Fuzzy Approximation Research Articles

Parameter-coupled state space models based on quasi-Gaussian fuzzy approximation

The accuracy of a fuzzy system’s approximation is closely tied to the performance of fuzzy control systems design, while this system’s interpretability depends on the description of a mechanical model using human language. This research introduces a quasi-Gaussian membership function characterized by a pair of parameters to achieve the sensitivity of a triangular membership function along with the interpretability of Gaussian membership functions. Consequently, a two-dimensional (2-D) quasi-Gaussian membership function is derived, and a method for establishing quasi-Gaussian fuzzy systems (QGFS) using a rectangular grid is proposed. After validating the approximation properties using the sine function for the one-dimensional (1-D) and 2-D QGFS, the systems are applied to approximate the depyrogenation tunnel, a significant piece of equipment in the pharmaceutical industry with various mechanical designs. Validation results indicate that the 1-D and 2-D QGFS can achieve an approximation error varying within a ± 5% range. Meanwhile, the 1-D and 2-D QGFSs are applied to mechanical models of the depyrogenation tunnel with satisfactory final approximation results. Lastly, the 2-D QGFS is capable of demonstrating an excellent description of models with coupled parameters.

Fuzzy Fourier Approximations for Uncertain Data Theory and Applications

This study explores the integration of fuzzy logic with Fourier series and transforms to address the challenges posed by uncertainty and imprecision in real-world data. By representing uncertain data through fuzzy numbers and applying fuzzy Fourier approximations, this method enhances the accuracy and robustness of signal processing, image reconstruction, and time-series forecasting, particularly in noisy environments. The comparative analysis demonstrates that fuzzy Fourier methods outperform traditional Fourier techniques in handling uncertainty, while recognizing the computational complexity introduced by fuzzification. The study also explores future research directions, including multi-dimensional data processing, hybrid approaches with machine learning, and the use of fuzzy logic in quantum Fourier transforms. These advancements offer promising solutions for improving data analysis in fields like telecommunications, medical imaging, and financial forecasting, where uncertainty is a critical factor.

A Cooperative Control Method for Wide-Range Maneuvering of Autonomous Aerial Refueling Controllable Drogue

In the realm of autonomous aerial refueling missions for unmanned aerial vehicles (UAVs), the controllable drogue represents a novel approach that significantly enhances both the safety and efficiency of aerial refueling operations. This paper delves into the issue of wide-range maneuverability control for the controllable drogue. Initially, a dynamic model for the variable-length hose–drogue system is presented. Based on this, a cooperative control framework that synergistically utilizes both the hose and the drogue is designed to achieve wide-range maneuverability of the drogue. To address the delay in hose retrieval and release, an open-loop control strategy based on neural networks is proposed. Furthermore, a closed-loop control method utilizing fuzzy approximation and adaptive error estimation is designed to tackle the challenges posed by modeling inaccuracies and uncertainties in aerodynamic parameters. Comparative simulation results show that the proposed control strategy can make the drogue maneuvering range reach more than 6 m. And it can accurately track the time-varying trajectory under the influence of model uncertainty and wind disturbance with an error of less than 0.1 m throughout. This method provides an effective means for achieving wide-range maneuverability control of the controllable drogue in autonomous aerial refueling missions.

Fuzzy adaptive fixed-time control for output-constrained uncertain nonstrict-feedback time-delay systems

This paper deals with the problem of fuzzy adaptive fixed-time control for a class of uncertain nonlinear nonstrict feedback systems with time delays and output constraints. The negative effects of time delays on system performance and stability are overcome by introducing a new Barrier Lyapunov–Krasovskii (BLK) function and an innovative inequality. In addition, the singularity problem that arises in the process of deriving the control and adaptive laws, is solved by the proposed BLK function. The controller is designed in the backstepping framework which ensures that the system output does not violate predefined constraints. The proposed controller needs less information from the system and therefore has less implementation complexity than previous references, which leads to a more straightforward implementation. A fuzzy system approximation is utilized to overcome the uncertainties of the model. It is guaranteed that all closed-loop system signals converge to small neighborhoods around zero in a fixed time. Finally, an example is presented to verify that the proposed method achieves the control objectives.

