Fuzzy Approximation Research Articles

Event-Triggered Robust Fuzzy Adaptive Finite-Time Control of Nonlinear Systems With Prescribed Performance

In this article, an event-triggered robust fuzzy adaptive prescribed performance finite-time control strategy is presented for a class of strict-feedback nonlinear systems with external disturbances. The relative-threshold-based event-triggered signal is introduced to reduce communication burden, and the dynamic surface control technique is applied to address the computational complexity problem. A disturbance observer is designed to estimate the compounded disturbances, which are composed of external disturbances and fuzzy approximation errors. The proposed control strategy can guarantee that the closed-loop system is semiglobally practically finite-time stable, and the tracking error converges to a small residual set by incorporating the prescribed performance bound in finite-time. Finally, simulation results are provided to verify the effectiveness of the proposed robust fuzzy control strategy.

Asynchronous Stabilization of Nonlinear Markov Jump Singularly Perturbed Systems via Fuzzy Static Output Feedback Control

This study focuses on the static output feedback control of nonlinear Markov jump singularly perturbed systems within the framework of Takagi–Sugeno fuzzy approximation. From a practical point of view, the phenomenon of asynchronous switching between the plant and the controller is considered and characterized by a finite piecewise-homogenous Markov process. Particularly, for facilitating the controller synthesis, the closed-loop system is transformed into a fuzzy Markov jump singularly perturbed descriptor system by adopting descriptor representation. In order to fully accommodate the system features, an appropriate stochastic Lyapunov function is constructed. Afterwards, by combining Finsler’s lemma, the mean square exponential admissibility of the system is analyzed. The conditions ensuring the existence of the predesigned controller are given and further solved by designing a brief search algorithm. Finally, a typical circuit system is used to demonstrate the application potential of the developed control technology and the effectiveness of the control strategy.

Adaptive Fuzzy Consensus Control Strategy for UAS-Based Load Transportation Tasks

A control strategy for enabling a team of unmanned aircraft systems (UASs) to cooperatively transport a cable-suspended rigid-bodypayload of unknown characteristics is proposed. Considering the rigid-body payload as a disturbance with unmodeled dynamics affecting the aircraft, a fuzzy approximator is designed to identify the lumped disturbances existing in the system, which include also the forces of interaction between the aircraft. Next, a nonlinear controller is designed, which makes use of the approximator to compensate such disturbances. Making use of the fuzzy approximator and the nonlinear controller, a novel consensus-type strategy is proposed, enabling the load transportation task by means of two cooperative quad rotorcraft UASs. The stability of the proposed methodology is demonstrated making use of Lyapunov theory. The nonlinear consensus controller enhanced with fuzzy compensation is compared with respect to a feedback controller with feedforward compensation, demonstrating superior performance of the proposed technique in terms of robustness and stability. Experimental results demonstrate the applicability of the theoretical framework for executing the real-time transportation of a cable-suspended rigid-body payload by means of a team of two cooperative quad rotor UASs.

A new robust decentralized approach based on adaptive fuzzy backstepping method for microgrid secondary voltage control

Secondary microgrid voltage control is one of the most important parts of hierarchical control in islanded mode. By using of fuzzy and adaptive control methods, this paper presents a new method for secondary voltage control of islanded microgrid and also uses the rules of the backstepping method to obtain a control signal. The backstepping method has been used as a robust control technique to overcome the uncertainty and the effects of nonlinear terms, and the fuzzy method has been used to reduce its complexity. The adaptive method also gives fuzzy approximation coefficients. Considering the effect of neighbor agents on each other as uncertainty, a robust decentralized control method has been implemented in islanded microgrid. The simulation results in MATLAB environment show the controller’s ability to eliminate voltage deviations and the appropriate dynamic response.

Adaptive Fuzzy Fault-Tolerant Control for a Class of Nonlinear Systems under Actuator Faults: Application to an Inverted Pendulum

This work investigates a fuzzy direct adaptive fuzzy fault-tolerant Control (FFTC) for a class of perturbed single input single output (SISO) uncertain nonlinear systems. The designed controller consists of two sub-controllers. One is an adaptive unit, and the other is a robust unit, whereas the adaptive unit is devoted to getting rid of the dynamic uncertainties along with the actuator faults, while the second one is developed to deal with fuzzy approximation errors and exogenous disturbances. It is proved that the proposed approach ensures a good tracking performance against faults occurring, uncertainties, and exogenous disturbances, and the stability study of the closed-loop is proved regarding the Lyapunov direct method in order to prove that all signals remain bounded. Simulation results are presented to illustrate the accuracy of the proposed technique.

