Asymptotic Stability In Probability Research Articles

Robust static output feedback stabilization of stochastic discrete-time systems using stochastic geometric dissipativity

This paper delves into the realm of geometrically mean square stable in probability (GMSSiP) for stochastic discrete-time systems (SDTS). To achieve this, we construct a stochastic geometric QSR-dissipativity (SG-QSR-D) inequalities and devise a linear static output feedback (SOF) controller. This framework is instrumental in establishing GMSSiP and asymptotical stability in probability. Additionally, we establish sufficient and necessity conditions for GMSSiP of the SDTS by using SG-QSR-D. Expanding on this foundation, a linear SOF controller is developed by establishing connections between GMSSiP and SG-QSR-D. Specifically, we demonstrate that the asymptotical stability in probability of the closed-loop linear time-invariant (LTI) SDTS is ensured through a linear matrix inequality (LMI) condition and a linear SOF controller. Two illustrative examples are given to show the effectiveness of the obtained results.

Fault-Tolerant Control of Stochastic High-Order Fully Actuated Systems.

In recent years, high-order fully actuated (HOFA) systems, founded by Prof. GR Duan, have recorded rapid progress for deterministic systems. However, the control issue of stochastic fully actuated systems is still an open problem. This study develops a novel stochastic HOFA system model that complements the existing HOFA methodology. Notably, stochastic signals can be considered in the proposed model, different from the case in the deterministic model. By adopting a high-order operator, equivalent control and stabilization control laws are realized to guarantee the global asymptotic stability in probability of the closed-loop system. For the system with sensor gain faults, an observer-based fault-tolerant control law is designed. Finally, the simulation results validate the effectiveness of the proposed control schemes.

Emulation-based stabilization for networked control systems with stochastic channels

This paper studies the stabilization problem of networked control systems (NCSs) with random packet dropouts caused by stochastic channels. To describe the effects of stochastic channels on the information transmission, the transmission times are assumed to be deterministic, whereas the packet transmission is assumed to be random. We first propose a stochastic scheduling protocol to model random packet dropouts, and address the properties of the proposed stochastic scheduling protocol. The proposed scheduling protocol provides a unified modelling framework for a general class of random packet dropouts due to different stochastic channels. Next, the proposed scheduling protocol is embedded into the closed-loop system, which leads to a stochastic hybrid model for NCSs with random packet dropouts. Based on this stochastic hybrid model, we follow the emulation approach to establish sufficient conditions to guarantee uniform global asymptotical stability in probability. In particular, an upper bound on the maximally allowable transmission interval is derived explicitly for all stochastic protocols satisfying Lyapunov conditions that guarantee uniform global asymptotic stability in probability. Finally, two numerical examples are presented to demonstrate the derived results.

Nonlinear Optimal Control for Stochastic Dynamical Systems

This paper presents a comprehensive framework addressing optimal nonlinear analysis and feedback control synthesis for nonlinear stochastic dynamical systems. The focus lies on establishing connections between stochastic Lyapunov theory and stochastic Hamilton–Jacobi–Bellman theory within a unified perspective. We demonstrate that the closed-loop nonlinear system’s asymptotic stability in probability is ensured through a Lyapunov function, identified as the solution to the steady-state form of the stochastic Hamilton–Jacobi–Bellman equation. This dual assurance guarantees both stochastic stability and optimality. Additionally, optimal feedback controllers for affine nonlinear systems are developed using an inverse optimality framework tailored to the stochastic stabilization problem. Furthermore, the paper derives stability margins for optimal and inverse optimal stochastic feedback regulators. Gain, sector, and disk margin guarantees are established for nonlinear stochastic dynamical systems controlled by nonlinear optimal and inverse optimal Hamilton–Jacobi–Bellman controllers.

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.

The novel impulsive switching conditions for stability of impulsive switched stochastic systems

This paper studies the problems of stability analysis and stabilisation synthesis for a class of impulsive switched stochastic systems. By constructing a novel impulsive switching condition, the sufficient conditions of globally asymptotic stability in probability and stochastic input-to-state stability for the considered systems are presented. Furthermore, the stabilising controller design method can be presented for the corresponding closed-loop systems. Two examples are given to illustrate the effectiveness of the obtained results.

