Incommensurate Fractional-order Systems Research Articles

Fuzzy adaptive control for consensus tracking in multiagent systems with incommensurate fractional-order dynamics: Application to power systems

This paper presents a novel approach to address the consensus tracking problem within a class of incommensurate fractional-order nonlinear non-affine systems. Our method employs an adaptive fuzzy technique that integrates newly developed fractional adaptive algorithms based on the Lyapunov method into the controller's design process. This method develops stability based on the global representation of the follower and leader systems, reducing assumptions on the system dynamics to address non-affinity. Additionally, it introduces a simplified approach to designing controllers for incommensurate fractional-order multiagent systems. The proposed controller effectively discerns uncertainties and external disturbances, compelling follower agents to seamlessly follow the desired trajectories set by the leader. Notably, compared to the existing literature, our method exhibits key advantages, including reduced assumptions regarding the system's non-affinity and a simpler design for controlling incommensurate systems. We demonstrate the efficacy of the proposed incommensurate fractional controller through simulations conducted using MATLAB on a fractional-order multiagent power system.

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.

Stability Results for Nonlinear Fractional Differential Equations with Incommensurate Orders

This paper aims to establish some novel conditions for revealing the stability of nonlinear incommensurate fractional-order systems formulated using Caputo differential operator. This will be accomplished through introducing satisfactory proofs for these conditions with the help of using Lyapunov function method. All theoretical findings will be verified numerically by observing the wished stability is occurring on the systems at hand.

A Novel Approach to Modeling Incommensurate Fractional Order Systems Using Fractional Neural Networks

This research explores the application of the Riemann–Liouville fractional sigmoid, briefly RLFσ, activation function in modeling the chaotic dynamics of Chua’s circuit through Multilayer Perceptron (MLP) architecture. Grounded in the context of chaotic systems, the study aims to address the limitations of conventional activation functions in capturing complex relationships within datasets. Employing a structured approach, the methods involve training MLP models with various activation functions, including RLFσ, sigmoid, swish, and proportional Caputo derivative PCσ, and subjecting them to rigorous comparative analyses. The main findings reveal that the proposed RLFσ consistently outperforms traditional counterparts, exhibiting superior accuracy, reduced Mean Squared Error, and faster convergence. Notably, the study extends its investigation to scenarios with reduced dataset sizes and network parameter reductions, demonstrating the robustness and adaptability of RLFσ. The results, supported by convergence curves and CPU training times, underscore the efficiency and practical applicability of the proposed activation function. This research contributes a new perspective on enhancing neural network architectures for system modeling, showcasing the potential of RLFσ in real-world applications.

Event-triggered adaptive neural networks tracking control for incommensurate fractional-order nonlinear systems with external disturbance

The event-triggered adaptive neural networks (NNs) tracking control issue is considered in this paper for a type of incommensurate fractional-order systems (IFOSs) with disturbance. Unlike the existing works for IFOSs, the hypothesis of disturbance-like function and the frequency distributed model are avoided in this study by utilizing the continuity of fractional-order derivative (FOD). Then, based on the radial basis function (RBF) NNs and backstepping method, the adaptive control scheme is proposed to ensure that the reference signal can be tracked by the system output, where the derivative order of adaptive laws does not depend on the one of IFOS. Additionally, to further reduce the burden of communication network, an exponentially convergent term is introduced into traditional dynamic event-triggered mechanism (ETM) and the Zeno behavior is avoided. Finally, two illustrative examples are exhibited to verify the performance of designed control scheme.

Event‐triggered fuzzy adaptive control of incommensurate fractional order nonlinear systems

AbstractThis paper focuses on the event‐triggered control of nonstrict‐feedback incommensurate fractional order systems with external disturbances. To release more communication resources and reduce the triggered time instant, we adopt the relative‐threshold‐based event‐triggered strategy and prove that the Zeno behavior does not exist. We utilize the fuzzy systems to approximate the nonlinear systems and design an adaptive update law to estimate the approximation errors and external disturbances. To conquer the dimension explosion phenomenon in the backstepping process, we design a fractional order filter and apply the dynamic surface control technique. The closed‐loop system is semiglobally stable and the tracking error converges to a small neighborhood of equilibrium point under the proposed control approach. Finally, the validity of the developed controller is verified by the simulation examples.

