Uncertain Nonlinear Dynamical Systems Research Articles

Adaptive fixed-time TSM for uncertain nonlinear dynamical system under unknown disturbance.

For nonlinear systems subjected to external disturbances, an adaptive terminal sliding mode control (TSM) approach with fixed-time convergence is presented in this paper. The introduction of the fixed-time TSM with the sliding surface and the new Lemma of fixed-time stability are the main topics of discussion. The suggested approach demonstrates quick convergence, smooth and non-singular control input, and stability within a fixed time. Existing fixed-time TSM schemes are often impacted by unknown dynamics such as uncertainty and disturbances. Therefore, the proposed strategy is developed by combining the fixed-time TSM with an adaptive scheme. This adaptive method deals with an uncertain dynamic system when there are external disturbances. The stability of a closed-loop structure in a fixed-time will be shown by the findings of the Lyapunov analysis. Finally, the outcomes of the simulations are shown to evaluate and demonstrate the efficacy of the suggested method. As a result, examples with different cases are provided for a better comparison of suggested and existing control strategies.

Actuator and sensor fault isolation in a class of nonlinear dynamical systems

Fault isolation in dynamical systems is a challenging task due to modeling uncertainty and measurement noise, interactive effects of multiple faults and fault propagation. This paper proposes a unified approach for isolation of multiple actuator or sensor faults in a class of nonlinear uncertain dynamical systems. Actuator and sensor fault isolation are accomplished in two independent modules, that monitor the system and are able to isolate the potential faulty actuator(s) or sensor(s). For the sensor fault isolation (SFI) case, a module is designed which monitors the system and utilizes an adaptive isolation threshold on the output residuals computed via a nonlinear estimation scheme that allows the isolation of single/multiple faulty sensor(s). For the actuator fault isolation (AFI) case, a second module is designed, which utilizes a learning-based scheme for adaptive approximation of faulty actuator(s) and, based on a reasoning decision logic and suitably designed AFI thresholds, the faulty actuator(s) set can be determined. The effectiveness of the proposed fault isolation approach developed in this paper is demonstrated through a simulation example.

Time-Varying Optimal Formation Control for Second-Order Multiagent Systems Based on Neural Network Observer and Reinforcement Learning.

This article addresses a distributed time-varying optimal formation protocol for a class of second-order uncertain nonlinear dynamic multiagent systems (MASs) based on an adaptive neural network (NN) state observer through the backstepping method and simplified reinforcement learning (RL). Each follower agent is subjected to only local information and measurable partial states due to actual sensor limitations. In view of the distributed optimized formation strategic needs, the uncertain nonlinear dynamics and undetectable states may jointly affect the stability of the time-varying cooperative formation control. Furthermore, focusing on Hamilton-Jacobi-Bellman optimization, it is almost incapable of directly dealing with unknown equations. Above uncertainty and immeasurability processed by adaptive state observer and NN simplified RL are further designed to achieve desired second-order formation configuration at the least cost. The optimization protocol can not only solve the undetectable states and realize the prescribed time-varying formation performance on the premise that all the errors are SGUUB, but also prove the stability and update the critics and actors easily. Through the above-mentioned approaches offer an optimal control scheme to address time-varying formation control. Finally, the validity of the theoretical method is proven by the Lyapunov stability theory and digital simulation.

Neuro-adaptive non-singular terminal sliding mode control for distributed fixed-time synchronization of higher-order uncertain multi-agent nonlinear systems

