Fractional-order Adaptive Law Research Articles

Adaptive fuzzy backstepping secure control for incommensurate fractional order cyber–physical power systems under intermittent denial of service attacks

This paper proposes a fuzzy adaptive control strategy, based on backstepping, for incommensurate fractional-order power systems under denial-of-service (DoS) attacks. To address a wider range of fractional-order systems, this study assumes that the agents’ dynamics where each state have non-identical fractional derivative orders. This assumption brings up additional complexity to the design of the secure control system compared to existing approaches. Our approach tackles the challenge of maintaining system stability during both normal and attack periods by integrating adaptive control technique. The method includes an attack compensator to manage unavailable output measurements during attacks and a fuzzy estimator to approximate unobservable states. It also employs a fractional-order adaptive law and a backstepping controller to ensure performance and security, maintaining bounded tracking errors. MATLAB simulations confirm the effectiveness and robustness of the proposed method, demonstrating its ability to protect power systems from cyber threats while ensuring operational integrity and stability. Compared to prior studies, the presence of an attack compensator reduces both overshoots and convergence time following each attack.

Generalization of Direct Adaptive Control Using Fractional Calculus Applied to Nonlinear Systems

This paper presents a new direct adaptive control (DAC) technique using Caputo’s definition of the fractional-order derivative. This is the first time a fractional-order adaptive law is introduced to work together with an integer-order stable manifold for approximating the uncertainty of a class of nonlinear systems. The DAC approach uses universal function approximators such as multi-layer perceptrons with one hidden layer or fuzzy systems to approximate the controller. This paper presents a new lemma, which elucidates and clarifies the link between the Caputo and the Riemann–Liouville definitions. The introduced lemma is useful in developing a Lyapunov candidate to prove the stability of using the proposed fractional-order adaptive law. This is further explained by a numerical example, which is provided to elucidate the practicality of using the fractional-order derivative for updating the approximator parameters. The main novelty of the results in this paper is a rigorous stability proof of the fractional DAC approach for a class of nonlinear systems that is subjected to unstructured uncertainty and deals with the adaptation mechanism using a traditional integer-order stable manifold. This makes the control scheme easier to implement in practice. The fractional-order adaptation law provides greater degrees of freedom and a potentially larger functional control structure than the conventional adaptive control. Finally, the paper demonstrates that traditional integer-order DAC is a special case of the more general fractional-order DAC scheme introduced here.

Adaptive T-S fuzzy synchronization for uncertain fractional-order chaotic systems with input saturation and disturbance

This paper proposes an adaptive Takagi-Sugeno (T-S) fuzzy control scheme to synchronize two uncertain fractional-order (FO) chaotic systems with input saturations and external disturbances. Under the FO Lyapunov criterion, a novel adaptive T-S FO controller with a simple architecture is developed. System constraints are handled by modeling a saturator as a fraction of unknown functions. Due to the presence of system uncertainties, FO adaptive laws are established to adjust controller parameters. The asymptotic convergence of tracking errors and the boundedness of all closed-loop signals can be ensured. Especially, the feedback control is achieved in T-S FO chaotic systems and the posed method avoids solving linear matrix inequalities, which relieves the calculation burden. The feasibility of the presented method is confirmed through numerical simulations.

A novel adaptive synchronization algorithm for a general class of fractional-order complex-valued systems with unknown parameters, and applications to circuit realization and color image encryption

This article proposes an adaptive synchronization (AS) algorithm to synchronize a general class of fractional-order complex-valued systems with completely unknown parameters, which may appear in physical and engineering problems. The analytical and theoretical concepts of the algorithm rely on the mathematical framework of the Mittag-Leffler global stability of fractional-order systems. A specific control system is established analytically based on the fractional-order adaptive laws of parameters, and the corresponding numerical results are executed to verify the accuracy of the AS algorithm. The proposed synchronization method is evaluated using the fractional-order complex Rabinovich system as an attractive example. The electronic circuits of the new system with different fractional orders are designed. By utilizing the Multisim electronic workbench software, various chaotic/hyperchaotic behaviors have been observed, and a good agreement is found between the numerical results and experimental simulation. In addition, the approximation of the transfer function for different fractional-order are presented. And the corresponding resistor and capacitor values in the chain ship model (CSM) are estimated, which can be utilized in designing electronic circuits for other fractional-order systems. Furthermore, two strategies for encrypting color images are proposed using the AS algorithm and fractional-order adaptive laws of parameters. In the first strategy, the color image is treated as a single package and divided into two vectors. The first vector is embedded into transmitter parameters, while the second vector is injected into the transmitter state signals. In the second strategy, the primary RGB channel components of the original color image are extracted and separated into two vectors, and the same process is followed as in the first strategy. These strategies complicate the decryption task for intruders. Different scales of white Gaussian noise are added to color images to examine the robustness of the proposed color images encryption strategies.