Fuzzy Adaptive Command-Filter Control of Incommensurate Fractional-Order Nonlinear Systems.

This paper focuses on the command-filter control of nonstrict-feedback incommensurate fractional-order systems. We utilized fuzzy systems to approximate nonlinear systems, and designed an adaptive update law to estimate the approximation errors. To overcome the dimension explosion phenomenon in the backstepping process, we designed a fractional-order filter and applied the command filter control technique. The closed-loop system was semiglobally stable, and the tracking error converged to a small neighbourhood of equilibrium points under the proposed control approach. Lastly, the validity of the developed controller is verified with simulation examples.

Consensus control of incommensurate fractional-order multi-agent systems: An LMI approach

This paper concentrates on the distributed consensus control of heterogeneous fractional-order multi-agent systems (FO-MAS) with interval uncertainties. Unlike previous methods, no restrictive assumptions are considered on the fractional-orders of the agents and they can have non-identical fractional-orders. Therefore, the closed-loop system becomes an incommensurate fractional-order system and its stability analysis is not easy. It makes consensus control more challenging. To design a systematic controller, new Lyapunov-based Linear Matrix Inequality (LMI) conditions are proposed which are suitable to determine the state feedback controller gains. Then, the consensus of heterogeneous fractional-order agents with an observer-based controller is provided. Finally, some numerical examples are provided to verify the effectiveness of our results.

A fractional PI observer for incommensurate fractional order systems under parametric uncertainties

The problem of state observation in incommensurate fractional order systems has been poorly studied. Currently some observers that have been proposed are based on a copy of the system, which causes them to be highly dependent on the system parameters, additionally they are redundant (estimate variables that are available). So this paper proposes a novel fractional observer against parametric uncertainties for a certain type of incommensurate fractional order systems. The fractional observer design is based on a property concerning observability in incommensurate fractional order systems which allows us to construct the observer only considering the available output and its fractional derivatives. On the other hand, the convergence analysis of the observation error is carried out using a particular approach of fractional order systems related to the Global Mittag-Leffler boundedness. We prove that there is a compact set GMLA (Globally Mittag-Leffler Attractive, according to Definition 4) where the system that represents the observation error dynamics is attractive and we also prove that the observation error is uniformly bounded. Additionally, the fractional observer is model-free i.e., a system copy is not required, this gives robustness in spite of parametric uncertainties and it is also reduced order therefore one observer must be designed for each variable that we want to estimate consequently the observer is non-redundant (no estimation of variables that are already available). Moreover, our proposed fractional observer can be designed for commensurate fractional order systems and we also show that if we consider integer derivative order, the proposed fractional observer presents certain properties. Finally, in order to show the effectiveness of the proposed fractional observer, an incommensurate fractional order Rössler hyperchaotic system is considered as a numerical example and an incommensurate fractional model of the COVID-19 pandemic as a real-world application.

Practical finite-time adaptive neural networks control for incommensurate fractional-order nonlinear systems

Practical finite-time adaptive neural networks control for incommensurate fractional-order nonlinear systems

Lag quasi-synchronization of incommensurate fractional-order memristor-based neural networks with nonidentical characteristics via quantized control: A vector fractional Halanay inequality approach