Finite-time stabilization of stochastic systems with varying parameters

<abstract><p>This research deals with the stabilization of the stochastic nonlinear systems. In order to achieve the asymptotic stability in probability with respect to unknown bounded disturbances, a control Lyapunov function is applied to present a modified Sontag's homogeneous controller. The obtained results reveal that the presented control achieves the desirable robust asymptotic stability in probability. The finite-time stability in probability for stochastic nonlinear systems is also discussed in this manuscript. Simulation examples are provided to demonstrate the effectiveness of the controllers.</p></abstract>

Output feedback stabilization of stochastic planar nonlinear systems with output constraint

This paper is concerned with the problem of output feedback stabilization of stochastic planar nonlinear systems with output constraint. By introducing some new nonlinear mappings, adding a power integrator and homogeneous domination techniques, according to different value ranges of the homogeneous degree in the same form of assumptions, we establish two new design and analysis frames of output feedback stabilization to achieve respectively stochastic asymptotic stability (or asymptotical stability in probability) and stochastic finite-time stability of the trivial solution of the closed-loop system while guaranteeing the requirements of almost sure boundedness of the closed-loop signals and the symmetric output constraint.

Global stabilization of low-order stochastic nonlinear systems with multiple time-varying delays by a continuous feedback control

The global stabilization problem is investigated in the present paper for a class of stochastic nonlinear systems, whose structure is in a p-normal form (0<p<1) and states are perturbed by multiple time-varying delays. By means of an extended stability theory of stochastic nonlinear time-delay systems, where the existence and uniqueness of strong solution of an initial value problem is not required, a nonsmooth but continuous delay-dependent global stabilizer is systematically constructed to achieve the global strong asymptotic stability in probability of the closed-loop system under some conditions. In order to overcome the obstacles caused by low-order nonlinear terms, multiple time-varying delays and stochastic disturbances, a novel nonlinear Lyapunov-Krasovskii functional is skillfully introduced, which plays an important role in the controller design. The effectiveness of the obtained results is verified by an illustrative example in the end.

Nonlinear–nonquadratic optimal and inverse optimal control for discrete‐time stochastic dynamical systems

AbstractIn this article, we investigate the role of Lyapunov functions in evaluating nonlinear–nonquadratic cost functionals for Itô‐type nonlinear stochastic difference equations. Specifically, it is shown that the cost functional can be evaluated in closed‐form as long as the cost functional is related in a specific way to an underlying Lyapunov function that guarantees asymptotic stability in probability. This result is then used to analyze discrete‐time linear as well as nonlinear stochastic dynamical systems with polynomial and multilinear cost functionals. Furthermore, a stochastic optimal control framework is developed by exploiting connections between stochastic Lyapunov theory and stochastic Bellman theory. In particular, we show that asymptotic and geometric stability in probability of the closed‐loop nonlinear system is guaranteed by means of a Lyapunov function that can clearly be seen to be the solution to the steady state form of the stochastic Bellman equation, and hence, guaranteeing both stochastic stability and optimality.

Dynamical analysis and numerical simulations on a crowley-Martin predator-prey model in stochastic environment

This paper systematically investigates a stochastic Crowley-Martin predator-prey model. Firstly, we derive sufficient conditions for the existence of positive T-periodic solution for the impulsive case. Secondly, distinguished from the previous papers, asymptotical stability in probability is investigated by combining Khasminskii theory of stability with Lyapunov method. Our conclusion also improves and extends the corresponding existing ones. Thirdly, we establish the sufficient criteria for the existence of a unique ergodic stationary distribution for the Markovian switching case. Finally, two numerical examples are presented to demonstrate the effectiveness and feasibility of our analytical results and reveal the respective effect of white noises and Markovian switching.

Stability criteria of random delay differential systems subject to random impulses

AbstractThis article investigates the stability of random impulsive delay differential systems with the kind of random impulsive intensity determined by an arbitrary random sequence or an irreducible aperiodic Markov chain. Moreover, the continuous dynamics are described by random delay differential equations whose random disturbances are driven by second‐order moment processes. Based on the approaches of average impulsive interval and mode‐dependent average impulsive interval, the criteria of global asymptotic stability in probability and exponential stability in th moment are constructed, respectively. In the end, the feasibility of the proposed theoretical results is verified by an example.

Output‐feedback stabilization of a class of stochastic high‐order nonlinear systems with stochastic inverse dynamics and multidelay

AbstractIn this article, the problem of output‐feedback stabilization for a class of stochastic high‐order nonlinear systems with stochastic inverse dynamics and multidelay is studied. In the control scheme, an output‐feedback controller with a reduced‐order observer is designed to globally stabilize the whole system. Different from the classical literature on stochastic upper‐triangular systems, a high‐gain‐based technique is utilized to eliminate the effect of uncertainties in z0‐subsystem. By skillfully choosing suitable forms of the control gains and the Lyapunov–Krasovskii functional, we analyze the global asymptotic stability in probability of the closed‐loop system. Finally, two simulation examples are presented to show the effectiveness of our theoretical results.

