Fractional-order Nonlinear Systems Research Articles

Exponential stability of impulsive conformable fractional-order nonlinear differential system with time-varying delay and its applications

This paper investigates the exponential stability of conformable fractional-order nonlinear differential systems with time-varying delay and impulses. The authors utilize the principle of comparison and the Lyapunov function method to establish sufficient conditions that guarantee the exponential stability of a specific class of conformable fractional-order nonlinear differential systems. These findings extend the existing results on systems with integer-order to a certain extent. Furthermore, the paper considers the effect of impulses in the delayed system, which was not addressed in previous literature The authors also apply the criterion for exponential stability to conformable fractional-order neural networks. Finally, two numerical examples are presented to illustrate the effectiveness of the proposed results.

Modeling and Initialization of Nonlinear and Chaotic Fractional Order Systems Based on the Infinite State Representation

Based on the infinite state representation, any linear or nonlinear fractional order differential system can be modelized by a finite-dimension set of integer order differential equations. Consequently, the recurrent issue of the Caputo derivative initialization disappears since the initial conditions of the fractional order system are those of its distributed integer order differential system, as proven by the numerical simulations presented in the paper. Moreover, this technique applies directly to fractional-order chaotic systems, like the Chen system. The true interest of the fractional order approach is to multiply the number of equations to increase the complexity of the chaotic original system, which is essential for the confidentiality of coded communications. Moreover, the sensitivity to initial conditions of this augmented system generalizes the Lorenz approach. Determining the Lyapunov exponents by an experimental technique and with the G.S. spectrum algorithm provides proof of the validity of the infinite state representation approach.

Fractional-order iterative learning control for fractional-order systems with initialization non-repeatability

The effect of initialization non-repeatability on iterative learning control performance for fractional-order systems has not been sufficiently investigated. It is a hidden deficiency that leads directly to the breaking of perfect tracking conditions in both theoretical analysis and real-world applications. Therefore, under the framework of general fractional-order nonlinear systems, this paper proposes an open-close loop Dα-type iterative learning control scheme based on system preconditioning and strictly derives two convergence conditions. By applying the preconditioning optimization strategy based on the short-memory principle, the tracking error due to initialization nonrepetition can converge to any desired range. Compared with the existing results, the proposed iteration scheme fully considers the complexity of the initialization and initial conditions of fractional-order systems, and provides several practical preconditioning methods to improve the tracking efficiency. Two numerical examples are presented to validate the above conclusions.

Event-triggered finite-time fuzzy control approach for fractional-order nonlinear chaotic systems with input delay

In this paper, a novel fuzzy event-triggered control approach with guaranteed performance of practical finite-time tracking convergence is constructed for uncertain fractional-order nonlinear chaotic systems in the presence of time-varying input delay. Firstly, fuzzy logic systems are employed to deal with immeasurable systematic information. Secondly, fractional-order command filters are incorporated to circumvent the problem of “complexity explosion” in backstepping controller construction. Meanwhile, a sequence of fractional-order finite-time compensation signals are introduced to counteract the adverse influence of time-varying input delay rapidly. Furthermore, a novel practical finite-time chaos control approach based on event-triggered strategy is proposed, which not only facilitates the decrease of sampling frequency and controller burden dynamically, but also ensures the convergence of all tracking errors towards a small region around the origin in finite time. Finally, through the simulation research, the transient response performance and the chaos-suppression ability of fractional-order chaotic systems stabilized via the suggested approach are rigorously testified.

Nonlinear Filter-Based Adaptive Output-Feedback Control for Uncertain Fractional-Order Nonlinear Systems with Unknown External Disturbance

This study is devoted to a nonlinear filter-based adaptive fuzzy output-feedback control scheme for uncertain fractional-order (FO) nonlinear systems with unknown external disturbance. Fuzzy logic systems (FLSs) are applied to estimate unknown nonlinear dynamics, and a new FO fuzzy state observer based on a nonlinear disturbance observer is established for simultaneously estimating the unmeasurable states and mixed disturbance. Then, with the aid of auxiliary functions, a novel FO nonlinear filter is given to approximately replace the virtual control functions, together with the corresponding fractional derivative, which not only erases the inherent complexity explosion problem under the framework of backstepping, but also completely compensates for the effects of the boundary errors induced by the constructed filters compared to the previous FO linear filter method. Under certain assumptions, and in line with the FO stability criterion, the stability of the controlled system is ensured. An FO Chua–Hartley simulation study is presented to verify the validity of the proposed method.