Fuzzy multiple objective fractional optimization in rough approximation and its aptness to the fixed-charge transportation problem

This article presents a multiple objective fractional fixed-charge transportation problem (MFFTP) in a rough decision-making framework. A transformation procedure is modified to convert non-linear multi-objective transportation problem to its linear version. The parameters of the designed model are considered to be fuzzy. We employ separate kinds of fuzzy scale, i.e., possibility, credibility and necessity measures, to deal with the fuzzy parameters. Using the fuzzy chance-constrained rough approximation (FCRA) technique, we extract the more preferable optimal solution from our suggested MFFTP. The initial result is compared with that of the robust ranking (RR) technique. We also use the theory of rough sets for expanding as well as dividing the feasible domain of the MFFTP to accommodate more information by considering two approximations. Employing these approximations, we introduce two variants, namely, the lower approximation (LA) and the upper approximation (UA), of the suggested MFFTP. Finally, by using these models, we provide the optimal solutions for our proposedproblem. We also associate our MFFTP with a real-world example to showcase its applicability as well as performance. Our core concept of this article is that it tackles an MFFTP using two separate kinds of uncertainty and expands its feasible domain for optimal solutions. Optimal solutions of the designed model (obtained from FCRA technique) belong to two separate regions, namely, “surely region” and “possible region”. The optimal solution which belongs to the “surely region” is better (as these are minimum values) than the one in the “possible region” and other cases. An interpretation of our approach along with offers about the intended future research work are provided at last.

An approach to the study of fuzzy rough prime ideal extensions of semigroups

Fuzzy rough set is a pair of lower and upper approximations of fuzzy set in a fuzzy approximation space. This paper studies the extension of fuzzy rough prime and semi-prime ideals in semigroups. Then we introduce the composition theory on fuzzy rough set. Finally some new characterization of semigroups by the properties of composition on fuzzy rough ideal extension are given.

Granular representation of OWA-based fuzzy rough sets

Granular representations of crisp and fuzzy sets play an important role in rule induction algorithms based on rough set theory. In particular, arbitrary fuzzy sets can be approximated using unions of simple fuzzy sets called granules. These granules, in turn, have a straightforward interpretation in terms of human-readable fuzzy “if..., then...” rules. In this paper, we are considering a fuzzy rough set model based on ordered weighted average (OWA) aggregation over considered values. We show that this robust extension of the classical fuzzy rough set model, which has been applied successfully in various machine learning tasks, also allows for a granular representation. In particular, we prove that when approximations are defined using a directionally convex t-norm and its residual implicator, the OWA-based lower and upper approximations are definable as unions of fuzzy granules. This result has practical implications for rule induction from such fuzzy rough approximations.

Approximations of fuzzy soft sets by fuzzy soft relations with image processing application

Fuzzy soft sets represent a generalization of fuzzy sets with considerable application potential in decision-making problems, optimization, and computer science. In the paper, we use the fact that the fuzzy soft sets powerset theory is defined by the monad and, using this theory, we introduce the concept of fuzzy soft relations defined by the monad. Using that general method, we can also introduce the fuzzy soft approximation of fuzzy soft sets defined by fuzzy soft relations. We show that fuzzy soft sets and fuzzy soft approximations can be naturally used in selective color segmentation problem, where reliable and fully automated methods do not yet exist.

Event-Based Adaptive Fuzzy Fixed-Time Secure Control for Nonlinear CPSs Against Unknown False Data Injection and Backlash-Like Hysteresis

This article investigates the event-triggered adaptive fuzzy fixed-time secure control problem for a class of nonlinear cyber-physical systems subject to unknown deception attacks and backlashlike hysteresis. Based on an improved fractional-order command filtered backstepping method, fuzzy approximation technique, and Nussbaum gain technique, a novel adaptive practical fixed-time secure control scheme is proposed. Theoretical analyses prove that the fixed-time stability of the resulting control system and boundedness of all signals in the closed-loop system can be guaranteed by using the presented resilient controller. Especially, the given convergence time is independent of the initial states of the system and better system performance consisting of higher control accuracy and faster convergence rate can be ensured. Finally, a chemical reaction system is carried out to show the validity and superiority of the developed secure control scheme.

Design of Prescribed Performance Controller for Load Disturbance Interconnected Power System Based on Fuzzy Adaptive Backstepping

The complex structure of interconnected power system and the existence of external load disturbance make it difficult to establish accurate system model and realize stability control with preset performance. In this paper, a fuzzy adaptive controller design method with preset performance is proposed for uncertain interconnected power systems. Firstly, the system approximation model is obtained by using the fuzzy approximation principle, then the preset performance control strategy is introduced to transform the steady-state preset performance of the system into an error performance function. Finally, the adaptive controller design with preset performance is realized based on the Backstepping method. For the validity of the algorithm, the Simulink simulation is carried out. The results show that the controller can make the output of the system uniformly bounded, that is, the tracking error can always be controlled within the specified performance range.

