Fractional-order Nonlinear Systems Research Articles

Finite-Time Synchronization of Fractional-Order Nonlinear Systems with State-Dependent Delayed Impulse Control

This paper delves into the topics of Finite-Time Stabilization (FTS) and Finite-Time Contractive Stabilization (FTCS) for Fractional-Order Nonlinear Systems (FONSs). To address these issues, we employ a State-Dependent Delayed Impulsive Controller (SDDIC). By leveraging both Lyapunov theory and impulsive control theory, we establish sufficient conditions for achieving stability criteria for fractional-order systems. Initially, we employ the aforementioned sufficient conditions to derive stability criteria for general FONSs within the SDDIC framework, employing Linear Matrix Inequality (LMI) techniques. Furthermore, we apply these theoretical findings to tackle the challenge of finite-time synchronization in fractional-order chaotic systems using the proposed SDDIC. We substantiate the efficacy of these theoretical advancements through numerical simulations that vividly demonstrate their capability to achieve finite-time synchronization in fractional-order cellular neural networks and fractional-order Chua’s circuits. Moreover, we introduce an innovative chaos-based multi-image encryption algorithm, thereby contributing significantly to the field. To ensure the algorithm’s robustness, we subject it to rigorous statistical tests, which confidently affirm its capacity to provide the requisite level of security.

Stochastic disturbance with finite-time chaos stabilization and synchronization for a fractional-order nonautonomous hybrid nonlinear complex system via a sliding mode control

This paper provides a novel approach to studying the chaos control and synchronization of fractional-order hybrid Darcy–Brinkman systems (FOHDBs). We use the truncated Galerkin method to transform the system of partial differential equations (PDEs) into ordinary differential equations (ODEs) based on Fourier modes in a nonlinear low-dimensional Brinkman model. In the nonlinear complex system, hybrid nanoparticles and fluid flow physical characteristics are considered to be external disturbances. As a result, the implications of model uncertainty and external disruptions are fully absorbed into the problem, further complicating it and resulting in highly complex and unpredictable behaviors. In addition, To assure the occurrence of the sliding motion in a finite amount of time, an appropriate robust fractional sliding mode technique is proposed. Sliding motion is shown to occur in finite time using the fractional Lyapunov stability. Following that, we discussed control techniques for achieving infinitesimally close to an equilibrium of chaotic states and tracking errors. It is important to note that FOHDBs and external stochastic disturbances can be controlled using the proposed fractional nonsingular terminal sliding mode control technique. Finally, the effectiveness and applicability of the suggested finite-time control technique are graphically illustrated by the analytical and numerical simulations.

Research on Electric Oil-Pneumatic Active Suspension Based on Fractional-Order PID Position Control.

In this study, an electric oil and gas actuator based on fractional-order PID position feedback control is proposed, through which the damping coefficient of the suspension system is adjusted to realize the active control of the suspension. An FOPID algorithm is used to control the motor's rotational angle to realize the damping adjustment of the suspension system. In this process, the road roughness is collected by the sensors as the criterion of damping adjustment, and the particle swarm algorithm is utilized to find the optimal objective function under different road surface slopes, to obtain the optimal cornering value. According to the mathematical and physical model of the suspension system, the simulation model and the corresponding test platform of this type of suspension system are built. The simulation and experimental results show that the simulation results of the fractional-order nonlinear suspension model are closer to the actual experimental values than those of the traditional linear suspension model, and the accuracy of each performance index is improved by more than 18.5%. The designed active suspension system optimizes the body acceleration, suspension dynamic deflection, and tire dynamic load to 89.8%, 56.7%, and 73.4% of the passive suspension, respectively. It is worth noting that, compared to traditional PID control circuits, the FOPID control circuit designed for motors has an improved control performance. This study provides an effective theoretical and empirical basis for the control and optimization of fractional-order nonlinear suspension systems.

