Uncertain Stochastic 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.

RL-based adaptive control for a class of non-affine uncertain stochastic systems with mismatched disturbances

This paper investigates the reinforcement learning (RL) adaptive tracking control design problem for a class of mismatched stochastic nonlinear systems with non-affine structure. The stochastic system studied in this paper is more generally representative due to the presence of non-affine inputs, internal uncertainties, and mismatched external disturbances. Firstly, in order to solve the non-affine structure of stochastic systems, an extended stochastic differential equation is constructed. Based on the actor–critic framework, by generating reinforcement signals, the network is stimulated to evolve more quickly towards the desired direction, while compensating internal uncertainties and induced uncertainties in the stochastic system. Furthermore, aiming at the approximation errors and external disturbances, while satisfying stochastic stability, the adaptive laws for disturbances boundary estimator are established with the usage of higher powers of tracking errors. As a result, a novel non-affine adaptive tracking controller has been proposed by integrating RL, disturbances boundary estimation, and dynamic surface method. Through the stability analysis, it is proved that all closed-loop signals are bounded in probability, and the system output converges to a small neighborhood near the desired trajectory. Numerical simulations demonstrate the effectiveness and superiority of the proposed controller.

Pareto-based guaranteed cost control of discrete-time uncertain stochastic systems

The Pareto-based guaranteed cost control (GCC) of discrete-time uncertain stochastic systems in the infinite horizon is studied. Firstly, the convexity of the weighted sum objective function is proved. Based on this convexity, we demonstrate the relationship between Pareto optimality and minimization of the weighted sum objective function. This relationship confirms that the Pareto-based GCC strategy can be obtained by optimizing the weighted sum objective function. Secondly, to address the Pareto-based GCC issue for discrete-time uncertain stochastic systems, we derive the generalized algebraic Riccati inequality (GARI) specific to these systems. Thirdly, the necessary condition for a Pareto-based guaranteed cost controller is derived using the Karush–Kuhn–Tucker (KKT) condition. Lastly, we introduce a method based on linear matrix inequality (LMI) to determine the Pareto-based GCC strategy. This method reduces computational complexity and establishes the sufficient conditions for a Pareto-based guaranteed cost controller. To validate our findings, we provide a example implemented in MATLAB, confirming the accuracy of the proposed approach.

Fusion estimation against mixed network attacks for systems with random parameter matrices, correlated noises, and quantized measurements

This paper is concerned with the information fusion estimation problems for stochastic uncertain systems with quantized measurements and mixed network attacks including random deception attack and denial-of-service (DoS) attack. The cross-correlated random parameter matrices and addictive noises simultaneously exist in the studied system. By resorting to the optimal prediction compensation mechanism for DoS attack, an optimal centralized fusion filter in the linear minimum variance sense is proposed using an innovation analysis approach. In addition, the Kalman-like recursive distributed optimal linear fusion predictor and filter without feedback are presented based on local estimators from single-sensor subsystems. The estimation error cross-covariance matrices between two arbitrary local estimators, and those between local and prior fusion estimators are derived. They have good flexibility due to the parallel structure. However, they have lower accuracy than the centralized fusion estimators. To further improve the estimation accuracy, the distributed optimal linear fusion predictor and filter with feedback are also presented. They avoid the calculation of cross-covariance matrices. Moreover, it has been mathematically proved that they have the same estimation accuracy as the centralized fusion estimators. A simulation example demonstrates the effectiveness of the proposed algorithms.

Saddle-Point Equilibrium Strategy for Linear Quadratic Uncertain Stochastic Hybrid Differential Games Based on Subadditive Measures

This paper describes a kind of linear quadratic uncertain stochastic hybrid differential game system grounded in the framework of subadditive measures, in which the system dynamics are described by a hybrid differential equation with Wiener–Liu noise and the performance index function is quadratic. Firstly, we introduce the concept of hybrid differential games and establish the Max–Min Lemma for the two-player zero-sum game scenario. Next, we discuss the analysis of saddle-point equilibrium strategies for linear quadratic hybrid differential games, addressing both finite and infinite time horizons. Through the incorporation of a generalized Riccati differential equation (GRDE) and guided by the principles of the Itô–Liu formula, we prove that that solving the GRDE is crucial and serves as both a sufficient and necessary precondition for identifying equilibrium strategies within a finite horizon. In addition, we also acquire the explicit formulations of equilibrium strategies in closed forms, alongside determining the optimal values of the cost function. Through the adoption of a generalized Riccati equation (GRE) and applying a similar approach to that used for the finite horizon case, we establish that the ability to solve the GRE constitutes a sufficient criterion for the emergence of equilibrium strategies in scenarios extending over an infinite horizon. Moreover, we explore the dynamics of a resource extraction problem within a finite horizon and separately delve into an H∞ control problem applicable to an infinite horizon. Finally, we present the conclusions.