This paper presents a novel design for a distributed fixed-time synchronization controller based on neuro-adaptive non-singular terminal sliding mode control for higher-order multi-agent nonlinear systems. In this study, a distributed control strategy is employed in which networked nonlinear systems, referred to as nodes, communicate through a predefined network topology. Within this framework, a leader-follower configuration is adopted, designating one node as the leader and the others as follower agents. Notably, the dynamics of the agents are assumed to be unknown, leading to the development of a generalized uncertain higher-order nonlinear dynamical system model that accounts for the disturbed dynamics of all agents, including the matched bounded uncertainty with an upper algebraic bound. The control design leverages radial basis function neural networks for approximation, effectively addressing unknown nonlinear terms, such as drift terms and input channels. Importantly, the proposed technique successfully mitigated matched-type uncertainty, resulting in rapid sliding-mode enforcement with reduced chattering and stress. This innovative approach establishes fixed-time synchronization by creating a sliding mode with respect to the sliding surface of networked synchronization mismatches. To underscore the effectiveness of this approach, a simulation example that vividly illustrates its captivating properties and contributions is presented.

Adaptive Critic Control With Knowledge Transfer for Uncertain Nonlinear Dynamical Systems: A Reinforcement Learning Approach

Adaptive Critic Control With Knowledge Transfer for Uncertain Nonlinear Dynamical Systems: A Reinforcement Learning Approach

Online accelerated data‐driven learning for optimal feedback control of discrete‐time partially uncertain systems

SummaryIn this paper, we develop an online learning algorithm for solving the Bellman equation for affine in the control discrete‐time nonlinear uncertain dynamical systems. To ensure accelerated learning of our algorithm in generating optimal control policies, we use an actor‐critic structure predicated on higher‐order tuner laws. More specifically, we construct a Nesterov‐like architecture involving momentum‐based learning laws leading to an accelerated convergence of the optimal control policy. The proposed online learning‐based optimal control framework guarantees uniform ultimate boundedness of the closed‐loop system under the assumption that the system is persistently excited. Finally, two illustrative numerical examples are provided to demonstrate the efficacy of the proposed approach.

Neural Network Trajectory Tracking Control on Electromagnetic Suspension Systems

A new adaptive-like neural control strategy for motion reference trajectory tracking for a nonlinear electromagnetic suspension dynamic system is introduced. Artificial neural networks, differential flatness and sliding modes are strategically integrated in the presented adaptive neural network control design approach. The robustness and efficiency of the magnetic suspension control system on desired smooth position reference profile tracking can be improved in this fashion. A single levitation control parameter is tuned on-line from a neural adaptive perspective by using information of the reference trajectory tracking error signal only. The sliding mode discontinuous control action is approximated by a neural network-based adaptive continuous control function. Control design is firstly developed from theoretical modelling of the nonlinear physical system. Next, dependency on theoretical modelling of the nonlinear dynamic system is substantially reduced by integrating B-spline neural networks and sliding modes in the electromagnetic levitation control technique. On-line accurate estimation of uncertainty, unmeasured external disturbances and uncertain nonlinearities are conveniently evaded. The effective performance of the robust trajectory tracking levitation control approach is depicted for multiple simulation operating scenarios. The capability of active disturbance suppression is furthermore evidenced. The presented B-spline neural network trajectory tracking control design approach based on sliding modes and differential flatness can be extended to other controllable complex uncertain nonlinear dynamic systems where internal and external disturbances represent a relevant issue. Computer simulations and analytical results demonstrate the effective performance of the new adaptive neural control method.

A Simple Learning Approach for Robust Tracking Control of a Class of Dynamical Systems

This paper studies the robust tracking control problem for a class of uncertain nonlinear dynamical systems subject to unknown disturbances. A robust trajectory tracking control law is designed via a simple learning-based control strategy. In the developed design, the cost function based on the desired closed-loop error dynamics is minimized by means of gradient descent technique. A stability proof for the closed-loop nonlinear system is provided based on the pseudo-linear system theory. The learning capability of the developed robust trajectory tracking control law allows the system to mitigate the adverse effects of the uncertainties and disturbances. The numerical simulation results for a planar PPR robot are included to illustrate the effectiveness of the developed control law.

