Fuzzy Approximation Research Articles

Observer-based dynamic surface control for flexible-joint manipulator system with input saturation and unknown disturbance using type-2 fuzzy neural network

In this paper, the nonlinear disturbance observer (NDO) based dynamic surface control (DSC) with interval type-2 fuzzy neural network (IT2FNN) approximator is proposed for flexible-joint manipulator with the input saturation and unknown nonlinear disturbance. The DSC technique has tremendous advantages in eliminating the ’explosion of complexity’ problem. The IT2FNN approximator is used to deal with parameter uncertainties. The NDO is applied to estimate the unknown external disturbance and compensate the saturation constrain. From Lyapunov stability analysis, it is proved that with the proposed control scheme, all signals of the closed-loop system are semiglobally uniformly ultimately bounded. Simulation results are carried out to demonstrate the effectiveness of the proposed scheme. Compared with the adaptive DSC with neural network (NN) approximator and type-1 fuzzy (T1F) approximator, the tracking error of the proposed control scheme converges to a sufficiently small value.

A predictive approach to adaptive fuzzy sliding-mode control of under-actuated nonlinear systems with input saturation

In this paper, a computationally efficient robust predictive control method is proposed for continuous-time under-actuated SISO systems in the presence of actuator saturation and state-dependent uncertainties. The proposition of this research is to employ the idea of model prediction together with the Adaptive Fuzzy Sliding-Mode Control (AFSMC) for tuning the sliding surface parameters by predicting the anticipated effects of uncertainties. In the proposed scheme, only after the trigger conditions are met, the coefficients of the sliding surface are updated and the AFSMC is applied. Hence, computational complexity can be controlled by adjusting the switching rule. In the AFSMC, a fuzzy system is used to approximate a nonlinear function, and a robust term to compensate for any possible mismatches. An adaptively tuned gain is also applied to the control signal to prevent instability caused by the actuator saturation. Based on the updating sliding surface, fuzzy singletons, the upper bound of the fuzzy approximation error, and the saturation gain are adaptively tuned. Closed-loop stability is shown to be guaranteed using the multiple Lyapunov functions theorem and the Barbalat’s lemma. Finally, the method is applied for the depth control of an Autonomous Underwater Vehicle (AUV), depicting the excellent performance of the proposed method.

Grey Wolf and Weighted Whale Algorithm Optimized IT2 Fuzzy Sliding Mode Backstepping Control with Fractional-Order Command Filter for a Nonlinear Dynamic System

In this study, a fractional-order sliding mode backstepping control method was proposed, which involved the use of a fractional-order command filter, an interval type-2 fuzzy logic system approximation method, and a grey wolf and weighted whale optimization algorithm for multi-input multi-output nonlinear dynamic systems. For designing the stabilizing controls of the backstepping control, a novel fractional-order sliding mode surface was suggested. Further, the transformed errors that occurred during the recursive design steps were easily compensated by the controllers constructed using a new fractional-order command filter. Thus, the differentiation issue of the virtual control in the conventional backstepping control design could be bypassed with a simpler controller structure. Subsequently, the unknown plant dynamics were approximated by an interval type-2 fuzzy logic system. The uncertainties, such as the approximation error and the external disturbance, were compensated by the fractional-order sliding mode control that was added in the backstepping controller. Furthermore, the controller parameters and the fuzzy logic system were optimized via a grey wolf and weighted whale optimization algorithm to obtain a faster tuning process and an improved control performance. Simulation results demonstrated that the fractional-order sliding mode backstepping control scheme provides enhanced control performance over the conventional backstepping control system. Thus, in this paper, a fractional-order sliding mode surface and fractional-order backstepping control are studied, which provide more rapid convergence and enhanced robustness. Furthermore, a hybrid grey wolf and weighted whale optimization algorithm are proposed to provide an improved learning performance than those of conventional grey wolf optimization and weighted whale optimization methods.

Approach-angle-based three-dimensional indirect adaptive fuzzy path following of under-actuated AUV with input saturation

In this paper an approach-angle-based three-dimensional path-following control scheme has been proposed for underactuated Autonomous Underwater Vehicle (AUV) which experiences unknown actuator saturation and environmental disturbance. First, the path-following error dynamic model is derived based on the principle of relative motion which was followed by the design of approach-angle-based guidance law in both horizontal and vertical profiles of AUV to transform the three-dimensional tracking errors into heading angle and elevation angle tracking errors. The kinematic control law is designed based on the Lyapunov theory and backstepping technique. Second, the kinetic controller is designed based on the Lyapunov theory, backstepping technique and fuzzy logic system approximation method. The indirect adaptive fuzzy logic system is applied to approximate unknown smooth functions which are composed of coupled AUV hydrodynamics and complex differentials of desired pitch and yaw velocities. Moreover, the application of fuzzy control completely free from dependence on accurate AUV kinetic model. Considering a disturbance-like term, which is comprised of fuzzy logic system approximation error and bounded ocean disturbance, an adaptive law is designed to estimate the bound of it. Finally, two sets of comparative numerical simulations, including straight path following with different initial posture and spatial helix path following with sudden disturbance, are studied to illustrate the effectiveness and robustness of the proposed control scheme.