This work realizes lag quasi-synchronization of incommensurate fractional-order memristor-based neural networks (FMNNs) with nonidentical characteristics via quantized control. The motivations behind this research work are threefold: (1) quantized controllers, which generate discrete control signals, can be more easily realized in computers than non-quantized controllers, and can consume smaller communication capacity; (2) incommensurate orders in a single FMNN and nonidentical characteristics in drive-response FMNNs are inescapable due to the differences among the circuit elements used to implement FMNNs; (3) convergence analysis of delayed incommensurate fractional-order nonlinear systems, which is the basis for the derivation of synchronization criterion, has not been handled perfectly. As an effective tool for convergence analysis of delayed incommensurate fractional-order nonlinear systems, especially for estimation of ultimate state bound, a vector fractional Halanay inequality is established at first. Then, a quantized synchronization controller, in which the dead-zone is introduced into some logarithmic quantizers to avoid chattering phenomenon, is designed. By means of vector Lyapunov function together with the newly derived vector fractional Halanay inequality, the synchronization criterion is proved theoretically. Lastly, numerical simulations supplementarily illustrate the correctness of the synchronization criterion. In contrast with the hypotheses in the relevant literature, the hypotheses in this paper are weaker.

Stability analysis of time-delay incommensurate fractional-order systems

In this paper, the stability of time-delay incommensurate fractional-order systems is investigated. Using the Nyquist Theorem, a non-conservative condition is obtained that guarantees the stability of these systems. A formula is proposed which shows the delay values that break this stability condition. Using this formula associated with the assumption that the zero-delay system is stable, one can find the maximum allowable values for delay that guarantee the stability. The proposed method can be used for incommensurate fractional-order systems without any restrictions on fractional-orders or the system dimension. The effectiveness of the proposed method is investigated by applying it on the gene regularity network model.

Analytical solutions of incommensurate fractional differential equation systems with fractional order $ 1 < \alpha, \beta < 2 $ via bivariate Mittag-Leffler functions

<abstract><p>In this paper, we derive the explicit analytical solution of incommensurate fractional differential equation systems with fractional order $ 1 < \alpha, \beta < 2 $. The derivation is extended from a recently published paper by Huseynov et al. in <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>, which is limited for incommensurate fractional order $ 0 < \alpha, \beta < 1 $. The incommensurate fractional differential equation systems were first converted to Volterra integral equations. Then, the Mittag-Leffler function and Picard's successive approximations were used to obtain the analytical solution of incommensurate fractional order systems with $ 1 < \alpha, \beta < 2 $. The solution will be simplified via some combinatorial concepts and bivariate Mittag-Leffler function. Some special cases will be discussed, while some examples will be given at the end of this paper.</p></abstract>

A new memristor-based fractional-order chaotic system

In this paper, a new four-dimensional incommensurate fractional-order system is proposed by introducing an ideal flux-controlled memristor into a three-dimensional chaotic system, and combining it with fractional-order calculus theory, which is solved by using the Adomian decomposition method (ADM). Through theoretical analysis we found the system has numerous equilibrium points. Compared with the original system, the modified system exhibits richer dynamical behaviors. The main manifestations are: (i) Antimonotonicity varying with the initial value. (ii) Three kinds of transient transition behaviors: transient asymptotically-period (A-period), transient chaos, and tri-state transition (chaos-A-period-chaos). (iii) Initial offset boosting behavior. (iv) Hidden extreme multistability. (v) As the order q changes, the system is capable of generating a variety of asymptotically periodic attractors and chaotic attractors. These behaviors above are analyzed in detail by means of numerical simulations such as phase diagram, bifurcation diagram, Lyapunov exponent spectrum (LEs), time-series diagram, and attraction basin. Finally, the system is implemented with a hardware circuit based on a digital signal processor (DSP), which in turn proved the correctness of the numerical analysis simulations and the physical realizability of the system.

Chaos in a Financial System with Fractional Order and Its Control via Sliding Mode

In this paper, the dynamical behaviors and chaos control of a fractional-order financial system are discussed. The lowest fractional order found from which the system generates chaos is 2.49 for the commensurate order case and 2.13 for the incommensurate order case. Also, period-doubling route to chaos was found in this system. The results of this study were validated by the existence of a positive Lyapunov exponent. Besides, in order to control chaos in this fractional-order financial system with uncertain dynamics, a sliding mode controller is derived. The proposed controller stabilizes the commensurate and incommensurate fractional-order systems. Numerical simulations are carried out to verify the analytical results.