Observer-Based adaptive neural inverse optimal consensus control of nonlinear multiagent systems

The objective of this article is to present an adaptive neural inverse optimal consensus tracking control for nonlinear multi-agent systems (MASs) with unmeasurable states. In the control process, firstly, to approximate the unknown state, a new observer is created which includes the outputs of other agents and their estimated information. The neural network is used to reckon the uncertain nonlinear dynamic systems. Based on a new inverse optimal method and the construction of tuning functions, an adaptive neural inverse optimal consensus tracking controller is proposed, which does not depend on the auxiliary system, thus greatly reducing the computational load. The developed scheme not only insures that all signals of the system are cooperatively semiglobally uniformly ultimately bounded (CSUUB), but also realizes optimal control of all signals. Eventually, two simulations provide the effectiveness of the proposed scheme.

Mittag-Leffler stability of random-order fractional nonlinear uncertain dynamic systems with impulsive effects

Mittag-Leffler stability of random-order fractional nonlinear uncertain dynamic systems with impulsive effects

Uncertain Multiagent Systems With Distributed Constrained Optimization Missions and Event-Triggered Communications: Application to Resource Allocation

This article deals with solving distributed optimization problems with equality constraints by a class of uncertain nonlinear heterogeneous dynamic multiagent systems. It is assumed that each agent with an uncertain dynamic model has limited information about the main problem and limited access to the information of the state variables of the other agents. A distributed algorithm that guarantees cooperatively solving of the constrained optimization problem by the agents is proposed. Via applying this algorithm, the agents do not need to continuously broadcast their data. It is shown that the proposed algorithm can be useful in solving resource allocation problems.

Adaptive Robust Servo Control for Vertical Electric Stabilization System of Tank and Experimental Validation

A tracking stability control problem for the vertical electric stabilization system of moving tank based on adaptive robust servo control is addressed. This paper mainly focuses on two types of possibly fast time-varying but bounded uncertainty within the vertical electric stabilization system: model parameter uncertainty and uncertain nonlinearity. First, the vertical electric stabilization system is constructed as an uncertain nonlinear dynamic system that can reflect the practical mechanics transfer process of the system. Second, the dynamical equation in the form of state space is established by designing the angular tracking error. Third, the comprehensive parameter of system uncertainty is designed to estimate the most conservative effects of uncertainty. Finally, an adaptive robust servo control which can effectively handle the combined effects of complex nonlinearity and uncertainty is proposed. The feasibility of the proposed control strategy under the practical physical condition is validated through the tests on the experimental platform. This paper pioneers the introduction of the internal nonlinearity and uncertainty of the vertical electric stabilization system into the settlement of the tracking stability control problem, and validates the advanced servo control strategy through experiment for the first time.

An Approach for Numerical Solutions of Caputo–Hadamard Uncertain Fractional Differential Equations

This paper is devoted to investigating a numerical scheme for solving the Caputo–Hadamard uncertain fractional differential equations (UFDEs) arising from nonlinear uncertain dynamic systems. In our approach, we define an α-path, which is a link between a Caputo–Hadamard UFDE and a Caputo–Hadamard fractional differential equation and is the inverse uncertainty distribution of a Caputo–Hadamard UFDE. Then, a formula for calculating the expected value of the Caputo–Hadamard UFDE is studied. With the help of the modified predictor–corrector method, some numerical algorithms for the inverse uncertainty distribution and the expected value of the solution of Caputo–Hadamard UFDEs are designed. Corresponding numerical examples are given to confirm the validity and accuracy of the proposed algorithms.