An adaptive unscented particle filter for a nonlinear fractional-order system with unknown fractional-order and unknown parameters

An unscented particle filter (UPF) is proposed for a nonlinear fractional-order system (NFOS) with an unknown order (UO) and unknown parameters. The Grünwald–Letnikov difference is used to discretize the continuous-time NFOS and the corresponding difference equation is acquired. For each sampled particle, a unscented transformation is applied, and the particles are afterwards optimized using a resampling algorithm. Furthermore, the augmented equations of the states, UO, and unknown parameters are established by an augmented vector method. The proposed fractional-order UPF is more accurate in estimating states than the fractional-order unscented Kalman filter and the fractional-order particle filter. Besides, the adaptive fractional-order UPF effectively estimate the UO and unknown parameters. Finally, two numerical examples and a practical example are used to verify the effectiveness of the proposed algorithm.

State estimation-based parameter identification for a class of nonlinear fractional-order systems

Parametric identification is an important part of system theory since knowledge of the parameters allows the analysis and control of the system. The aim of this paper is to propose a novel robust (against measurement noise) parameter identification method for a class of nonlinear fractional-order systems. In order to solve the parametric identification we carry out this problem to a state estimation problem, we introduce a Fractional Algebraic Identifiability (FAI) property which allows to represent the system parameters as a function of the inputs and outputs of the system, this parameter identification method provides an on-line identification process (while the system is operating), we also propose a fractional-order differentiator which allows to reduce the effect of measurement noise as well as to provide the estimation of a fractional-order derivative of the system output. Moreover, we use the Mittag–Leffler boundedness to demonstrate the convergence of this method, a different approach for this stability analysis method is given in this paper. Finally, we illustrate the accuracy and robustness of our proposed method by means of the parametric identification of two nonlinear fractional-order systems: a time-varying nonlinear fractional-order system and a nonlinear fractional-order mathematical model of a simple pendulum.

Examining reachability of fractional dynamical systems with delays in control utilizing ψ-Hilfer pseudo-fractional derivative

This research focuses on the reachability criteria of fractional dynamical systems with delays in control of order ϑ ∈ (0, 1) and type ϱ ∈ [0, 1] in the context of control theory. The study introduces the utilization of the ψ-Hilfer pseudo fractional derivative to describe the dynamics of systems with delays in control. We obtained the necessary and sufficient conditions for the reachability criteria of fractional order linear dynamical systems by employing the positive definiteness of Gramian matrices. We arrived the sufficient conditions for the reachability criteria of fractional order nonlinear dynamical systems using Banach’s fixed point theorem. A few numerical examples are made available for a better understanding of the theoretical conclusions.

Finite-time stability of fractional-order nonlinear systems.

The paper studies the finite-time stability (FTS) of equilibrium points (EPs) in fractional-order nonlinear systems (FONSs). Classifying the EPs into initial EP and finite-time EP innovatively, equations of the EPs in FONSs are investigated comprehensively, and a unified definition depicting the EPs in the FONSs is proposed. Furthermore, sufficient conditions of the FTS of EPs in FONSs are given. The proposed results are verified with an illustrative example. Thus, different from existing works that declare non-existence of finite-time stable equilibria in FONSs, the existence of finite-time stable EPs in FONSs is confirmed in this paper.

Event-Triggered Adaptive Finite-Time Containment Control for Fractional-Order Nonlinear Multiagent Systems.

This article investigates the finite-time containment control problem for fractional-order nonlinear multiagent systems (FOMASs) with event-triggered inputs. First, a new Lyapunov stability lemma is developed, which provides a basic approach for the completely unknown nonlinear fractional-order system to realize finite-time convergence. Considering the containment control for FOMASs, the restricted assumption that the derivatives of leaders' trajectories are known to the followers is removed, and only the boundedness of derivatives is required in this article, whose upper bound need not be known. To reduce the burden of communication, an event-triggered condition consisting of the control input and a decreasing function related to the containment errors is devised, which can provide more design freedom to balance the system performance and communication resources. According to the proposed fractional-order finite-time convergence lemma, a distributed adaptive containment control scheme is developed, such that each follower can be steered to the convex hull spanned by the leaders in finite time. Simulation examples further demonstrate the effectiveness of our proposed method.