Linear quadratic zero‐sum game for time‐delayed uncertain stochastic systems

SummaryThis study focuses on the analysis of a linear quadratic zero‐sum game (LQZSG) for time‐delayed uncertain stochastic systems. To begin, we introduce a general time‐delayed zero‐sum game. Employing an algebraic transformation method, we transform the time‐delayed zero‐sum game into an equivalent uncertain random zero‐sum game without time delay. Subsequently, we present equilibrium equations that streamline the transformation of the uncertain random zero‐sum game into problems solvable as deterministic difference equations. We then investigate the LQZSG involving a linear time‐delayed uncertain stochastic system with a quadratic objective function. Within this framework, we provide a unified framework for solving this type of game and obtaining the analytic expression for the saddle‐point equilibrium of the LQZSG. Additionally, we present a numerical example and a counter‐terrorism economic game to illustrate the applicability of our findings.

Adaptive stabilization of stochastic systems with polynomial nonlinear conditions, unknown parameters, and dead‐zone actuator

AbstractMany practical dynamic models always suffer from complex nonlinearities, parameter uncertainties and random noises. Such kind of models are always characterized by uncertain stochastic nonlinear systems. Besides, many factors will lead to dead‐zone actuator, which further adds more challenges to control engineering. This paper studies stabilization issue of stochastic systems with polynomial nonlinear conditions, unknown parameters and dead‐zone actuator. An adaptive control approach is proposed by exploring homogeneous domination control and using dynamic‐gain‐based adaptive design strategy. It shows that all the signals are bounded almost surely and the system states converge to the origin almost surely. The presented method is successfully applied to the mass spring mechanical system.

Command filtered adaptive fuzzy tracking control for uncertain stochastic nonlinear systems with event-triggered input

Command filtered adaptive fuzzy tracking control for uncertain stochastic nonlinear systems with event-triggered input

Understanding the Capability of PD Control for Uncertain Stochastic Systems

In this article, we focus on the global stabilizability problem for a class of uncertain stochastic control systems, where both the drift term and the diffusion term are nonlinear functions of the state variables and the control variables. We will show that the widely applied proportional-derivative (PD) control in engineering practice has the ability to globally stabilize such systems in the mean square sense, provided that the upper bounds of partial derivatives of the nonlinear functions satisfy a certain algebraic inequality. It will also be proved that the stabilizing PD parameters can only be selected from a two dimensional bounded convex set, which differs significantly from the existing results for PD controlled stochastic systems where the diffusion term does not depend on the control variables. Moreover, a particular polynomial on these bounds is introduced, which can be used to determine under what conditions the system is not stabilizable by the PD control, and thus demonstrating the fundamental limitations of PD control.

Zero-sum games subject to time-delayed uncertain stochastic systems

Zero-sum games subject to time-delayed uncertain stochastic systems

A Fast Finite-Time Neural Network Control of Stochastic Nonlinear Systems.

This article takes a fast finite-time control of stochastic nonlinear systems into account. The presence of unknown stochastic disturbance terms makes the traditional fast finite-time control approaches unavailable. To deal with this difficulty, by establishing an auxiliary function and using Jensen's inequality, in Lemma 6, a new criterion of fast finite-time stability is first established for the uncertain stochastic system. Based on the approximation ability of neural networks (NNs), an innovative fast finite-time strategy is put forward for stochastic nonlinear systems. Furthermore, by adopting the presented fast finite-time stability criterion, the stability of the stochastic systems is confirmed. Finally, two simulations are implemented to validate the feasibility of the presented NN control strategy.

Guaranteed cost fault-tolerant control for uncertain stochastic systems via dynamic event-triggered adaptive dynamic programming

It is difficult to apply the event-triggered adaptive dynamic programming (ET-ADP) algorithm to Itô-type stochastic systems due to stochastic disturbances. Therefore, few studies have addressed the ET-ADP algorithm for Itô-type stochastic systems. To fill this gap, this paper investigates a guaranteed cost fault-tolerant control problem for uncertain stochastic systems via a dynamic ET-ADP algorithm. First, the original control problem is transformed into the optimal control problem of a nominal system, and sufficient criteria are presented to verify the feasibility of the problem transformation. Second, a dynamic event-triggered condition is designed. Then, the optimal control of the nominal system is achieved via an ET-ADP algorithm. Finally, a numerical simulation and a comparative experiment are presented to illustrate the effectiveness of the developed control scheme.