Adaptive Fuzzy Fast Finite-Time Formation Control for Second-Order MASs Based on Capability Boundaries of Agents

This article addresses a new adaptive fuzzy fast finite-time state-constraint protocol for leader-follower formation control. Each agent in uncertain nonlinear dynamic multiagent systems is represented by second-order integrator, which synchronously governs its position and velocity. The fuzzy logic systems are employed to compensate and approximate uncertain functions. On the premise of maintaining formation structure and coupling communication topology, time-varying transformation equations containing exponential signals are introduced to ensure that state capability boundaries for different physical quantities of agents are not violated. It not only guarantees own state performance and collision avoidance among agents, but also realizes the specified transient and steady formation performance. Furthermore, focusing on convergence rate, the adaptive fuzzy fast finite-time strategy is designed that can guarantee all agents will follow the desired formation configuration in fast finite-time. Through the abovementioned approaches provide a good way to improve the convergence and ensure the security for decentralized formation control. Finally, the validity of the theoretical method is proved by fast finite-time stable theory and Lyapunov stability theory. The effectiveness of the protocol is verified by digital simulation and simulation comparison.

Towards a theoretical foundation of PID control for uncertain nonlinear systems

As is well-known, the classical PID control plays a dominating role in various control loops of industrial processes. However, a theory that can explain the rationale why the linear PID can successfully deal with the ubiquitous uncertain nonlinear dynamical systems and a method that can provide explicit design formulae for the PID parameters are still lacking. This paper is a continuation of the authors recent endeavor towards establishing a theoretical foundation of PID for nonlinear uncertain systems. In contrary to most of the existing literature on linear or affine nonlinear systems, we will consider a class of non-affine nonlinear uncertain systems, and will show that a three dimensional parameter set can be constructed explicitly, such that whenever the PID parameters are chosen from this set, the closed-loop systems will be globally stable and the regulation error will converge to zero exponentially fast, under some suitable conditions on the system uncertainties. Moreover, we will also consider the simpler PI and PD control, and provide a necessary and sufficient condition for the choice of the PI parameters for a class of one dimensional non-affine uncertain systems, by applying the Markus–Yamabe theorem in differential equations. These theoretical results show explicitly that the ubiquitous PID control does indeed have strong robustness with respect to both the system nonlinear uncertainties and the selection of the controller parameters.

Efficient Implementation of Continuous-Discrete Extended Kalman Filters for State and Parameter Estimation of Nonlinear Dynamic Systems

The extended Kalman filter (EKF), one of the most popular state estimators for nonlinear uncertain dynamic systems, is considered here in its continuous-discrete (CD) form. The CD version enhances the EKF's accuracy but requires robust and computationally efficient numerical integration methods. The purpose of this article is to present several new efficient integration methods suited to the implementation of the CD-EKF and to compare them to classical implementations such as the Dormand–Prince 4/5 and Euler methods. They are first applied on two proposed nonlinear parameter estimation testbenches and to the stiff Curtiss and Hirschfelder model for state and parameter estimation. Then, they are applied to the estimation of the iron losses of a high-speed permanent-magnet synchronous motor. The results show that for highly nonlinear dynamic systems or for a low to medium sampling frequency, the implicit implementation of the EKF provides a better accuracy and convergence safety. For less nonlinear functions and a higher sampling frequency, classical explicit methods obviously give similar results in a lower computation time.

Bayesian updating of failure probability curves with multiple performance functions of nonlinear structural dynamic systems

System failure often involves multiple failure modes which require considering multiple performance functions. Based on measured system response data, Bayesian updating of multiple failure probability curves needs to be performed. In this paper, a new approach based on extending ISS is proposed which can update multiple failure probability curves by simultaneously considering all the multiple performance functions in one run. A new scheme of how the generated samples are used is proposed to improve the estimation of failure probability curves. Discontinuity issues on the failure curves in ISS are resolved by the proposed method. The proposed method is applied to two illustrative examples involving uncertain model parameters and nonlinear structural dynamic systems subjected to future uncertain earthquake excitations. The computational efficiency of the proposed method is compared with direct application of Subset Simulation. Significant computational savings are observed while the coefficient of variation of the failure probability estimators is kept at the same level.

