Stochastic Singular Systems Research Articles

Zero-sum games subject to uncertain stochastic noncausal systems considering polynomial control functions

An uncertain stochastic noncausal system (USNS) is an uncertain stochastic singular system that is expected to be regular but not impulse-free. This study investigates zero-sum games (ZSGs) under the constraint of nonlinear uncertain stochastic noncausal systems (USNSs) considering polynomial control functions. Firstly, a method is introduced to convert USNSs into subsystems, including forward uncertain stochastic difference equations as well as backward uncertain stochastic difference equations. Equilibrium equations are then derived to determine the saddle-point equilibrium solution of the zero-sum game (ZSG) for USNSs. Subsequently, a unified framework is developed to find the saddle-point equilibrium and equilibrium value of the ZSG. Additionally, a numerical example and an analysis of the treatment of industrial wastewater are provided to demonstrate the effectiveness of the proposed framework.

Strong convergence of the tamed Euler-Maruyama method for stochastic singular initial value problems with non-globally Lipschitz continuous coefficients

In our previous works [1] and [2], we delved into numerical methods for solving stochastic singular initial value problems (SSIVPs) that involve coefficients satisfying the global Lipschitz condition. The paper addresses the limitations of our previous work by introducing an explicit method, called the tamed Euler-Maruyama method, for numerically solving SSIVPs with non-globally Lipschitz continuous coefficients, which is both easy-to-implement and exceptionally efficient. We prove the existence and uniqueness theorem and the boundedness of the moments of the solution to SSIVPs under the non-globally Lipschitz condition. Moreover, we provide a sharp analysis of the strong convergence of the proposed method, along with the uniform boundedness of numerical solutions. We also apply our results to the stochastic singular Ginzburg-Landau system and provide numerical simulations to illustrate our theoretical findings.

Multi-sensor information fusion filtering for delayed nonlinear stochastic singular systems with packet disorders

The fusion filtering problem is investigated for a class of time-varying nonlinear singular systems with packet disorders (PDs). The key idea is to transfer the original singular systems with multiple sensors to full-order augmented nonsingular systems by utilizing the matrix transformation method. The phenomenon of PDs accompanied with random transmission delays (RTDs) is considered, which exists in the channel from sensor to filter. A set of independent and identically distributed random variables obeying known probability distribution are introduced to model the RTDs. A centralized fusion filter is designed, which employs the rounding function of the mathematical expectation of the RTDs. Furthermore, the nonlinear function is linearized by utilizing the method of Taylor series expansion. Then, the desired upper bounds of the estimation error covariance are guaranteed and then locally minimized through appropriately designing the corresponding filter gains. Finally, a numerical example with comparisons is carried out to verify the feasibility of the proposed fusion filtering method.

Uniform in time propagation of chaos for the 2D vortex model and other singular stochastic systems

We adapt the work of Jabin and Wang (2018) to show the first result of uniform in time propagation of chaos for a class of singular interaction kernels. In particular, our models contain the Biot–Savart kernel on the torus and thus the 2D vortex model.

Control for Stochastic Singular Systems With Time-Varying Delays via Sampled-Data Controller.

In this article, control for stochastic singular time-varying delay systems under arbitrarily variable samplings is addressed via designing a sampled-data controller. The first and foremost, a novel time-dependent discontinuous Lyapunov-Krasovskii (L-K) functional is built, which takes good advantage of the factual sampling pattern's available properties. Then, based on the refined input delay method by utilizing the constructed time-dependent L-K functional, the free-weighting matrix method, and the auxiliary vector function approach are adopted to develop conditions ensuring the stochastic admissibility for the studied stochastic singular systems with time-varying delays. On the basis of the derived conditions, the sampled-data control issue is tackled, and an unambiguous expression for the sampled-data controller design method is obtained. Finally, simulation examples manifest that our proposed results are correct and effective.

H ∞ performance analysis for 2D discrete singular stochastic systems

This paper addresses the problem of H∞ performance analysis of 2D discrete singular stochastic system described by Roesser model which is challenging since it involves 2D random variables and the disturbance simultaneously. Sufficient conditions are established for the regularity, causality and stability of the system. The proposed results are expressed in terms of strict linear matrix inequalities. Furthermore, a mean square asymptotic stability with an H ∞ disturbance level is developed. Simulation example is provided in order to illustrate the relevance of our approach.

Controllability of Semilinear Stochastic Generalized Systems in Hilbert Spaces by GE-Evolution Operator Method

Controllability is a basic problem in the study of stochastic generalized systems. Compared with ordinary stochastic systems, the structure of stochastic singular systems is more complex, and it is necessary to study the controllability of stochastic generalized systems in the context of different solutions. In this paper, the controllability of semilinear stochastic generalized systems was investigated by using a GE-evolution operator for integral and impulsive solutions in Hilbert spaces. Some sufficient and necessary conditions were obtained. Firstly, the existence and uniqueness of the integral solution of semilinear stochastic generalized systems were discussed using the GE-evolution operator theory and Banach fixed point theorem. The existence and uniqueness theorem of the integral solution was obtained. Secondly, the approximate controllability of semilinear stochastic generalized systems was studied in the case of the integral solution. Thirdly, the existence and uniqueness of the impulsive solution of semilinear stochastic generalized systems were considered, and some sufficient conditions were provided. Fourthly, the approximate controllability of semilinear stochastic generalized systems was studied for the impulsive solution. At last, the exact controllability of linear stochastic systems was studied in the case of the impulsive solution, with some necessary and sufficient conditions given. The obtained results have important theoretical and practical value for the study of controllability of semilinear stochastic generalized systems.