Observer-Based Decentralized Control for Non-Strict-Feedback Fractional-Order Nonlinear Large-Scale Systems With Unknown Dead Zones.

This article addresses the output-feedback decentralized control issue for the fractional-order nonlinear large-scale nonstrict-feedback systems with states immeasurable and unknown dead zones. The unknown nonlinear functions are identified by neural networks (NNs), and immeasurable states are estimated by establishing an NNs' decentralized state observer. The algebraic loop issue is solved by using the property of NN basis functions and designing the fractional-order adaptation laws. In addition, the fractional-order dynamic surface control (FODSC) design technique is introduced in the adaptive backstepping control algorithm to avoid the issue of ``explosion of complexity.'' Then, by treating the nonsymmetric dead zones as the time-varying uncertain systems, an adaptive NNs' output-feedback decentralized control scheme is developed via the fractional-order Lyapunov stability criterion. It is proven that the controlled fractional-order systems are stable, and the tracking and observer errors can converge to a small neighborhood of zero. Two simulation examples are given to confirm the validity of the put forward control scheme.

Event-Triggered Distributed Sliding Mode Control of Fractional-Order Nonlinear Multi-Agent Systems

In this study, the state consensus problem is investigated for a class of nonlinear fractional-order multi-agent systems (FOMASs) by using a dynamics event-triggered sliding mode control approach. The main objective is to steer all agents to some bounded position based on their own information and the information of neighbor agent. Different from the existing results, both asymptotic consensus problem and Zeno-free behavior are ensured simultaneously. To reach this objective, a novel event-triggered sliding mode control approach is proposed, composed of distributed dynamic event-triggered schemes, event-triggered sliding mode controllers, and auxiliary switching functions. Moreover, to implement the distributed control scheme, the fractional-order adaptive law is also developed to tuning the coupling weight, which is addressed in distributed protocol. With the improved distributed control scheme, all signals in the fractional-order closed-loop systems are guaranteed to be consensus and bounded, and a novel approach is developed to avoid the Zeno behavior. Finally, the availability and the effectiveness of the above-mentioned approach are demonstrated by means of a numerical example.

Analysis and design of control systems via parameter‐based approach

Analysis and design of control systems via parameter‐based approach

Neuro-adaptive finite-time control of fractional-order nonlinear systems with multiple objective constraints

<abstract><p>This paper presents a neuro-adaptive finite-time control strategy for uncertain nonstrict-feedback fractional-order nonlinear systems with multiple-objective constraints. To stabilize the uncertain nonlinear fractional-order systems, neural networks (NNs) are employed to identify the unknown nonlinear functions, and dynamic surface control is used to avoid the computational complexity of the backstepping design procedure. The effect caused by the algebraic loop problem can be solved via establishing fractional-order adaptive laws. Introducing a new barrier function, the system output is always limited to the predefined time-varying acceptable range while effectively solving the multi-objective constraint problem. Utilizing fractional-order finite-time stability theory, a finite-time control scheme is constructed to drive the system output to the reference signal in finite time, which ensures better tracking performance. Two examples are given to illustrate the availability and superiority of the presented control scheme.</p></abstract>

A Discrete Fractional Order Adaptive Law for Parameter Estimation and Adaptive Control

In this article, a discrete fractional order adaptive law (DFOAL) is designed based on the Caputo fractional difference to perform parameter estimation of structured uncertainties. The paper provides a rigorous stability analysis of the DFOAL parameter estimation method. The DFOAL is then modified in order to improve parameter estimator performance to show that, under certain conditions, it provides asymptotic convergence to the true parameter values even when the regressor is not persistently exciting. A method to allow for practical implementation of the DFOAL and the modified DFOAL is developed. Finally, the modified DFOAL is used to identify the plant parameters in an indirect adaptive control law for a class of nonlinear discrete-time systems with structured uncertainty.

Adaptive synchronization of fractional-order complex-valued coupled neural networks via direct error method

In this paper, without dividing the fractional-order complex-valued coupled neural networks (FCVCNNs) into two real-valued systems, the problem of synchronization is investigated for FCVCNNs with time-varying coupling strength. To achieve synchronization, by updating coupling strength, two feasible adaptive protocols are designed, 1) a fractional-order adaptive strategy depending on global information; 2) a fractional-order adaptive law relying on the local information based on the connected dominating set theory. Additionally, instead of making the weighted average technique or the solution of the isolated node equation as synchronization reference value, direct error method is adopted to realize synchronization, which offers a flexible way in analysis of FCVCNNs. At last, a numerical simulation is provided to illustrate the validity of theoretical results.