Further stability results for random nonlinear systems with stochastic impulses

In this paper, the global asymptotic stability in probability and the exponential stability in mth moment are investigated for random nonlinear systems with stochastic impulses, whose occurrence is determined by a Poisson process. The stochastic disturbances in the impulsive random nonlinear systems are driven by second-order processes, which have bounded mean power. Firstly, the improved Lyapunov approaches for the global asymptotic stability in probability and the exponential stability in mth moment are established for impulsive random nonlinear systems based on the uniformly asymptotically stable function. Secondly, the improved results are further extended to the impulsive random nonlinear systems with Markovian switching. Finally, two examples are provided to verify the feasibility and effectiveness of the obtained results.

Dynamic event-triggered output feedback control for stochastic nonlinear systems

Abstract In this paper, the event-triggered stabilization problem for stochastic nonlinear systems is addressed. A dynamic event-triggered transmission strategy is proposed for general stochastic nonlinear systems without the linear growth assumption and the proposed transmission strategy can be applied to the output-based dynamic controller. Zeno phenomenon is avoided in the transmission strategy. The resulted closed-loop system is modeled into a stochastic hybrid system. Uniformly globally asymptotic stability in probability is concluded by showing that the equilibrium set is uniformly globally stable in probability and arbitrarily small neighborhood of the equilibrium set is uniformly globally recurrent in probability.

Energy‐based output regulation for stochastic port‐Hamiltonian systems

AbstractThis article investigates the output regulation for stochastic port‐Hamiltonian systems (SPHSs) subject to sinusoidal disturbances. An energy‐based regulation scheme with an internal model unit is proposed by exploiting the stochastic Hamiltonian structure, which drives the tracking error to the origin while maintaining asymptotical stability in probability of the closed‐loop system. An energy‐based robust regulation scheme as well as an alternative condition is then developed without solving Hamilton–Jacobi–Issacs inequalities. The proposed regulators preserve the stochastic Hamiltonian structure of the disturbed SPHS by coordinate transformation. Hence the output regulation problems fall into the stabilization framework for SPHSs and there is no need to solve regulator equations. These results cover the stabilization of SPHSs and the output regulation of deterministic port‐Hamiltonian systems. Simulations on an inverted pendulum show the effectiveness of the proposed methods.

Стiйкiсть розв’язку стохастичних динамiчних систем випадкової структури iз зовнiшнiми збуреннями, пуассоновими перемиканнями i всiєю передiсторiєю

The Ito-Skorokhod stochastic dynamical system with external random disturbances, with Markov switching and all prehistory is considered in this paper. Basic definitions of stability of a strong solution for such systems are introduced and sufficient conditions for asymptotic stability in probability in general and the asymptotic mean square stability in general are received.

Asymptotic Behavior of Multigroup SEIR Model with Nonlinear Incidence Rates under Stochastic Perturbations

In this paper, the asymptotic behavior of a multigroup SEIR model with stochastic perturbations and nonlinear incidence rate functions is studied. First, the existence and uniqueness of the solution to the model we discuss are given. Then, the global asymptotical stability in probability of the model with R0&lt;1 is established by constructing Lyapunov functions. Next, we prove that the disease can die out exponentially under certain stochastic perturbation while it is persistent in the deterministic case when R0&gt;1. Finally, several examples and numerical simulations are provided to illustrate the dynamic behavior of the model and verify our analytical results.

Asymptotic Stability Analysis for Switched Stochastic Nonlinear Systems Using Mode-dependent Uniformly Stable Functions

In this paper, we intend to investigate uniform global asymptotic stability in probability (UGAS-P) for a class of time-varying switched stochastic nonlinear systems. Conventional criteria on stability for switched stochastic systems are based on the negativity of the infinitesimal generator of Lyapunov functions, it is demonstrated that these criteria are conservative. Taking this fact into account, the infinitesimal generator for each active subsystem acting on Lyapunov functions is relaxed to be indefinite with the help of uniformly stable function (USF). Subsequently, improved criteria on asymptotic stability are proposed by applying the weakened condition and mode-dependent average dwell time (MDADT) technique. In addition, numerical examples are presented to verify the effectiveness of the obtained results.

Practical asymptotic stability of stochastic systems driven by Lévy processes and its application to control of TORA systems

Two Lyapunov-type theorems are developed to investigate global (practical) asymptotic stability in probability for SDEs driven by (non-vanishing) Lévy processes, which can be either affine or non-affine or both in the states. These theorems provide sufficient conditions for the stability, which are relatively easy to be verified, and thus have a potential application in control design. The theoretical development is then applied to design feedback stabilisers for global (practical) asymptotic stability in probability for a translating oscillator with a rotating actuator (TORA) system.