THE USE OF NONLINEAR DYNAMIC SYSTEM AND DEEP LEARNING IN PRODUCTION CONDITION MONITORING AND PRODUCT QUALITY PREDICTION

To improve the product quality and production efficiency in industrial production, the industrial production process is explored. First, for the production state in industrial production, the radio frequency identification technology (RFID) is used to monitor the state of the nodes in the production process and collect data. Based on the problem of blind area in industrial production, an adaptive controller design based on uncertain nonlinear dynamic systems is introduced; second, based on this, for the problem that product quality is difficult to guarantee in the process flow, the improved multi-layer feedforward network (back propagation, BP) algorithm is used to predict it. Finally, the control diagnosis algorithm based on multi-order least squares support vector machine (LS-SVM) is used to analyze the abnormal cause of the product. The results show that the use of the RFID can monitor the production process in real time and obtain real-time product quality information data. The proposed adaptive controller based on uncertain nonlinear dynamic system has good applicability in solving the problem of blind area. The improved BP neural network algorithm improves the prediction accuracy in the process of product quality prediction and increases the convergence speed of the algorithm. The control diagnosis algorithm based on multi-order least squares that support vector machine (SVM) can classify and learn the historical data of products, as well as perform multi-order search and analysis on the causes of abnormal quality. It helps relevant personnel to carry out process improvement and control, thereby achieving the purpose of improving product quality. Through this investigation, it is found that the method proposed can monitor the state of industrial production in real time and help relevant personnel to control and predict product quality. It can be applied in actual processes.

Zonotopic Filtering for Uncertain Nonlinear Systems: Fundamentals, Implementation Aspects, and Extensions [Applications of Control

This tutorial provides a comprehensive perspective on the study of zonotopic filtering for nonlinear uncertain dynamical systems. Unlike stochastic methods that characterize the states by means of random vectors, set-based observers estimate deterministic sets that are guaranteed to contain the system states. Among many classes of sets, the focus here is on zonotopes due to their appealing properties that pursue a good tradeoff between precision and computational cost. We start by reviewing the fundamentals of zonotopic filtering together with illustrative examples. As nonlinear systems bring up many challenges for this topic, we choose one of the first nonlinear zonotopic observers proposed in the literature to guide a series of discussions. After presenting this algorithm and understanding its limitations, we extend it by incorporating state inequality constraints. In addition, we motivate the use of linear programs when solving minimum-volume intersections. Additionally, we give a practical discussion on the implementation of discrete-time zonotopic algorithms for sampled-data nonlinear dynamical systems. To better introduce the zonotopic approaches, we briefly review unscented Kalman filtering to trace parallels between the stochastic and set-theoretic frameworks. Then the presented algorithms are experimented and compared in a case study involving a quadrotor unmanned aerial vehicle.

Adaptive Fuzzy Consensus Tracking Control for Nonlinear Multiagent Systems with Time-Varying Delays and Constraints

This paper proposes an adaptive fuzzy distributed consensus tracking control scheme for a class of uncertain nonlinear dynamic multiagent systems (MASs) with state time-varying delays and state time-varying constraints. The existing controllers with Lyapunov–Krasovskii functions (LKFs) were not suitable to address time-varying delays and time-varying constraints in nonlinear MASs simultaneously. State constraints further increase the difficulty of controller design and stability analysis, especially for nonstrict feedback systems. Fuzzy logic systems (FLSs) tackle the approximation of unknown dynamics functions and parameters. Especially when the distributed consensus tracking error is infinitely close to the origin, although there is no singular value, it would lead to the rapid growth of control rate or uncontrollability. Constructing appropriate piecewise functions can effectively avoid the above occurrence and accelerate convergence. Based on Lyapunov stability theory and algebraic graph theory, the constructed tracking control can ensure states within defined time-varying constraint bounds and eliminate the influence of time delays. All signals in closed-loop systems can be guaranteed semiglobally uniformly ultimately bounded (SUUB). Finally, the validity of the theoretical method is verified by the simulation.