Disturbance observer-based nonsingular fixed-time fuzzy adaptive event-triggered output feedback control of uncertain nonlinear systems

This article examines the fuzzy adaptive event-triggered output feedback control issue of eliminating fixed-time singularity for a type of uncertain nonlinear systems with disturbance observer structures. By introducing novel bounded time-varying functions (TVFs), a set of more general intermediate-variable-based disturbance observers (IVBDOs) is constructed. Combining the corresponding auxiliary reduced-power function and the quartic Lyapunov method, a co-design scheme for a fuzzy approximation high-order controller and an improved event-triggered mechanism with a judgment indicator threshold is proposed. Especially, the developed framework not only removes the singularity during the fixed-time design process but also further reduces the communication resources. By means of theoretical analysis, the devised solution achieves fixed-time stable control (FTSC) of the system and the tracking error converges to the desired neighborhood around the origin. After verifying the absence of Zeno behavior, the efficacy of the presented methodology is supported by two simulation examples.

Comparison of the Performance of Type-1 and Interval Type-2 Fuzzy PI Controllers for Liquid Level Control in a Coupled Tank System

In this study, liquid level control in a coupled tank system is realized with PI controllers whose parameters are adapted using fuzzy logic approximation methods. Type-1 fuzzy PI control, interval type-2 fuzzy PI control and classical PI+Feedforward control methods are applied to the tank system in real time for different reference liquid levels. The performances of the controllers are compared in terms of system response parameters such as rise time, settling time and overshoot percentage. In addition, the performance evaluation of the controllers over a certain period of time is also compared by calculating integral-based concepts that express error performance metrics such as integral square error (ISE), integral time square error (ITSE), integral absolute error (IAE) and integral time absolute error (ITAE). The obtained results show that the type-2 fuzzy PI controller shows the best performance, followed by type-1 fuzzy PI control and classical PI+FF controller, respectively.

Fuzzy disturbance observer-based fuzzy swing suppression control of a ‘mooring-heavy lift crane-cargo’ coupled system

In this paper, an adaptive fuzzy disturbance observer (FDO)-based control is proposed for the swing suppression of the mooring-heavy lift crane (HLC)-cargo system in the presence of a wave disturbance. First, the dynamic model of the HLC system is determined by employing the Lagrangian method, and the wave force is modeled as an exosystem with unknown terms based on the Jonswap spectrum. Then, based on the HLC model and the wave force exosystem, an FDO is established to determine the wave disturbances, and a fuzzy approximator is developed to estimate the unknown terms. A novel disturbance estimation error observer is first developed to facilitate the parameter adaptive updating law. Subsequently, by augmenting the HLC system and disturbance estimation error system, an FDO-based fuzzy antiswing control method is proposed in terms of the linear matrix inequality technique to suppress the swing. The closed-loop system stability is examined by using the Lyapunov method. Finally, the effectiveness of the proposed method is validated by numerical simulation.

Exploring the CIEDE2000 thresholds for lightness, chroma, and hue differences in dentistry

ObjectiveTo evaluate the perceptibility and acceptability CIEDE2000 (KL:KC:KH) thresholds for lightness, chroma and hue differences in dentistry. MethodA Python-based program was developed to conduct a psychophysical experiment based on visual assessments of computer-simulated images of human teeth. The experiment was performed on a calibrated display. A 40-observer panel: dentists and laypersons (male and female; n=10), evalu- ated three subsets of simulated human teeth: the lightness subset (%∆L00 ≥ 98 %), the chroma subset (%∆C00 ≥ 98 %) and the hue subset (%∆H00 ≥ 98 %), using ∆E00 ≥ 5 units. A Takagi-Sugeno-Kang Fuzzy Approximation model was used as fitting procedure, and 50:50 % lightness, chroma and hue CIEDE2000 (1:1:1) and CIEDE2000 (2:1:1) perceptibility (PT00) and accept- ability (AT00) thresholds were calculated. Data was statistically analyzed using t-test (p < 0.05). ResultsThe 50:50 % PT00 for KL=1 were ∆L00=1.04, ∆C00=1.58 and ∆E00=1.01; and for KL=2 were ∆L00=0.51, ∆C00=1.58 and ∆E00=0.71. The 50:50 % AT00 for KL=1 were ∆L00=2.82, ∆C00=3.04 and ∆E00=2.66; and for KL=2 were ∆L00=1.40, ∆C00=3.04 and ∆E00=1.78. PT00 and AT00 ∆H00 may be considered no computable. A significant difference was found between CIEDE2000(1:1:1) and CIEDE2000(2:1:1), between lightness and chroma metrics, and between observer groups. No differences for lightness and chroma PT00 were found between male and female groups. ConclusionsIt is important to use PT00 and AT00 for lightness, chroma and hue specific to evaluate perceptual sensitivity for color changes in the tooth color space. AT00 for lightness and chroma are influenced by the observer's experience and gender. Males and laypersons show more tolerance for changes in chroma (∆C00) and in lightness (∆L00). SignificanceHue and chroma mismatch are more difficult to be accepted in dental color space.