New stochastic convergence theorems: Overcoming the limitations of LaSalle theorems

LaSalle theorems, sometimes called as invariance-like theorems, have witnessed abundant applications as powerful tools of analysing system convergence. However, these theorems have twofold limitations: On the one hand, the uniformity of convergence with respect to initial state values cannot be offered, which indicates that the possibility of excessively slow convergence could not be ruled out for a given bounded set of initial state values. On the other hand, unbounded time-variations are not allowed in the systems, and meanwhile, the boundedness of all the states are required even if one merely concerns the convergence of partial states, precluding many scenarios with the unboundedness (e.g., induced by stochastic noise) for some states defined on the whole time horizon. Towards the limitations of LaSalle theorems, this paper seeks to further develop convergence theorems in the stochastic framework. First, a Lyapunov-like function based theorem on the convergence owning the uniformity with respect to initial state values is presented for Itô-type stochastic systems, with the infinitesimal of Lyapunov-like functions exploited more delicately than those in stochastic LaSalle theorems. Based on this, a framework of adaptive stabilization with the uniformity of convergence is established for uncertain stochastic nonlinear systems. In particular, the performance specifications involved are impossible to achieve by applying LaSalle theorems as in the related results. Second, an enlarged convergence theorem is established with Lyapunov-like conditions moderately relaxed, which allows the unboundedness for partial states defined on the whole time horizon and has potential applications in the presence of unbounded time-variations. What is notable, the enlarged convergence theorem is nontrivial to verify, which compels us to propose an extended Barbălat lemma with the conventional requirement of uniform continuity relaxed in the stochastic framework.

Adaptive NN Fixed-Time Fault-Tolerant Control for Uncertain Stochastic System With Deferred Output Constraint via Self-Triggered Mechanism.

For a class of nonstrict-feedback stochastic nonlinear systems with the injection and deception attacks, this article explores the problem of adaptive neural network (NN) fixed-time control ground on the self-triggered mechanism in a pioneering way. After developing the self-triggered mechanism and the delay-error-dependence function, a neural adaptive delay-constrained fault-tolerant controller is proposed by employing the backstepping technique. The self-triggered mechanism does not require an additional observer to determine the time of the data transmission, which reduces the consumption of the system resources more efficiently. In addition, the whole Lyapunov function with the delay-error-dependence term is developed to solve the deferred output constraint problem. Under the proposed controller, it can be proven that all the signals within the closed-loop system are semiglobally uniformly bounded in probability, while the convergence time is independent of the initial state and the deferred output constraint control performance is achieved. The feasibility and the superiority of the proposed control strategy are shown by some simulations.

Multi-Sensor Data Fusion Method Based on Improved Evidence Theory

To achieve autonomous navigation in complex marine environments, unmanned surface vehicles are equipped with a variety of sensors for sensing the surrounding environment and their own state. To address the issue of unsatisfactory multi-sensor information fusion in stochastic uncertain systems with unknown disturbances, an improved evidence theory multi-sensor data fusion method is proposed in this article. First, the affiliation function in fuzzy set theory is introduced as a support function to assign initial evidence for multi-sensor data, and the initial evidence is corrected according to the degree of data bias. Second, a divergence measure is employed to measure the degree of conflict and discrepancy among the evidence, and each piece of evidence is allocated proportional weight based on the conflict allocation principle. Finally, the evidence is synthesized through the evidence combination rule, and the data are weighted and summed to obtain the data fusion results. Since it is difficult to obtain dynamic information from multiple sensors carried by unmanned surface vehicles in practical applications, and considering that the proposed method has universal applicability, practical application experiments using previous research demonstrate that the proposed method has higher fusion accuracy than other existing data fusion methods.

Vision-based particle filtering for quad-copter attitude estimation using multirate delayed measurements.