Robust [formula omitted] sliding mode control for uncertain polynomial fuzzy stochastic singular systems

This paper considers the robust H∞ sliding mode control (SMC) problem for uncertain polynomial fuzzy stochastic singular systems. The interest is focused on designing a polynomial fuzzy sliding mode controller, and two assumptions in most integral SMC methods for fuzzy stochastic systems are removed. By constructing an integral sliding surface, the equivalent control law is obtained, and the sliding mode dynamics is derived. With the help of the sum-of-squares method, for sliding mode dynamics, the mean-square admissibility and the H∞ performance are guaranteed. Furthermore, the sliding mode controller is established. The effectiveness and advantages of the proposed design method are finally revealed via a practical example of the modified bio-economic model and a numerical example.

具有状态时滞的奇异随机系统基于滚动时域控制的镇定

具有状态时滞的奇异随机系统基于滚动时域控制的镇定

An Enhanced Input-Delay Approach to Sampled-Data Stabilization for Nonlinear Stochastic Singular Systems Based on T-S Fuzzy Models

The sampled-data stabilization problem of nonlinear stochastic singular systems on the basis of the Takagi–Sugeno fuzzy models under variable samplings is discussed in this article. A new piecewise Lyapunov–Krasovskii functional is constructed, which can capture the actual sampling mode’s available features more fully, and an enhanced input-delay method is presented. By using the proper augmented scheme based on the auxiliary vector function, the new mean square admissibility criteria are derived by making good use of the convex combination techniques and the free weighting matrix approach. It is shown that the obtained results in this article contain less conservatism when compared with the existing ones. The superiority and correctness of our results are verified by an application example of a truck–trailer model.

On numerical methods to second-order singular initial value problems with additive white noise

In this work, we investigate the strong convergence of the Euler–Maruyama method for second-order stochastic singular initial value problems with additive white noise. The singularity at the origin brings a big challenge that the classical framework for stochastic differential equations and numerical schemes cannot work. By converting the problem to a first-order stochastic singular differential system, the existence and uniqueness of the exact solution is studied. Moreover, under some suitable assumptions, it is proved that the Euler–Maruyama method is of (1/2−ɛ) order convergence in mean-square sense, where ɛ is an arbitrarily small positive number, which is different from the consensus that the Euler–Maruyama method is convergent with first order in strong sense when solving stochastic differential equations with additive white noise. While, it is found that if the diffusion coefficient vanishes at the origin, the convergence order in mean-square sense will be increased to 1−ɛ. Our theoretical findings are well verified by numerical examples.

Adaptive fuzzy observer-based mismatched faults and disturbance design for singular stochastic T-S fuzzy switched systems

This paper deals with the problem of observer-based mismatched faults for switched singular fuzzy systems subject to disturbances. First, in order to obtain the estimation of unmeasurable or partially measurable states, fuzzy adaptive observer is obtained, eliminating the influence of external disturbances and mismatched faults, simultaneously. Second, by constructing appropriate piecewise Lyapunov functions and average dwell time, the switched singular fuzzy stochastic system with external parameter perturbations and faults is guaranteed to be robustly stable Further, by solving the linear matrix inequalities a sufficient condition to maintain the stability of error dynamic system is presented. Finally, to illustrate the proposed method is available, numerical examples are presented.

Optimal control of stochastic singular affine systems with Markovian jumps

We consider an optimal control problem for a class of stochastic singular affine systems with Markovian jumps. We establish the existence and uniqueness of the solution to stochastic singular affine systems with Markovian jumps for the first time. Via square completion technique and the generalized Itô’s formula, we derive new kinds of generalized differential Riccati equations (GDREs) and generalized backward differential equations (GBDEs), which give sufficient conditions for the well-posedness of the optimal control problem, and present an explicit representation of optimal control. Also, we discuss the solvability of the GDREs in two cases. As an application, we present a leader-follower differential game to demonstrate the practicability of our results.

Sliding‐mode‐based admissible consensus of nonlinear singular stochastic uncertain semi‐Markov multi‐agent systems with time‐varying delay

In this paper, the sliding mode control of nonlinear singular stochastic uncertain semi-Markov multi-agent systems with time-varying delay are studied. Firstly, a new integral sliding surface function is designed. Secondly, a new sufficient condition in the form of linear matrix inequality is derived to guarantee the stochastic admissibility by using the semi-Markov–Lyapunov function. Thirdly, a sliding mode control law can be synthesised to ensure the reachability of the predefined sliding surface. Finally, an example is given to demonstrate the effectiveness of the result.