Generalized Fuzzy Rough Approximations on Hypergroups

In this paper, we define the fuzzy set-valued homomorphisms of the canonical hypergroups as a generalization of fuzzy congruences and investigate some of their features. This structure is an extension of the definition of set-valued homomorphism defined for groups to hypergroups. With this extension, it has become possible to study generalized fuzzy rough approximations in hyperalgebraic structures such as semihypergroups, polygroups, hyperrings, hypermodules, etc. This paper presents the generalized fuzzy rough approximations based on two-universe (I,T)-fuzzy model on canonical hypergroups.

A novel fixed-time fuzzy adaptive tracking control for nonlinear high-order systems with state constraints

A fixed time fuzzy adaptive tracking controller is presented for high-order nonlinear systems, which includes unknown nonlinear functions and satisfies full state constraints. This is a constructive control scheme based on integrator barrier Lyapunov function, unique dominant adaptive parameter related to fuzzy approximation, and complex high-order back-stepping technique, which ensures that the tracking error can be confined to a preassigned neighbourhood including the origin within a fixed time independent of initial values, and all the states remain the given constraints. A practical application to single joint manipulator tracking control is provided, where comparative simulation results are illustrated for two different preassigned tracking errors. Another numerical example is offered to further manifest the feasibility of the theoretical results and the effectiveness of the controller in this paper.

Multigranulation variable-scale fuzzy neighborhood measures and corresponding Choquet-like integrals for feature selection

Choquet-like integrals, a nonlinear fuzzy aggregation function, are diffusely applied in several real problems. However, the corresponding fuzzy measures in them are provided by human intervention and not by data-driven methods. As an effective knowledge representation tool, rough set models in different approximation spaces are established for data processing. Particularly, the multigranulation fuzzy β-covering approximation space has been concerned with only a single scale β∈(0,1]. However, different granulation structures should have different scales. Therefore, establishing a multigranulation approximation space with a variable scale by inducing fuzzy measures through data-driven methods and using corresponding Choquet-like integrals in practical problems is a significant challenge. To address this challenge, the notion of multigranulation variable-scale fuzzy Θ-covering group approximation space is presented here, as well as multigranulation variable-scale fuzzy neighborhood measures in it. Furthermore, Choquet-like integrals with the presented measures are constructed to solve the issue of feature selection (i.e., attribute reduction) under the new approximation spaces. Firstly, the concept of multigranulation variable-scale fuzzy Θ-covering group approximation space is presented, where different fuzzy β-coverings have different scales β∈Θ. Moreover, multigranulation variable-scale fuzzy neighborhood measures in it, as fuzzy measures, are presented. Subsequently, Choquet-like integrals with multigranulation variable-scale fuzzy neighborhood measures are constructed. A novel attribute reduction method under Choquet-like integrals with multigranulation variable-scale fuzzy neighborhood measures is proposed for any decision information table. Finally, the presented method is selected for solving problems of classification and diagnosis of rolling bearing faults. Several public data sets are used to demonstrate the effectiveness and feasibility of the presented methods mentioned above.

Outlier detection for heterogeneous data via fuzzy [formula omitted] covering

Outlier detection is an important research topic in data mining. The existing outlier detection method does not consider ordinal data, such as heterogeneous data. Fuzzy β covering is an information granularity representation tool, which can better fit the distribution of objects in real data and characterize the differences of objects. This paper studies outlier detection for heterogeneous data via fuzzy β covering. Firstly, some new properties of fuzzy β covering are proposed. Then, the fuzzy similarity relation between any two objects in heterogeneous data is defined, and fuzzy information granules are constructed by fuzzy β covering to obtain the new fuzzy approximation accuracy. Next, the outlier factor based on fuzzy β covering is established by integrating the outlier degree and weight function of fuzzy information granules. Moreover, the corresponding algorithm (called FBCOD algorithm) is designed. Finally, the FBCOD algorithm is compared with eight existing outlier detection algorithms on 12 UCI data sets. The experimental results of AUC value, statistical test and F1 measurement show that the FBCOD algorithm has better effectiveness and flexibility than some existing algorithms.