Adaptive Fuzzy Echo State Network Control of Fractional-Order Large-Scale Nonlinear Systems With Time-Varying Deferred Constraints

Adaptive Fuzzy Echo State Network Control of Fractional-Order Large-Scale Nonlinear Systems With Time-Varying Deferred Constraints

Evaluation of various controllers in fractional order non-linear systems with actuator fault

Evaluation of various controllers in fractional order non-linear systems with actuator fault

Observer-Based Adaptive Fuzzy Quantized Control for Fractional-Order Nonlinear Time-Delay Systems with Unknown Control Gains

This paper investigates the observer-based adaptive fuzzy quantized control problem for a class of fractional-order nonlinear time-delay systems with unknown control gains based on a modified fractional-order dynamic surface control (FODSC) technique and an indirect Lyapunov method. First, a fractional-order, high-gain state observer is constructed to estimate unavailable state information. Furthermore, the Nussbaum gain technique and a fractional-order filter are adopted to cope with the problem of unknown control gains and to reduce the computational complexity of the conventional recursive procedure, respectively. Moreover, through integration with the compensation mechanism and estimation model, the adaptive fuzzy quantized controllers and adaptive laws are designed to ensure that all the signals of the closed-loop system are bounded. In the end, the proposed controller is applied to a numerical example and a single-machine-infinite bus (SMIB) power system; the simulation results show the validity, superiority, and application potential of the developed control strategy.

Adaptive iterative learning control for a class of parameterized fractional-order systems subjected to backlash-like hysteresis

In this article, a fractional-order adaptive iterative learning control scheme is proposed for a class of parameterized fractional-order systems with unknown control gain and backlash-like hysteresis nonlinearity under disturbance. Based on the sufficient condition for the stability of linear fractional-order systems, a sliding mode surface of tracking errors is constructed to facilitate the controller design and stability analysis. A new boundary layer function is designed by using Mittag-Leffler function to relax the restriction of the identical initial condition of iterative learning control design. The fractional-order differential-type adaptive laws and difference-type learning law are designed to estimate unknown constant parameters and time-varying parameter, respectively. To deal with the influence of backlash-like hysteresis nonlinearity reminder term and unknown bounded external disturbance, a robust control term is designed by employing the hyperbolic tangent function with a convergent series sequence which can guarantee the learning convergence along iteration axis. By constructing Lyapunov-like composite energy function, the stability analysis is presented to prove the convergence of the system output to a small neighborhood of the desired trajectory and the boundedness of all the closed-loop signals. Finally, a simulation example of second-order nonlinear fractional-order system is presented, which demonstrates the effectiveness of the proposed fractional-order adaptive iterative learning control scheme.

Fault-Tolerant Control and Diagnosis for Trajectory Tracking in a Class of Nonlinear Fractional Order Systems

This article deals with the Fault Tolerant Control (FTC) and Fault Diagnosis (FD) for trajectory tracking in a class of nonlinear Fractional Order (FO) systems. The proposed control scheme consists of a set of FO differential equations describing a FO dynamic controller, the estimation of the unknown pseudo-states of the system and faults, the estimators that are designed have different structures, therefore, this scheme can be considered as a hybrid type. In addition, the existence and uniqueness of the solution for the controller is proved using the Banach fixed point theorem. On the other hand, it is shown that the dynamic controller, estimators and the tracking error are globally Mittag-Leffler bounded ensuring the desired trajectory tracking, in consequence a separation principle for nonlinear FO systems is presented. This FO control scheme is illustrated by considering a FO chaotic system as an example.

Leaderless Consensus Control of Fractional-Order Nonlinear Multi-Agent Systems With Measurement Sensitivity and Actuator Attacks

Leaderless Consensus Control of Fractional-Order Nonlinear Multi-Agent Systems With Measurement Sensitivity and Actuator Attacks

Approximate Optimized Backstepping Control of Uncertain Fractional-Order Nonlinear Systems Based on Reinforcement Learning

Approximate Optimized Backstepping Control of Uncertain Fractional-Order Nonlinear Systems Based on Reinforcement Learning

Fuzzy Weight-Based Reinforcement Learning for Event-Triggered Optimal Backstepping Control of Fractional-Order Nonlinear Systems