Direct adaptive control for nonlinear systems using a TSK fuzzy echo state network based on fractional-order learning algorithm

This paper presents a new Takagi-Sugeno-Kang fuzzy Echo State Neural Network (TSKFESN) structure to design a direct adaptive control for uncertain SISO nonlinear systems. The proposed TSKFESN structure is based on the echo state neural network framework containing multiple sub-reservoirs. Each sub-reservoir is weighted with a TSK fuzzy rule. The adaptive law of the TSKFESN-based direct adaptive controller is derived by using a fractional-order sliding mode learning algorithm. Moreover, the Lyapunov stability criterion is employed to verify the convergence of the fractional-order adaptive law of the controller parameters. The evaluation of the proposed direct adaptive control scheme is verified using two case studies, the regulation problem of a torsional pendulum and the speed control of a direct current (DC) machine as a real-time application. The simulation and the experimental results show the effectiveness of the proposed control scheme.

Fractional-Order Nonsingular Terminal Sliding Mode Tension Control for the Deployment of Space Tethered Satellite

In this article, a novel fractional-order nonsingular terminal sliding mode (FONTSM) control scheme is developed, which guarantees the desired deployment performance of space tethered satellite (STS) system. The presented control scheme is to stabilize the underactuated deployment mission with only tension regulation. Based on the underactuated model, a practicable integer-order nonsingular terminal sliding mode (IONTSM) controller is designed to realize the deployment of the STS system, with the finite-time stability. Then, to improve the deployment performance, the fractional-order calculus is introduced to raise the FONTSM controller, whose Mittag-Leffler-based uniform ultimate boundedness is proved to ensure the finite-time convergence and steady-state performance. In the FONTSM controller, a fractional-order disturbance observer and a fractional-order adaptive compensation law are proposed, to eliminate the adverse impacts of external disturbances and input saturation, respectively. Moreover, compared to the reported fractional-order sliding mode control strategy of the STS system, a better deployment performance can be obtained by the IONTSM and FONTSM controllers in this work. Finally, the effectiveness of the proposed controllers is verified by groups of deployment simulations of the STS system.

T-S Fuzzy Control of Uncertain Fractional-Order Systems with Time Delay

In this article, the adaptive control of uncertain fractional-order time-delay systems (FOTDSs) with external disturbances is discussed. A Takagi-Sugenu (T-S) fuzzy model with if-then rules is adopted to characterize the dynamic equation of the FOTDS. Besides, a fuzzy adaptive method is proposed to stabilize the model. By utilizing the Lyapunov functions, a robust controller is constructed to stabilize the FOTDS. Due to the uncertainty of system parameters, some fractional-order adaptation laws are designed to update these parameters. At the same time, some if-then rules with linear structure based on the fuzzy T-S adaption concept are established. The designed method not only guarantees that the state of closed-loop system asymptotically converges to origin but also keeps the signal in the FOTDS bounded. Finally, the applicability of the control method is proved by simulation examples.

General type-2 fuzzy multi-switching synchronization of fractional-order chaotic systems

In this study, the problem of multi-switching synchronization of the chaotic systems (CSs) with fractional-order dynamics is considered. Unlike to the most studies, the dynamics of slave chaotic systems are unknown and also the master systems are not fixed but are switched between several synchronization modes. A general type-2 (GT2) fuzzy neural network (FNN) is proposed to approximate the unknown nonlinearities. The stability and robustness is investigated by the Lyapunov theorem and the fractional-order adaptation laws are obtained to optimize the rules of GT2FNNs. The robustness of the proposed scheme against approximation errors and variation of synchronization modes is guaranteed by the proposed compensators. The good performance of schemed method is demonstrated by several simulations and comparison with recent presented control techniques.

Adaptive Fuzzy Consensus Tracking Control for Uncertain Fractional-Order Multiagent Systems With Event-Triggered Input

This article focuses on the problem of adaptive fuzzy consensus tracking control with event-triggered input for a class of fractional-order multiagent systems, where systems are considered to be with unmeasurable states, mismatched uncertain external disturbances, and unknown nonlinearities. An event-triggered mechanism with a decreasing threshold function is proposed in this article, which mitigates communication burden and avoids large control impulse. To achieve better tracking performance in distributed control strategy, a compensating fractional-order adaptation law associated with consensus tracking error is introduced when agents cannot obtain the information of reference trajectory directly. Under the fractional-order stability theory, a novel distributed adaptive event-triggered controller with fuzzy observers is constructed, which ensures the boundedness of tracking errors. A simulation example demonstrates the correctness of the presented scheme.