On the lattice of fuzzy rough sets

By the means of lower and upper fuzzy approximations we define quasiorders. Their properties are used to prove our main results. First, we characterize the pairs of fuzzy sets that form fuzzy rough sets w.r.t. a t-similarity relation θ on U, for certain t-norms and implicators. If U is finite or the range of θ and of the fuzzy sets is a fixed finite chain, we establish conditions under which fuzzy rough sets form lattices. We show that this is the case for the min t-norm and any S-implicator defined by an involutive negator and the max co-norm.

Fuzzy finite‐time adaptive control of switched nonlinear systems with input nonlinearities

SummaryThis article addresses the fuzzy finite‐time command filtering tracking control problem for switched nonlinear systems with input nonlinearities via using backstepping technique. An equivalent transformation method is proposed for the purpose of handling the impediment problem caused by the nonlinearity of control input. Finite‐time command filtering control technology can solve the “explosion of complexity” issue, which is caused by the multiple derivation of virtual controllers appeared in the classic backstepping control. A filtering compensation scheme is developed to reduce the filtering error. By combining the fuzzy logic system's approximation ability and the finite‐time theory, a new fast convergence adaptive control scheme is proposed so that the output of the system can converge to a small area near the desired signal and the boundedness of all signals of the controlled system can be assured in finite time. Finally, a simulation example successfully verifies the feasibility of the developed control scheme.

A new control method for leader–follower consensus problem of uncertain constrained nonlinear multi-agent systems

The design of Adaptive Fuzzy Sliding-Mode Control (AFSMC) is discussed in this study for the distributed leader–follower consensus problem of nonlinear uncertain multi-agent systems with input and state constraints. Avoiding the critical circumstances related to the violation of input and state constraints is an important issue in the design of consensus control systems. The limitations of existing adaptive consensus controllers developed for full-state constrained systems motivated the present novel method without restrictive structural assumptions. To convert the original problem to an unconstrained consensus control problem, the proposed control system uses a barrier function-based state transformation method and the input-state linearization methodology. As a result, each agent’s adaptive fuzzy sliding-mode control input, which consists of a fuzzy system and a robust term, is constructed based on the derived representation of the system dynamics. The fuzzy system is used to approximate the dynamic equations of the agent, its neighbors, and maybe the leader in each local feedback control law structure, and the robust term is designed to compensate for the fuzzy approximation error. The magnitudes of the robust terms and the output vectors of the fuzzy systems are also determined using the proposed adaptation laws. The second Lyapunov theorem and Barbalat’s lemma are used to verify the closed-loop system’s asymptotic stability. The effectuality of the proposed methodology is confirmed by numerical simulations for several Autonomous Underwater Vehicles (AUVs). Simulation results show that by means of the proposed AFSMC, the leader–follower consensus is achieved without state constraint violation and performance degradation.

Adaptive event-triggered tracking control for nonlinear systems with prescribed performance: A time-domain mapping approach

This paper proposed an event-triggered adaptive fuzzy prescribed time control strategy for a class of nonlinear systems with uncertain external disturbances and unknown virtual control coefficients under prescribed performance. A novel time-domain mapping technique is proposed to transform the prescribed time tracking control problem into an asymptotic convergence problem. An observer is designed to estimate for the external disturbance and fuzzy approximation error of fuzzy logic systems (FLSs). The problem of “explosion of complexity” is addressed by applying dynamic surface control (DSC) technique to backstepping. An adaptive fuzzy event-triggered strategy is designed using the Lyapunov stability theorem. This strategy ensures that, after time-domain mapping, the tracking error asymptotically converges to zero. Additionally, there is no Zeno behavior and all system signals are bounded. Finally, the effectiveness of the proposed scheme is verified through three simulation examples.

