonlinear system example is provided to illustrate the effectiveness of the proposed results.

Synthesis of ℋ∞ Control for Descriptor Hybrid Systems with Actuator Saturation

This paper addresses a mode-dependent state-feedback H∞ control for stochastic descriptor hybrid systems, considering both the absence and presence of actuator saturation. Firstly, the necessary and sufficient conditions for the stochastic admissibility criterion with H∞ performance γ of the closed-loop system are proposed. Given the proposed non-convex condition, the author reformulates it into linear matrix inequalities (LMIs). Then, to extend the result to the systems with actuator saturation, the actuator-saturated control input is expressed as a linear combination of a given state-feedback control input and a virtual control input that always remains under the saturation level. To verify this expression, the set invariant condition is also suggested by using the singular mode-dependent Lyapunov function candidate. Therefore, the conditions for the existence of both the mode-dependent state-feedback H∞ control and the ellipsoidal shape invariant sets are successfully derived in terms of LMIs. Two numerical examples demonstrate the effectiveness of the proposed method by solving optimization problems subject to the proposed LMIs that minimize H∞ performance γ and maximize the invariant set, respectively.

Note on control for hybrid stochastic systems by intermittent feedback rooted in discrete observations of state and mode with delays

<abstract> <p>For a hybrid stochastic system, most existing feedback controllers need to observe modes at continuous times, which is feasible when the system's mode is observable and does not incur any cost. However, in most cases, the mode is not readily apparent, and identifying it always incurs a certain expense. Therefore, in order to reduce control costs, when designing a feedback controller, both the state and the mode should be observed at discrete moments. This paper introduces an intermittent feedback controller for stabilizing an unstable hybrid stochastic system through discrete delayed observations of state and mode. By utilizing M-matrix theory, intermittent control approach, and the comparison principle, we propose sufficient conditions for the stabilization theory of hybrid stochastic systems. An illustrative example is taken to validate the proposed theory.</p> </abstract>

Variational Integrators for Stochastic Mechanical Hybrid Systems

Variational Integrators for Stochastic Mechanical Hybrid Systems

Periodically Intermittent Control of Hybrid Stochastic Delay Systems with Asynchronous Switching by Sampled-data Feedback

Periodically Intermittent Control of Hybrid Stochastic Delay Systems with Asynchronous Switching by Sampled-data Feedback

Timely Cache Updating in Parallel Multi-Relay Networks

We consider a system consisting of a server, which receives updates for <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> files according to independent Poisson processes. The goal of the server is to deliver the latest version of the files to a user through a parallel network of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> caches. We consider an update received by the user successful, if the user receives the same file version that is currently prevailing at the server. We derive an analytical expression for information freshness at the user. We observe that freshness for a file increases with increase in consolidation of rates across caches. To solve the multi-cache problem, we first solve the auxiliary problem of a single-cache system. We then rework this auxiliary solution to our parallel-cache network by consolidating rates to single routes as much as possible. This yields an approximate (sub-optimal) solution for the original problem. We provide an upper bound on the gap between the sub-optimal solution and the optimal solution. We present counterpart expressions and policies for version age of information by employing a stochastic hybrid system approach. Numerical results for both timeliness metrics show that the proposed sub-optimal policy closely follows the optimal policy.