Weighted (λ,μ)-statistical convergence and statistical summability methods of double sequences of fuzzy numbers with application to Korovkin-type fuzzy approximation theorem

The paper aims to investigate different types of weighted statistical convergence and weighted statistical summability of double sequences of fuzzy numbers. Relations connecting statistical convergence and statistical summability have been investigated in the environment of the newly defined classes of double sequences of fuzzy numbers. Relevant inclusion relations and related results concerning the new fuzzy summability methods have also been examined in detail. Finally, a Korovkin-type approximation theorem for double sequences of fuzzy positive linear operators has been established using the most generalized version of the summability method and showed its efficiency by means of a concrete example.

Research on predictive sliding mode control strategy for horizontal vibration of ultra-high-speed elevator car system based on adaptive fuzzy

In order to effectively suppress horizontal vibration of the ultra-high-speed elevator car system. Firstly, considering the nonlinearity of guide shoe, parameter uncertainties, and uncertain external disturbances of the elevator car system, a more practical active control model for horizontal vibration of the 4-DOF ultra-high-speed elevator car system is constructed and the rationality of the established model is verified by real elevator experiment. Secondly, a predictive sliding mode controller based on adaptive fuzzy (PSMC-AF) is proposed to reduce the horizontal vibration of the car system, the predictive sliding mode control law is achieved by optimizing the predictive sliding mode performance index. Simultaneously, in order to decrease the influence of uncertainty of the car system, a fuzzy logic system (FLS) is designed to approximate the compound uncertain disturbance term (CUDT) on-line. Furthermore, the continuous smooth hyperbolic tangent function (HTF) is introduced into the sliding mode switching term to compensate the fuzzy approximation error. The adaptive laws are designed to estimate the error gain and slope parameter, so as to increase the robustness of the system. Finally, numerical simulations are conducted on some representative guide rail excitations and the results are compared to the existing solution and passive system. The analysis has confirmed the effectiveness and robustness of the proposed control method.

A hybrid GA-FS algorithm with fuzzy approximation scheme for solving credibilistic reliability optimization problem

In this paper, credibilistic safety-based design optimization (CSDO) model is firstly formulated as structural reliability optimization framework with credibilistic safety constraint, and fuzzy simulation (FS) approximation is presented to handle the credibilistic safety constraint and approximate the feasible domain. Then, we introduce GA to design the hybrid GA-FS algorithm by integrating FS scheme into GA, which enables the hybrid GA-FS algorithm to solve CSDO model effectively. Finally, two numerical examples are conducted to evidence the applicability of the proposed CSDO formulation and the hybrid GA-FS algorithm in practical engineering. By using out-of-sample back-testing, we provide the comparison between the predetermined reliability level and the backtesting reliability level, which shows the effectiveness of the hybrid GA-FS algorithm for the proposed CSDO formulation.

Hexagonal fuzzy approximation of fuzzy numbers and its applications in MCDM

Numerous research papers and several engineering applications have proved that the fuzzy set theory is an intelligent effective tool to represent complex uncertain information. In fuzzy multi-criteria decision-making (fuzzy MCDM) methods, intelligent information system and fuzzy control-theoretic models, complex qualitative information are extracted from expert’s knowledge as linguistic variables and are modeled by linear/non-linear fuzzy numbers. In numerical computations and experiments, the information/data are fitted by nonlinear functions for better accuracy which may be little hard for further processing to apply in real-life problems. Hence, the study of non-linear fuzzy numbers through triangular and trapezoidal fuzzy numbers is very natural and various researchers have attempted to transform non-linear fuzzy numbers into piecewise linear functions of interval/triangular/trapezoidal in nature by different methods in the past years. But it is noted that the triangular/trapezoidal approximation of nonlinear fuzzy numbers has more loss of information. Therefore, there is a natural need for a better piecewise linear approximation of a given nonlinear fuzzy number without losing much information for better intelligent information modeling. On coincidence, a new notion of Generalized Hexagonal Fuzzy Number has been introduced and its applications on Multi-Criteria Decision-Making problem (MCDM) and Generalized Hexagonal Fully Fuzzy Linear System (GHXFFLS) of equations have been studied by Lakshmana et al. in 2020. Therefore, in this paper, approximation of nonlinear fuzzy numbers into the hexagonal fuzzy numbers which includes trapezoidal, triangular and interval fuzzy numbers as special cases of Hexagonal fuzzy numbers with less loss/gain of information than other existing methods is attempted. Since any fuzzy information is satisfied fully by its modal value/core of that concept, any approximation of that concept is expected to be preserved with same modal value/core. Therefore, in this paper, a stepwise procedure for approximating a non-linear fuzzy number into a new Hexagonal Fuzzy Number that preserves the core of the given fuzzy number is proposed using constrained nonlinear programming model and is illustrated numerically by considering a parabolic fuzzy number. Furthermore, the proposed method is compared for its efficiency on accuracy in terms of loss of information. Finally, some properties of the new hexagonal fuzzy approximation are studied and the applicability of the proposed method is illustrated through the Group MCDM problem using an index matrix (IM).