Event-Triggered Adaptive Saturated Fault-Tolerant Control for Unknown Nonlinear Systems With Full State Constraints

This paper investigates the issue of event-triggered adaptive saturated fault-tolerant control (ESFC) for uncertain nonlinear systems with time-varying full state constraints (TFSCs), actuator saturation and faults as well as unknown control direction. A bounded function with an auxiliary variable is constructed by utilizing a novel dynamics of the auxiliary system, which contributes to reducing the adverse impact of actuator saturation. Different from the previous backstepping-based event-triggered control methods such specifications by either using fuzzy approximation or by employing neural approximation techniques, this paper skillfully addresses the unknown nonlinearities, actuator saturation and faults without involving any approximation structures, and thus, we proposes the ESFC on the basis of low-complexity design framework as contributing to communication and computational resource reduction. A rigorous theoretical analysis shows that the proposed control method is an effective way to handle with the problems of actuator saturation and faults, full state constraints, and unknown system uncertainties, while simultaneously simplifying the backstepping design and avoiding the issue of explosion of complexity. The asymptotic stability of the closed-loop system is guaranteed and the Zeno behavior can be effectively removed. We present an application example of a linear motor scenario to illustrate the effectiveness of the method. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —Since state constraints, actuator saturation and faults, unknown mechanism model, and limited bandwidths exist extensively in practical engineering systems, which constantly degrade the operation performance of the plant. To handle these disadvantages, this paper is focus on providing simple but effective ESFC methods to ensure the asymptotic stability and enhance reliability. Compared to existing results, the presented method only uses the state signals of system without using system dynamic functions under mild conditions, which provides a theoretical basis, and has the advantages of low-complexity design, and easy implementation in practical engineering. Preliminary physical experimental comparisons demonstrate that this method is applicable to practical liner-motor platform, and achieves satisfactory control performance.

Adaptive Projection and Fuzzy Tracking Design for Unknown Control Coefficients and References.

This article addresses the tracking control problem of nonlinear pure-feedback systems, where the control coefficients and the dynamics of the references are unknown. Fuzzy-logic systems (FLSs) are used to approximate the unknown control coefficients and at the same time the adaptive projection law is designed to allow each fuzzy approximation to cross zero, which yields that the proposed method avoids the assumption of using Nussbaum function, that is, the unknown control coefficients never cross zeros. Another adaptive law is designed to estimate the unknown reference and then it is intergraded into the saturated tracking control law to achieve the uniformly ultimately bounded (UUB) performance of the resulting closed-loop system. Simulations show the feasibility and effectiveness of the proposed scheme.

A Hybrid Approach for Sustainable and Resilient Farmer Selection in Food Industry: Tunisian Case Study

This paper presents an innovative approach for supplier selection (SS) within the Tunisian dairy industry, with a specific focus on sustainability and resilience. The proposed method is a hybrid model that integrates multi-criteria decision-making (MCDM) with fuzzy logic. Initially, twelve distinct criteria are identified, categorized into social, environmental, economic, and resilience indexes through a combination of literature review and expert input. To establish the importance of these criteria, a fuzzy analytic hierarchy process (FAHP) is employed to calculate their weights. These weights are then integrated into a new method called fuzzy multi-attribute border approximation area comparison (FMABAC) to rank the suppliers. The methodology aims to provide decision-makers with a more intuitive means of expressing their preferences and evaluating supplier criteria, ultimately identifying the most effective dairy farmers. The research concludes by evaluating the results through two sensitivity analyses: one involving varying criteria weights and the other involving a comparative assessment with alternative fuzzy MCDM techniques.

A Fuzzy Fractional Power Series Approximation and Taylor Expansion for Solving Fuzzy Fractional Differential Equation

Fuzzy fractional differential has the strength to capture the senses of memory and uncertainty simultaneously involved in dynamical systems. However, a solution for fuzzy fractional differential equations is not always found regularly. This paper discusses a numerical solution approach for the fuzzy fractional differential equation using power series approximation with a fuzzy fractional counterpart of Taylor’s theorem. Caputo’s definition of the fractional derivative and generalized Hukuhara difference are used to describe the fuzzy differential equation in this paper. Utilization of the generalized Hukuhara difference for the fuzzy valued function ensures the uniqueness and boundedness of the fuzzy solution in parametric form.