Fractal and Fractional

Research

17 pages, 2361 KiB

Open AccessArticle

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

by Zhiyao Ma and Ke Sun

Fractal Fract. 2023, 7(9), 694; https://doi.org/10.3390/fractalfract7090694 - 18 Sep 2023

Cited by 3 | Viewed by 1425

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Robust and Adaptive Control of Fractional-Order Systems)

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20 pages, 840 KiB

Open AccessArticle

Control Design for Fractional Order Leader and Follower Systems with Mixed Time Delays: A Resilience-Based Approach

by Asad Khan, Azmat Ullah Khan Niazi, Waseem Abbasi, Airish Jamil and Jaleel Ahsan Malik

Fractal Fract. 2023, 7(5), 409; https://doi.org/10.3390/fractalfract7050409 - 18 May 2023

Cited by 5 | Viewed by 1269

Abstract

In this article, we consider the problem of resilient base containment control for fractional-order multi-agent systems (FOMASs) with mixed time delays using a reliable and simple approach, where the communication topology among followers is a weighted digraph. A disturbance term is introduced into [...] Read more.

In this article, we consider the problem of resilient base containment control for fractional-order multi-agent systems (FOMASs) with mixed time delays using a reliable and simple approach, where the communication topology among followers is a weighted digraph. A disturbance term is introduced into the delayed and non-delayed controller part to make it more practical. Our method involves proposing algebraic criteria by utilizing non-delayed and delayed protocols, applying the Razumikhin technique and graph theory respectively. The presented method can well overcome the difficulty resulting from fractional calculus, time delays and fractional derivatives. To demonstrate the validity and effectiveness of our findings, we provide an example at the end of our study. Full article

(This article belongs to the Special Issue Robust and Adaptive Control of Fractional-Order Systems)

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24 pages, 6344 KiB

Open AccessArticle

Pitch Angle Control of an Airplane Using Fractional Order Direct Model Reference Adaptive Controllers

by Gustavo E. Ceballos Benavides, Manuel A. Duarte-Mermoud, Marcos E. Orchard and Juan Carlos Travieso-Torres

Fractal Fract. 2023, 7(4), 342; https://doi.org/10.3390/fractalfract7040342 - 20 Apr 2023

Cited by 4 | Viewed by 2309

Abstract

This paper deals with the longitudinal movement control of an airplane (pitch angle) using fractional order adaptive controllers (FOACs). It shows the improvements achieved in the plane’s behavior, in terms of the minimization of a given performance index. At the same time, less [...] Read more.

This paper deals with the longitudinal movement control of an airplane (pitch angle) using fractional order adaptive controllers (FOACs). It shows the improvements achieved in the plane’s behavior, in terms of the minimization of a given performance index. At the same time, less control effort is needed to accomplish the control objectives compared with the classic integer order adaptive controllers (IOACs). In this study, fractional order direct model reference adaptive control (FO-DMRAC) is implemented at the simulation level, and exhibits a better performance compared with the classic integer order (IO) version of the DMRAC (IO-DMRAC). It is also shown that the proposed control strategy for FO-DMRAC reduces the resultant system control structure down to a relative degree 2 system, for which the control implementation is simpler and avoids oscillations during the transient period. Moreover, it is interesting to note that this is the first time that an FOAC with fractional adaptive laws is applied to the longitudinal control of an airplane. A suitable model for the longitudinal movement of the airplane related to the pitch angle θ as the output variable with the lifter angle (δe) as the control variable, is first analyzed and discussed to obtain a reliable mathematical model of the plane. All of the other input variables acting on the plane are considered as perturbations. For certain operating conditions defined by the flight conditions, an FO-DMRAC is designed, simulated, and analyzed. Furthermore, a comparison with the implementation of the classical adaptive general direct control (relative degree ≥ 2) is presented. The boundedness and convergence of all of the signals are theoretically proven based on the new Lemma 3, assuring the boundedness of all internal signals ω(t). Full article

(This article belongs to the Special Issue Robust and Adaptive Control of Fractional-Order Systems)

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23 pages, 815 KiB

Open AccessArticle

Finite-Time Synchronization for Fractional Order Fuzzy Inertial Cellular Neural Networks with Piecewise Activations and Mixed Delays

by Yihong Liu and Yeguo Sun

Fractal Fract. 2023, 7(4), 294; https://doi.org/10.3390/fractalfract7040294 - 29 Mar 2023

Cited by 2 | Viewed by 1428

Abstract

This paper investigates a class of finite-time synchronization problems of fractional order fuzzy inertial cellular neural networks (FFICNNs) with piecewise activations and mixed delays. First, the Caputo FFICNNs are established. A suitable transformation variable is constructed to rewrite FFICNNs with mixed delays into [...] Read more.