Novel results on asymptotic stability and synchronization of fractional-order memristive neural networks with time delays: The [formula omitted] case

This paper investigates the asymptotic stability and synchronization of fractional-order (FO) memristive neural networks with time delays. Based on the FO comparison principle and inverse Laplace transform method, the novel sufficient conditions for the asymptotic stability of a FO nonlinear system are given. Then, based on the above conclusions, the sufficient conditions for the asymptotic stability and synchronization of FO memristive neural networks with time delays are investigated. The results in this paper have a wider coverage of situations and are more practical than the previous related results. Finally, the validity of the results is checked by two examples.

An event-triggered sliding mode control mechanism to exponential consensus of fractional-order descriptor nonlinear multi-agent systems

The problem of leader-following exponential consensus is addressed for fractional-order descriptor nonlinear multi-agent systems by designing an event-triggered sliding mode control strategy. Such an exponential consensus is achieved by defining a suitable sliding surface and a controller under the Mittag–Leffler stability of fractional-order systems. Due to the memory nature of the fractional-order calculus operator, a new condition is established to avoid Zeno behaviors happening, which is different from existing event-triggered schemes. Finally, to illustrate the effectiveness and feasibility of the proposed method, two examples are provided.

Noise-boosted weak signal detection in fractional nonlinear systems enhanced by increasing potential-well width and its application to mechanical fault diagnosis

Noise is seen as the annoying something for weak signal detection, but noise is beneficial to enhance weak signals of interest embedded in noise in nonlinear systems. Moreover, the fractional-order derivative can reinforce noise-boosted weak signal detection. However, the role of varying potential-well depth and width individually on noise-boosted weak signal detection in overdamped and underdamped fractional-order nonlinear systems has not been investigated yet. For this purpose, this paper designs a special bistable potential with varying potential-well depth and width individually to study their influences on weak signal detection numerically. Then, based on simulated conclusions a noise-boosted weak fault diagnosis method enhanced by increasing potential-well width is proposed to enhance weak fault characteristics of machinery, where the amplitude amplification factor is seen as an indicator to quantify the performance of weak signal detection. Finally, some simulations and a bearing fault experiment with an outer race defect were performed to demonstrate the proposed method. Simulated conclusions are identical to experimental ones and indicate that increasing potential-well width would improve noise-boosted weak signal detection but increasing potential-well depth would weaken it regardless of underdamped or overdamped fractional-order nonlinear systems. Nonetheless, increasing the potential-well width endlessly would make the amplitude at the outer race fault characteristic frequency unchanged. That is because the signal with the constant energy would not accelerate the particles to jump across the wide potential wells adequately, where it is not an optimal SR. The compared results with a classical denoising method demonstrate the superiority of the proposed method further.

Event-triggered impulsive control for synchronization in finite time of fractional-order reaction–diffusion complex networks

This paper is concerned with the finite-time synchronization (FTS) issue for fractional-order reaction–diffusion complex networks (RDCNs). A finite-time stability principle, which plays a key role in the synchronization analysis later, is developed for fractional-order nonlinear impulsive system. A novel hybrid controller, which consists of a event-triggered controller and a impulsive controller, is designed to realize the global FTS objective for the considered network. By applying the Lyapunov stability theory and the fractional calculus, the global FTS conditions are addressed in the form of algebraic inequalities. In addition, the exclusion of Zeno behavior are proved for the designed event-triggered strategy. Finally, a numerical example is provided to illustrate the feasibility of the proposed control approach and the correctness of the theoretical results.

Embedded adaptive fractional-order sliding mode control based on TSK fuzzy system for nonlinear fractional-order systems

An adaptive fractional-order sliding mode control (AFOSMC) is proposed to control a nonlinear fractional-order system. This scheme combines the features of sliding mode control and fractional control for improving the response of nonlinear systems. The structure of AFOSMC includes two units: fractional-order sliding mode control (FOSMC) and the tuning unit that employs a certain Takagi–Sugeno–Kang fuzzy logic system for online adjusting the parameters of FOSMC. Tuning the parameters of the FOSMC improves its performance with various control problems. Moreover, stability analysis of the proposed controller is studied using Lyapunov theorem. Finally, the developed control scheme is introduced for controlling a fractional-order gyroscope system. The proposed AFOSMC is implemented practically using a microcontroller where the test is carried out using the hardware-in-the-loop simulation. The practical results indicate the improvements and enhancements introduced by the developed controller under external disturbance, uncertainties and random noise effects.

Cluster consensus of general fractional-order nonlinear multi agent systems via adaptive sliding mode controller

Cluster consensus of general fractional-order nonlinear multi agent systems via adaptive sliding mode controller

Saturated delayed impulsive effects for fractional order nonlinear system with piecewise Caputo derivative and its application

Fractional order nonlinear systems with saturated delayed impulses and short-term memory is investigated in this paper. Models in this paper are described by piecewise Caputo derivative. First, sufficient conditions for local exponential stability are derived based on the fractional order Lyapunov function method and saturated control theory. Then, the optimization model to estimate the domain of attraction is built, which is constrained by some LMIs. Third, the application of synchronization for fractional order complex dynamical networks via distributed saturated delayed impulsive control is studied. Finally, two numerical examples are given to illustrate the effectiveness of the theoretical results.