In this paper, the problem of attitude estimation of a quad-copter system equipped with a multi-rate camera and gyroscope sensors is addressed through extension of a sampling importance re-sampling (SIR) particle filter (PF). Attitude measurement sensors, such as cameras, usually suffer from a slow sampling rate and processing time delay compared to inertial sensors, such as gyroscopes. A discretized attitude kinematics in Euler angles is employed where the gyroscope noisy measurements are considered the model input, leading to a stochastic uncertain system model. Then, a multi-rate delayed PF is proposed so that when no camera measurement is available, the sampling part is performed only. In this case, the delayed camera measurements are used for weight computation and re-sampling. Finally, the efficiency of the proposed method is demonstrated through both numerical simulation and experimental work on the DJI Tello quad-copter system. The images captured by the camera are processed using the ORB feature extraction method and the homography method in Python-OpenCV, which is used to calculate the rotation matrix from the Tello's image frames.

Uncertain stochastic hybrid differential game system with V-n jumps: Saddle point equilibrium strategies and application to advertising duopoly game

This paper focuses on the saddle point equilibrium strategies for uncertain stochastic hybrid differential game systems under chance theory. We consider systems with dynamics described by both an uncertain differential equation with V-n jumps (UDEV) and a stochastic differential equation (SDE), with linear and quadratic performance index functions. Based on chance theory and the equation of optimality of uncertain stochastic hybrid optimal control model, the saddle equilibrium equation for general uncertain stochastic hybrid differential game system is established. Furthermore, three classes of differential game models are discussed by using equilibrium equation. Finally, an advertising duopoly game problem as an application is provided, the corresponding optimal profit difference value and advertising strategies are analyzed.

An order insensitive optimal generalised sequential fusion estimation for stochastic uncertain multi‐sensor systems with correlated noise

AbstractThe globally optimal generalised sequential fusion (GSF) algorithm in the sense of linear minimum variance for multi‐sensor stochastic uncertain systems is investigated by the authors. Specifically, in the GSF algorithm, the estimation of measurement noise is considered, and ma (ma ≥ 1) sensors' measurement data are fused at the ath reception instant, which makes it very flexible and suitable for practical applications. The centralised and sequential fusion algorithms are special cases of the proposed GSF algorithm. Furthermore, for any ma, a = 1, 2, …, M, the estimated values of the GSF algorithm remain invariant and globally optimal. Moreover, the independence between the estimated values and fusion order is proved in the proposed GSF algorithm. Finally, simulation results are given to demonstrate the usefulness of the developed algorithm.

Intelligent decision support tool for optimizing stochastic inventory systems under uncertainty

Pricing and production policies play a key role in ensuring the added value of supply chain systems. For perishable inventory management, the pricing and production lines must be manipulated dynamically since several uncertainties are involved in the system’s behavior. This study discusses the impact of dynamic pricing and production policies on an uncertain stochastic inventory system with perishable products. The mathematical model of the inventory management system under external disturbance is formulated using a continuous differential equation in which the price and production rates are considered as control factors to optimize total profits, which is described as an objective function. An analytical solution for the optimal pricing and production rate was obtained using the Hamilton-Jacobi-Bellman equation. The unknown disturbance was approximated using an intelligent approach called radial basis function neural network. Finally, extensive numerical simulations were presented to validate the theoretical results and optimization solutions (including the efficiency of the approximation of the unknown disturbance) for the dynamic pricing and production management strategy of an uncertain stochastic inventory system against volatile markets. The performance of the proposed method was analyzed under different stock level conditions, which highlighted the importance of keeping the inventory levels at an optimal range to ensure the profitability of business operations. This management strategy can assist a business with solutions for inventory policies while supporting decision-making processes to facilitate coping with production management disruptions.

Reduced-order K-filters-based event-triggered adaptive command filtered tracking control for stochastic constrained nonlinear systems

To ensure better performance and simultaneously save resources, an event-triggered adaptive command filtered dynamic surface control (ACFDSC) method for uncertain stochastic nonstrict-feedback nonlinear systems with dynamic output constraints and prescribed performance is designed in this article. Firstly, with the help of reduced-order K-filters, linearly parameterized neural networks and specific coordinate transformation technique, the unmeasurable states, nonlinearities, two types of unmodeled dynamics and output constraints are dealt with respectively. Then, an event-triggered ACFDSC strategy is proposed to ensure that the tracking error reaches a specific bound within a finite time. By introducing the compensated signal into the complete Lyapunov function, and with the assistance of the compact set defined in the stability analysis, all signals are strictly demonstrated to be semi-globally uniformly ultimately bounded. The simulation results verify the effectiveness of the proposed method.