This paper investigates a class of finite-time synchronization problems of fractional order fuzzy inertial cellular neural networks (FFICNNs) with piecewise activations and mixed delays. First, the Caputo FFICNNs are established. A suitable transformation variable is constructed to rewrite FFICNNs with mixed delays into a first-order differential system. Secondly, some new effective criteria are constructed on the basis of the finite-time stability theory and Lyapunov functionals to realize the synchronization of the drive-response system. Finally, two numerical simulation examples show that the proposed method is effective. Full article

(This article belongs to the Special Issue Robust and Adaptive Control of Fractional-Order Systems)

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19 pages, 4713 KiB

Open AccessArticle

Compound Adaptive Fuzzy Synchronization Controller Design for Uncertain Fractional-Order Chaotic Systems

by Fengyan Liu and Xiulan Zhang

Fractal Fract. 2022, 6(11), 652; https://doi.org/10.3390/fractalfract6110652 - 5 Nov 2022

Cited by 3 | Viewed by 1205

Abstract

In this paper, the synchronization of two fractional-order chaotic systems with uncertainties and external disturbances is considered. A fuzzy logic system is utilized to estimate uncertain nonlinearity, and its estimation accuracy is improved by constructing a series-parallel model. A disturbance observer is implemented [...] Read more.

In this paper, the synchronization of two fractional-order chaotic systems with uncertainties and external disturbances is considered. A fuzzy logic system is utilized to estimate uncertain nonlinearity, and its estimation accuracy is improved by constructing a series-parallel model. A disturbance observer is implemented to estimate bounded disturbance. To solve the “explosion of complexity” problem in the backstepping scheme, fractional-order command filters are employed to estimate virtual control inputs and their derivatives, and error compensation signals are devised to reduce filtering errors. Based on the fractional-order Lyapurov criterion, the proposed compound adaptive fuzzy backstepping control strategy can guarantee that the synchronization error converges to a small neighborhood of the origin. At last, the validity of the proposed control strategy is verified via a numerical simulation. Full article

(This article belongs to the Special Issue Robust and Adaptive Control of Fractional-Order Systems)

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17 pages, 10979 KiB

Open AccessArticle

Model-Free Fractional-Order Sliding Mode Control of Electric Drive System Based on Nonlinear Disturbance Observer

by Yingxin Yu and Xudong Liu

Fractal Fract. 2022, 6(10), 603; https://doi.org/10.3390/fractalfract6100603 - 16 Oct 2022

Cited by 10 | Viewed by 1863

Abstract

A model-free fractional-order sliding mode control (MFFOSMC) method based on a non-linear disturbance observer is proposed for the electric drive system in this paper. Firstly, the ultra-local model is established by using the mathematical model of electric drive system under parameter perturbation. Then, [...] Read more.

A model-free fractional-order sliding mode control (MFFOSMC) method based on a non-linear disturbance observer is proposed for the electric drive system in this paper. Firstly, the ultra-local model is established by using the mathematical model of electric drive system under parameter perturbation. Then, aiming at reducing the chattering of the sliding mode controller and improving the transient response, a model-free fractional-order sliding mode controller is designed based on fractional-order theory. Next, considering that the traditional sliding mode control can only suppress matched disturbance and that it is sensitive to mismatched disturbance, a non-linear disturbance observer is used to estimate disturbance, and the estimated variables are used in the design of a sliding mode surface to improve the tracking accuracy of the system. Finally, the experiment is completed on an asynchronous motor drive platform. Compared with the model-free integer-order sliding mode control (MFIOSMC), the results show that the proposed method has good dynamic response and strong robustness. Meanwhile, the proposed method reduces the dependence on mathematical models. Full article

(This article belongs to the Special Issue Robust and Adaptive Control of Fractional-Order Systems)

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15 pages, 717 KiB

Open AccessArticle

Local Stabilization of Delayed Fractional-Order Neural Networks Subject to Actuator Saturation

by Yingjie Fan, Xia Huang and Zhen Wang

Fractal Fract. 2022, 6(8), 451; https://doi.org/10.3390/fractalfract6080451 - 19 Aug 2022

Cited by 4 | Viewed by 1525

Abstract

This paper investigates the local stabilization problem of delayed fractional-order neural networks (FNNs) under the influence of actuator saturation. First, the sector condition and dead-zone nonlinear function are specially introduced to characterize the features of the saturation phenomenon. Then, based on the fractional-order [...] Read more.