Adaptive Fuzzy Approximation Control of PV Grid-Connected Inverters

Three-phase inverters are widely used in grid-connected renewable energy systems. This paper presents a new control methodology for grid-connected inverters using an adaptive fuzzy control (AFC) technique. The implementation of the proposed controller does not need prior knowledge of the system mathematical model. The capabilities of the fuzzy system in approximating the nonlinear functions of the grid-connected inverter system are exploited to design the controller. The proposed controller is capable to achieve the control objectives in the presence of both parametric and modelling uncertainties. The control objectives are to regulate the grid power factor and the dc output voltage of the photovoltaic systems. The closed-loop system stability and the updating laws of the controller parameters are determined via Lyapunov analysis. The proposed controller is simulated under different system disturbances, parameters, and modelling uncertainties to validate the effectiveness of the designed controller. For evaluation, the proposed controller is compared with conventional proportional-integral (PI) controller and Takagi–Sugeno–Kang-type probabilistic fuzzy neural network controller (TSKPFNN). The results demonstrated that the proposed AFC showed better performance in terms of response and reduced fluctuations compared to conventional PI controllers and TSKPFNN controllers.

On (IO,O)-fuzzy rough sets based on overlap functions

In the last past years, as a class of continuous binary aggregation functions, there are many scholars keeping a watchful focus on overlap functions for their widely applicability in various actual problems. On the other side, after that rough sets were presented as a formal tool to handle indetermination and inaccuracy in the analysis of data, there arise many works concerning the extension study of rough approximation operators in rough set model. This paper mainly discusses the so-called (IO,O)-fuzzy rough set model obtained through extending the classical conjunction operator in rough approximation operator to overlap functions. Firstly, we investigate a number of essential properties of (IO,O)-fuzzy rough sets, especially for the connections between the obtained new fuzzy rough approximation operators and fuzzy relations. Secondly, we give the topological properties of (IO,O)-fuzzy rough sets. Finally, we show a brief comparison of the (IO,O)-fuzzy rough sets with other common fuzzy rough set models.

Disturbance-observer-based finite-time adaptive fuzzy control for non-triangular switched nonlinear systems with input saturation

This paper investigates finite-time adaptive fuzzy control for a class of non-triangular switched nonlinear systems with asymmetric input saturation and mismatched external disturbances. Firstly, the mismatched external disturbances and fuzzy approximation errors are regarded as total disturbances of the system, which are estimated accurately via the finite-time exact disturbance observer. Secondly, the prescribed performance function is adopted to constrain the system output to meet the prescribed dynamic properties. To cope with the obstacle arising from asymmetric input saturation, a smooth nonlinear function is applied to approximate the actuator nonlinearity by utilizing the mean-value theorem. Thirdly, on the basis of the fast practical finite-time stability criteria, a finite-time adaptive fuzzy control where the adaptive parameter laws contain fractional-order item is developed. Meanwhile, the stability analysis verifies that all the signals in closed-loop system are bounded and error signals can converge to the prescribed small compact sets in finite time. Finally, a simulation example is included to further demonstrate the effectiveness of the proposed control strategies.

Probability granular distance-based fuzzy rough set model

Fuzzy rough set theory is sensitive to noisy samples as the fuzzy approximations are proposed based on sensitive statistics, i.e. minimum and maximum. Here, we develop a robust fuzzy rough set model called probability granular distance-based fuzzy rough sets (PGDFRS), in which the similarity between samples is substituted by that between granules to reduce the impact of noise on the statistical minimum and maximum. The robust principle is to take the probability density values of samples as weights for computing probability distances between granules. By using PGDFRS, a feature selection algorithm is created. This algorithm limits feature selection to two-dimensional space and avoids the difficulty of parameter setting in high-dimensional space. The experimental results indicate that the designed feature selection algorithm is effective and robust. Additionally, it confirms that the proposed PGDFRS model is more robust than some existing fuzzy rough set models.

Adaptive interval type-2 fuzzy control for multi-legged underwater robot with input saturation and full-state constraints

In this paper, a trajectory tracking control strategy is proposed for multi-legged underwater robot in the presence of input saturation and full-state constraints. An anti-windup compensator is introduced to solve the input saturation problem, which can compensate for the saturation difference directly. Besides, the state constrains are addressed by a new state saturation function and the external unknown disturbances are solved by interval type-2 fuzzy neural network approximator. The dynamic surface control method combines with backstepping method is utilised to construct Lyapunov function. The control law and signals of the closed-loop system can be guaranteed to achieve semi-global uniform boundedness. Numerical simulations are presented to show the effectiveness of the proposed algorithm.