Novel convenient conditions for the stability of nonlinear incommensurate fractional-order difference systems

The absence of satisfactory results that address the stability analysis of nonlinear incommensurate Fractional-order Difference Systems (FoDSs) as well as the impossibility of establishing a proper Lyapunov function that helps us to reveal any stability outcome for those systems, are considered some significant nonlinear problems that should be taken into account. To overcome such dilemmas, the present work provides some novel convenient conditions that can be utilized to ensure the asymptotic stability of those systems. In particular, through utilizing some features of the Z-transform scheme, we intend to propose certain results that aim to reveal the stability of the nonlinear incommensurate fractional-order difference system formulated in the sense of the Caputo difference operator. These novel results are completely confirmed by demonstrating the stability of the numerical solutions for two nonlinear difference systems with incommensurate fractional-orders.

Backstepping-Based Adaptive Control of a Class of Uncertain Incommensurate Fractional-Order Nonlinear Systems With External Disturbance

Normally, achieving asymptotic convergence in the presence of unknown arbitrarily bounded external disturbance requires noncontinuous control signals. In this article, an adaptive backstepping controller for a class of incommensurate fractional-order nonlinear systems with parametric uncertainties and external time-varying disturbance is proposed to realize asymptotic perfect output tracking. Appropriate adaptive laws are designed to deal with the uncertainties and unknown bounded disturbance. In the designed adaptive backstepping control scheme, an auxiliary function is adopted to obtain a smooth control signal. It is theoretically shown that asymptotic output tracking to a given reference signal is achieved while ensuring global system stability in the sense that all the closed-loop signals are bounded. Simulation and experimental studies demonstrate the effectiveness of the proposed backstepping control scheme.

Corrigendum to “Chaos in a Financial System with Fractional Order and Its Control via Sliding Mode”

In this paper, the dynamical behaviors and chaos control of a fractional‐order financial system are discussed. The lowest fractional order found from which the system generates chaos is 2.49 for the commensurate order case and 2.57 for the incommensurate order case. Also, the period‐doubling route to chaos was found in this system. The results of this study were validated by the existence of a positive Lyapunov exponent. Besides, in order to control chaos in this fractional‐order financial system with uncertain dynamics, a sliding mode controller is derived. The proposed controller stabilizes the commensurate and incommensurate fractional‐order systems. Numerical simulations are carried out to verify the analytical results.

Robust stability analysis of incommensurate fractional-order systems with time-varying interval uncertainties

This paper focuses on the stability analysis of the incommensurate fractional-order systems describing by the pseudo-state-space form in the presence of interval uncertainties. By using the generalized small-gain theorem and the properties of M-matrices, a condition is achieved for robust finite-gain L2 stability of these systems. In addition, satisfying this condition ensures the asymptotic stability of uncertain fractional-order systems, based on the Caputo definition. Unlike the methods which are based on the eigenvalue argument checking, the proposed robust stability analysis method can be used for systems with irrational fractional orders and time-varying uncertainties. Compared with the Lyapunov based methods, the conservatism due to the lack of relation between stability condition and fractional-order of the system does not occur in the proposed method. An illustrative example is presented to confirm this method does not lead to conservatism generated by reformulation of the interval uncertainty, which is the basis of the commonly used methods in the literature. Finally, the suggested method is employed for robust stability checking of the Sallen–Key filter including fractional capacitors with irrational orders.

On robust stability of incommensurate fractional-order systems

This paper investigates the robust stability of fractional-order systems described in pseudo-state space model with incommensurate fractional orders. An existing non-conservative robust stability criterion for integer-order systems is extended to incommensurate-order fractional systems by using the generalized Nyquist Theorem. Some robust stability conditions for various uncertainty structures are proposed by employing the proposed criterion. Moreover, we focus on the interval uncertainty structure to discuss the conservatism of the common methods. A numerical example is provided to investigate that how the fractional-order of each pseudo-state can affect the robustness of the system. The effectiveness of the proposed methods is compared by applying them on the problem of space tether deployment with fractional-order control law.