Adaptive observer-based control of arbitrarily switched fractional-order uncertain nonlinear system subject to input nonlinearities and asymmetric output constraints

This paper addresses an adaptive observer-based control for a switched fractional-order uncertain nonlinear system subject to input nonlinearities and asymmetric output constraints. The switching signal is arbitrary, and the designed controller needs no information regarding this signal. By employing a universal framework, the designed controller can cope with different input nonlinearities in different modes without having any information about their characteristics. It is assumed that only system output is measured, and a designed observer estimates the system states for control implementation. The uncertainties are approximated by intelligent approximators like fuzzy logic systems (FLS) or neural networks (NN). The time-varying asymmetric output constraints have been satisfied using a common barrier Lyapunov function (BLF). Using the common Lyapunov function (CLF) method and the backstepping control technique, the virtual controllers, adaptive law and control input have been designed, and the boundedness of closed-loop signals has been guaranteed. The effectiveness of the designed control scheme has been demonstrated through a numerical and two practical simulation examples.

Lyapunov theorem for stability analysis of nonlinear nabla fractional order systems

This paper addresses the issue of stability analysis of nabla fractional order systems, which is crucial when applying fractional calculus to control problems in engineering applications. In contrast to the conventional Lyapunov theorem, the newly developed theorems presented here are practical and versatile. Firstly, two enhanced stability criteria are established for Mittag-Leffler stability and asymptotic stability, respectively. Subsequently, Lyapunov theorems based on class K functions are derived to assess boundedness and attractiveness. Finally, illustrative examples are also provided to substantiate the theoretical results.

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.

Distributed output-based bipartite consensus for nonlinear fractional-order multi-agent systems with continuous and intermittent communications

This paper studies the distributed output-based bipartite consensus problem in a nonlinear fractional-order multi-agent system (FOMAS) subjected to heterogeneous nonlinear functions and disturbances over a signed directed graph. Two types of distributed bipartite consensus algorithms are proposed to solve the bipartite consensus problem by utilizing continuous and intermittent output measurements, respectively. A novel discontinuous algorithm with a nominal controller is first developed to derive that all agents reach the sliding-mode surface in finite time, and meanwhile, each agent’s dynamics can be described by a linear FOMAS with a nominal controller. When each agent can access its neighbors’ output measurements continuously all the time, a continuous output-based nominal controller is designed for the linear FOMAS to realize bipartite consensus asymptotically. When each agent can only access its neighbors’ output measurements intermittently in some disconnected time intervals, an intermittent output-based nominal controller is further designed, where the communication rate is required to be greater than a derived threshold value. Numerical simulations are performed to validate the effectiveness of the main results.

SOS Approach for Practical Stabilization of Tempered Fractional-Order Power System

Fractional systems have been widely utilized in various fields, such as mathematics, physics and finance, providing a versatile framework for precise measurements and calculations involving partial quantities. This paper aims to develop a novel polynomial controller for a power system (PS) with fractional-order (FO) dynamics. It begins by studying the practical stability of a general class of tempered fractional-order (TFO) nonlinear systems, with broad applicability and potential for expanding its applications. Afterward, a polynomial controller is designed to guarantee the practical stability of the PS, encompassing the standard constant controller as a specific instance. The design conditions for this controller are resolved using the sum of squares (SOS) approach, a powerful technique for guaranteeing stability and control design. To showcase the practical value of the analytical findings, simulations of the PS are conducted utilizing SOSTOOLS.

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.

Quantized iterative learning control for singular nonlinear fractional-order time-delay multi-agent systems with iteration-varying reference trajectories and switching topologies

This paper addresses the consensus tracking of the leader-following singular nonlinear fractional-order system with state time-delay in face of iteration-varying communication topologies and reference trajectories via an iterative learning control scheme. Firstly, by means of the neighbor information, the closed-loop Dα-type quantized iterative learning protocol is constructed based on output tracking errors. Secondly, sufficient conditions of consensus errors between each follower agent and leader agent are given in a fixed time interval and convergence analysis is also presented under the fixed topology and reference trajectory. Thirdly, the extension to iteration-varying reference trajectory tracking and switching topologies cases are investigated. It is shown that the developed protocol can also effectively work along the iteration axis on a finite time interval, although the communication topologies and reference trajectories vary dynamically with respect to iteration. Finally, numerical examples are shown to verify the effectiveness of the obtained results.

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.