Neural-Network-Based Adaptive DSC Design for Switched Fractional-Order Nonlinear Systems.

Due to the particularity of the fractional-order derivative definition, the fractional-order control design is more complicated and difficult than the integer-order control design, and it has more practical significance. Therefore, in this article, a novel adaptive switching dynamic surface control (DSC) strategy is first presented for fractional-order nonlinear systems in the nonstrict feedback form with unknown dead zones and arbitrary switchings. In order to avoid the problem of computational complexity and to continuously obtain fractional derivatives for virtual control, the fractional-order DSC technique is applied. The virtual control law, dead-zone input, and the fractional-order adaptive laws are designed based on the fractional-order Lyapunov stability criterion. By combining the universal approximation of neural networks (NNs) and the compensation technique of unknown dead-zones, and stability theory of common Lyapunov function, an adaptive switching DSC controller is developed to ensure the stability of switched fractional-order systems in the presence of unknown dead-zone and arbitrary switchings. Finally, the validity and superiority of the proposed control method are tested by applying chaos suppression of fractional power systems and a numerical example.

Mixed Fractional Order Adaptive Control: Theory and Applications

Abstract In this paper we study the adaptive control problem of integer order plants using fractional order adaptive laws in the controller. The study is based on a general methodology developed recently to establish boundeness and asymptotic behavior of solutions to multi-order systems (set of differential equations with different derivation orders) having multiple time-varying delays. Also it is based on recent results for fractional order systems under the perspective of the so called ”Error Models”. The method relies on vector Lyapunov-like functions and on comparison arguments. Boundedness and convergence of the solutions are theoretically analyzed and applications to fractional adaptive schemes are presented towards the end of the paper, including numerical simulations to verify the analytical results.

Adaptive fractional‐order control of power system frequency in the presence of wind turbine

Inertial and droop control are common methods for improving grid frequency control. Using the first-order derivative of the frequency in the inertial control loop only investigates the rate of frequency variation, while other useful information is not taken into consideration. Accordingly, in this study, the application of a fractional-order derivative (between 1 and 2) in the inertial control loop is proposed to have more useful frequency-based information. To achieve this goal, firstly continued fraction approximations and secondly, the discretization is adopted by using the Tustin Method with Prewarping. Inertia loop and droop loop gain have a significant impact on the operation of the wind turbines and the power system frequency control. Therefore, in this paper, a novel method for the adaptive adjusting of droop and inertia control loops gains in DFIG is proposed. To utilize more frequency error information, fractional-order adaptive law is proposed to update the loop gains and the inertia control loop. In order to have a robust system, which is permanently stable for every fractional order (FO), is used parameter space method. Simulation results demonstrate the proper performance of the FO-based adaptive approach to increase the frequency nadir (FN) and decrease the frequency variations.

Adaptive fractional fuzzy sliding mode control of microgyroscope based on backstepping design.

In this paper, a robust sliding mode control (SMC) based on backstepping technique is studied for a microgyroscope in the presence of unknown model uncertainties and external disturbances using adaptive fuzzy compensator and fractional calculus. At first, the dynamic of microgyroscope is transformed into analogically cascade system to guarantee the application of backstepping design. Then a novel fractional differential sliding surface is proposed which integrates the capacities of the fractional calculus and SMC. In order to reduce the chattering in SMC, a fuzzy logical system is utilized to approximate the external disturbances. In addition, fractional order adaptive laws are derived to estimate the damping and stiffness coefficients and angular velocity online based on Lyapunov stability theory which also guarantees the stability of the closed loop system. Finally, simulation results signify the robustness and effectiveness of the proposed control schemes and the comparison of root mean square error under different fractional orders and integer order are given to demonstrate the better performance of proposed controller.

Fractional-order fuzzy adaptive controller design for uncertain robotic manipulators

A sliding mode adaptive fractional fuzzy control is provided in this article to achieve the trajectory tracking control of uncertain robotic manipulators. By adaptive fractional fuzzy control, we mean that fuzzy parameters are updated through fractional-order adaptation laws. The main idea of this work consists in using fractional input to control complex integer-order nonlinear systems. An adaptive fractional fuzzy control that guarantees tracking errors tend to an arbitrary small region is established. To facilitate the stability analysis, fractional-order integral Lyapunov functions are proposed, and the integer-order Lyapunov stability criterion is used. Finally, simulation results are presented to show the effectiveness of the proposed method.