Optimized distributed fusion filtering for singular systems with fading measurements and stochastic nonlinearity

<abstract><p>In this paper, the problem of optimized distributed fusion filtering is considered for a class of multi-sensor singular systems in the presence of fading measurements and stochastic nonlinearity. By utilizing the standard singular value decomposition, the multi-sensor stochastic singular systems are simplified to two reduced-order nonsingular subsystems (RONSs). The local filters (LFs) with corresponding error covariance matrices are proposed for RONSs via the innovation analysis approach. Then, on the basis of the matrix-weighted fusion estimation algorithm, the distributed fusion filters (DFFs) are designed for RONSs with multiple sensors in the linear minimum variance sense. Moreover, the DFFs are obtained by utilizing the state transformation for original singular systems. It can be observed that the DFFs have better accuracy in contrast with the LFs. Finally, an illustrate example is put forward to verify the feasibility of the proposed fusion filtering scheme.</p></abstract>

On the stochastic singular Cucker–Smale model: Well-posedness, collision-avoidance and flocking

We study the Cucker–Smale (CS) flocking systems involving both singularity and noise. We first show the local strong well-posedness for the stochastic singular CS systems before the first collision time, which is a well-defined stopping time. Then, for communication with higher order singularity at origin (corresponding to [Formula: see text] in the case of [Formula: see text]), we establish the global well-posedness by showing the collision-avoidance in finite time, provided that there is no initial collisions and the initial velocities have finite moment of any positive order. Finally, we study the large time behavior of the solution when [Formula: see text] is of zero lower bound, and provide the emergence of conditional flocking or unconditional flocking in the mean sense, for constant and square integrable intensity, respectively.

Review of the Latest Progress in Controllability of Stochastic Linear Systems and Stochastic GE-Evolution Operator

According to the spatial dimension, equation type, and time sequence, the latest progress in controllability of stochastic linear systems and some unsolved problems are introduced. Firstly, the exact controllability of stochastic linear systems in finite dimensional spaces is discussed. Secondly, the exact, exact null, approximate, approximate null, and partial approximate controllability of stochastic linear systems in infinite dimensional spaces are considered. Thirdly, the exact, exact null and impulse controllability of stochastic singular linear systems in finite dimensional spaces are investigated. Fourthly, the exact and approximate controllability of stochastic singular linear systems in infinite dimensional spaces are studied. At last, the controllability and observability for a type of time-varying stochastic singular linear systems are studied by using stochastic GE-evolution operator in the sense of mild solution in Banach spaces, some necessary and sufficient conditions are obtained, the dual principle is proved to be true, an example is given to illustrate the validity of the theoretical results obtained in this part, and a problem to be solved is introduced. The main purpose of this paper is to facilitate readers to fully understand the latest research results concerning the controllability of stochastic linear systems and the problems that need to be further studied, and attract more scholars to engage in this research.

Event-triggered output quantized control of discrete Markovian singular systems

This paper is concerned with the event-triggered H∞ admissible analysis and controller synthesis for discrete delayed singular Markovian jump network systems with output quantizations. Based on event-triggered communication mechanism and quantized control strategy, a singular network system model with network-induced delays is firstly established, and sufficient criteria are then given to guarantee that the singular stochastic network systems are stochastically H∞ admissible. Furthermore, according to variable separation and matrix decomposition techniques, output quantized controllers and event-triggered matrices are tactfully co-designed for the stochastic H∞ admissibility of the singular network model. An example is also provided to illustrate the effectiveness of the proposed approach.

Almost surely exponential stability of semi‐Markovian switched singular stochastic systems with mode‐dependent ranks

This paper investigated the almost surely exponential stability (ES a.s.) for semi‐Markovian switched singular stochastic systems (SSSSs) with mode‐dependent ranks, in which the derivative matrices Ei have different ranks. Due to the constraint of consistent initial conditions, state jumps may occur at switching instants in switched singular systems. Some more general conditions for the existence and uniqueness of the solution of semi‐Markovian SSSSs with mode‐dependent ranks are given together with a detailed analysis of state jumps. Based on the equivalent systems transformation and multiple Lyapunov functions method, novel conditions of ES a.s. are presented for addressed systems. An example is given to explain the effectiveness of developed results.

Finite-Time Admissibility and Controller Design for T-S Fuzzy Stochastic Singular Systems with Distinct Differential Term Matrices

The finite-time admissibility analysis and controller design issues for extended T-S fuzzy stochastic singular systems (FSSSs) with distinct differential term matrices and Brownian parameter perturbations are discussed. When differential term matrices are allowed to be distinct in fuzzy rules, such fuzzy models can describe a wide class of nonlinear stochastic systems. Using fuzzy Lyapunov function (FLF), a new and relaxed sufficient condition is proposed via strict linear matrix inequalities (LMIs). Different from the existing stability conditions by FLF, the derivative bounds of fuzzy membership functions are not required in this condition. Based on admissibility analysis results, a design method for parallel distribution compensation (PDC) controller of FSSSs is given to guarantee the finite-time admissibility of the closed-loop system. Finally, the feasibility and effectiveness of the proposed methods in this article are illustrated with three examples.