Fuzzy Weight-Based Reinforcement Learning for Event-Triggered Optimal Backstepping Control of Fractional-Order Nonlinear Systems

NOVEL UNIFIED STABILITY CRITERION FOR FRACTIONAL-ORDER TIME DELAY SYSTEMS WITH STRONG RESISTANCE TO FRACTIONAL ORDERS

In this study, a novel unified stability criterion is first proposed for general fractional-order systems with time delay when the fractional order is from 0 to 1. Such a new unified criterion has the advantage of having an initiative link with the fractional orders. A further advantage is that the corresponding asymptotic stability theorem, derived from the proposed criterion used to analyze the asymptotic stability, is only slightly affected by the change of the fractional order. In addition, the unified stability criterion is applied to general multi-dimensional nonlinear fractional-order systems with time delays, the corresponding asymptotic stability criterion is applied by combining the vector Lyapunov function with the M-matrix method. Compared with the traditional stability criterion, the unified stability criterion is slightly influenced by the changing fractional order and large time delays. The reliability and effectiveness of the novel uniform stability criterion were verified through three representative examples.

Robust tracking control for nonlinear fractional order delayed systems: Modified repetitive control approach

AbstractThis study focuses on the problem of perturbation suppression and tracking control of nonlinear fractional order systems with time‐varying delay. To simplify the control process for such systems, the Takagi–Sugeno fuzzy model is used to describe the nonlinear fractional order delayed systems. Additionally, a modified repetitive controller is applied to develop an output feedback technique for the proposed systems. To ensure system stability, the study establishes new delay‐dependent stabilization conditions using a novel Lyapunov–Krasovskii functional (LKF). These stability conditions are expressed in formation of linear matrix inequalities (LMIs) by adopting a frequency‐distributed model transformation and some slack matrices. The proposed method demonstrates that the system output can track the desired input with ignorable error, while effectively rejecting disturbances. Finally, the efficacy of the proposed method is evaluated through simulations of the modified Chua circuit and the hydraulic turbine governing system (HTGS).

Predefined Time Fuzzy Adaptive Control of Switched Fractional-Order Nonlinear Systems with Input Saturation

Article Predefined Time Fuzzy Adaptive Control of Switched Fractional-Order Nonlinear Systems with Input Saturation Lusong Ding, and Weiwei Sun * 1 Institute of Automation, Qufu Normal University, Qufu 273165, China * Correspondence: wwsun@hotmail.com Received: 31 August 2023 Accepted: 31 October 2023 Published: 21 December 2023 Abstract: This article investigates the predefined-time fuzzy adaptive tracking control problem for a class of nonlinear switched fractional-order systems with input saturation and external disturbances under a nonstrict feedback structure. By combining the backstepping technique and the common Lyapunov function method, a predefined-time switching control method is constructed based on a novel fractional-order auxiliary function. The fuzzy logic system and the adaptive method are introduced to identify unknown compounded continuous functions. Moreover, the issue of calculating explosion and the problem of singularity are tackled through the newly proposed predefined-time and filter-based dynamic surface control. Especially, the construction of a continuous term in the controller eliminates possible chattering. The developed control strategy achieves that the closed-loop system is practically predefined-time stable under arbitrary switchings, where the upper bound of the settling-time can be defined by users in advance. Finally, two simulation examples are illustrated to prove the feasibility and effectiveness of the presented scheme.

Anti-windup scheme-based control for fractional-order systems subject to actuator and sensor saturation

This paper addresses the problem of dissipative-based stabilization for fractional-order nonlinear control systems with uncertainties, time-delay, external disturbances, actuator faults, actuator and sensor saturations. Specifically, a non-fragile reliable dynamic output feedback controller is designed for the purpose of stabilization of the addressed system. Due to the effect of saturation phenomena occurring in both the actuator and sensor, the anti-windup loops framed with the help of the observer design are incorporated into the controller design. Some sufficient conditions are formulated in terms of linear matrix inequalities by choosing a suitable Lyapunov function thereby assuring the dissipative-based stabilization. The derived results are thus validated by two numerical examples including the practical model of two-stage chemical reactor.