This paper investigates the local stabilization problem of delayed fractional-order neural networks (FNNs) under the influence of actuator saturation. First, the sector condition and dead-zone nonlinear function are specially introduced to characterize the features of the saturation phenomenon. Then, based on the fractional-order Lyapunov method and the estimation technique of the Mittag–Leffler function, an LMIs-based criterion is derived to guarantee the local stability of closed-loop delayed FNNs subject to actuator saturation. Furthermore, two corresponding convex optimization schemes are proposed to minimize the actuator costs and expand the region of admissible initial values, respectively. At last, two simulation examples are developed to demonstrate the feasibility and effectiveness of the derived results. Full article

(This article belongs to the Special Issue Robust and Adaptive Control of Fractional-Order Systems)

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13 pages, 386 KiB

Open AccessArticle

The Passivity of Uncertain Fractional-Order Neural Networks with Time-Varying Delays

by Song Xu, Heng Liu and Zhimin Han

Fractal Fract. 2022, 6(7), 375; https://doi.org/10.3390/fractalfract6070375 - 2 Jul 2022

Cited by 3 | Viewed by 1492

Abstract

In this paper, we study the passive problem of uncertain fractional-order neural networks (UFONNs) with time-varying delays. First, we give a sufficient condition for the asymptotic stability of UFONNs with bounded time-varying delays by using the fractional-order Razumikhin theorem. Secondly, according to the [...] Read more.

In this paper, we study the passive problem of uncertain fractional-order neural networks (UFONNs) with time-varying delays. First, we give a sufficient condition for the asymptotic stability of UFONNs with bounded time-varying delays by using the fractional-order Razumikhin theorem. Secondly, according to the above stability criteria and some properties of fractional-order calculus, a delay-dependent condition that can guarantee the passivity of UFONNs with time-varying delays is given in the form of a linear matrix inequality (LMI) that can be reasonably solved in polynomial time using the LMI Control Toolbox. These conditions are not only delay-dependent but also order-dependent, and less conservative than some existing work. Finally, the rationality of the research results is proved by simulation. Full article

(This article belongs to the Special Issue Robust and Adaptive Control of Fractional-Order Systems)

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11 pages, 459 KiB

Open AccessArticle

On Sliding Mode Control for Singular Fractional-Order Systems with Matched External Disturbances

by Shubin Song, Bo Meng and Zhen Wang

Fractal Fract. 2022, 6(7), 366; https://doi.org/10.3390/fractalfract6070366 - 30 Jun 2022

Cited by 4 | Viewed by 1339

Abstract

In this paper, we investigate the problem of sliding mode control for singular fractional-order systems that have matched uncertainties. We design an innovative integral sliding mode function and controller based on the normalizable condition. A strict linear matrix inequality-based sufficient condition is obtained [...] Read more.

In this paper, we investigate the problem of sliding mode control for singular fractional-order systems that have matched uncertainties. We design an innovative integral sliding mode function and controller based on the normalizable condition. A strict linear matrix inequality-based sufficient condition is obtained for the system’s stability. The normalizable condition is eliminated by updating and developing the control method, and a sufficient and necessary condition is developed for the admissibility of the system. Lastly, verification of our method’s effectiveness is numerically conducted in two instances. Full article

(This article belongs to the Special Issue Robust and Adaptive Control of Fractional-Order Systems)

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23 pages, 375 KiB

Open AccessArticle

Local and Global Existence and Uniqueness of Solution for Time-Fractional Fuzzy Navier–Stokes Equations

by Kinda Abuasbeh, Ramsha Shafqat, Azmat Ullah Khan Niazi and Muath Awadalla

Fractal Fract. 2022, 6(6), 330; https://doi.org/10.3390/fractalfract6060330 - 14 Jun 2022

Cited by 21 | Viewed by 2439

Abstract

Navier–Stokes (NS) equation, in fluid mechanics, is a partial differential equation that describes the flow of incompressible fluids. We study the fractional derivative by using fractional differential equation by using a mild solution. In this work, anomaly diffusion in fractal media is simulated [...] Read more.

Navier–Stokes (NS) equation, in fluid mechanics, is a partial differential equation that describes the flow of incompressible fluids. We study the fractional derivative by using fractional differential equation by using a mild solution. In this work, anomaly diffusion in fractal media is simulated using the Navier–Stokes equations (NSEs) with time-fractional derivatives of order

β \in (0, 1)

. In

H^{γ, ℘}

, we prove the existence and uniqueness of local and global mild solutions by using fuzzy techniques. Meanwhile, we provide a local moderate solution in Banach space. We further show that classical solutions to such equations exist and are regular in Banach space. Full article

(This article belongs to the Special Issue Robust and Adaptive Control of Fractional-Order Systems)

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