Robust adaptive fuzzy control for single-chamber single-population microbial fuel cell

Energy supply from renewable sources is one of the most attractive fields of study in recent years. Ensuring the efficient operation of the microbial fuel cell (MFC) due to its capabilities requires the development of appropriate control strategies. This paper presents a combination of fuzzy and adaptive control methods to control single-chamber single-population MFC. For estimating nonlinear terms as well as the parametric uncertainty in the model, fuzzy approximation method has been used and the effects of disturbances as well as fuzzy approximation gains have been covered by adaptive method. Using Liapunov theory, the robust stability of the system under the proposed controller is shown. The simulation results clearly show the ability of the proposed method to achieve MFC control goals.

Statistical weighted mean summability of double sequences of fuzzy numbers

In this paper, we present statistical weighted mean ( briefly, (N?,p,q,1,1)) summability method for double sequences of fuzzy numbers and give necessary and sufficient Tauberian conditions under which statistical convergence of a double sequence of fuzzy numbers follows from its statistical (N?,p,q,1,1) summability. Furthermore, we apply our new method of summability to prove a fuzzy Korovkin type approximation theorem for a double sequence of fuzzy positive linear operators.

Soft Rough q-Rung Orthopair m-Polar Fuzzy Sets and q-Rung Orthopair m-Polar Fuzzy Soft Rough Sets and Their Applications

The notion of a q-rung orthopair fuzzy soft rough set ( $^{q}ROFSRS$ ) appeared as an extension of q-rung orthopair fuzzy set ( $^{q}ROFS$ ) and q-rung orthopair fuzzy soft set ( $^{q}ROFSS$ ) with the aid of rough set (RS) definition. Thus, $^{q}ROFSRS$ and m-polar fuzzy set ( $_{m}PFS$ ) are convenient to deal with uncertain knowledge which helps us to solve many problems in decision making. In this paper, we define the soft rough q-rung orthopair m-polar fuzzy sets ( $^{\text {q}}RO_{\text {m}}PFS$ ) and q-rung orthopair m-polar fuzzy soft rough sets ( $^{q}RO_{\text {m}}PFSRS$ ) through crisp soft and q- rung orthopair (q-RO) m-polar fuzzy soft approximation space. The related characteristics of these models are also studied. Then, we construct two new algorithms for these models to solve MADM issues. The successful application and corresponding comparative analyses proves that our proposed models are rational and effective.

Topological properties of multigranular rough sets on intuitionistic fuzzy approximation spaces

Two of the extensions of the basic rough set model introduced by Pawlak in 1982 is the rough sets on fuzzy approximation spaces introduced by De et al. in 2003 and its extension rough sets on intuitionistic fuzzy approximation spaces in 2006 by Tripathy. The model of Tripathy is dependent upon an intuitionistic fuzzy proximity relation. In an attempt to handle multiple granulations simultaneously two types of multigranulations called the optimistic and pessimistic multigranulation were introduced by Qian et al. in 2006 and 2010 respectively. The notion of multigranular rough sets over intuitionistic fuzzy approximation spaces were introduced by Tripathy et al. in 2018. Topological properties of rough sets, introduced by Pawlak have been studied extensively for different types of rough sets by Tripathy et al. Continuing this, here we introduce and study the concept of topological properties of multigranular rough sets on intuitionistic fuzzy approximation spaces.

Secondary control of islanded microgrid using robust finite time decentralized method based on adaptive fuzzy sliding mode controller

Secondary voltage control of islanded microgrid is one of the main needs of microgrid hierarchical control from one perspective and the realization of stable smart grids from another aspect. This article aims to achieve several goals simultaneously: First, it provides a decentralized structure for controlling and regulating the voltage of DGs, in which the effects of nonlinear terms on the dynamics of the microgrid system are considered without elimination or linearization. The fuzzy method has been used to approximate the uncertainties, the interactions of agents and nonlinear terms, and the adaptive method has been used to obtain the optimal fuzzy approximation gains. Sliding mode rules are also used to obtain a control signal and using Lyapunov's theory, the finite time and robust stability of the closed-loop control system is guaranteed. The finite time approach ensures the achievement of convergence in a finite and definite time, and the robust approach realizes the dynamic stability of the system against uncertainties and disturbances. The results of the microgrid system simulation show well the proposed controller capability in different operating conditions.

Adaptive Fuzzy Control for Input Restriction Airbreathing Hypersonic Vehicle

For the problem of actuator limitation in the control of aspirating hypersonic vehicle (AHV), this paper proposes a control strategy combined fuzzy approximation system with adaptive control. Firstly, the system is divided into velocity subsystem and altitude subsystem. Then, a new control law is designed. Finally, the uncertainty of the system is estimated based on the universal approximation theory of fuzzy system. This control strategy can both allow the control system with better robustness, and a good restraining effect for the uncertain disturbance. As shown from the simulation results, the method proposed in this paper has more stable than the traditional method in different situations and verifies the effectiveness of the control strategy.

Design of an Adaptive Fuzzy Observer-Based Fault Tolerant Controller for Pneumatic Active Suspension With Displacement Constraint

This paper proposes an adaptive fuzzy observer based fault tolerant controller for a pneumatic active suspension system considering unknown parameters, actuator failures, and displacement constraints. A pneumatic spring is used for a quarter car model to enhance the vibration attenuation performance. Since the pneumatic system contains uncertain nonlinearities, fuzzy logic systems are utilized to approximate unknown nonlinear functions of unmodeled dynamics and various masses of passengers. Besides, a nonlinear disturbance observer is proposed to estimate the effects of the actuator failures, approximation errors, and external disturbances. By utilizing the disturbance estimation and fuzzy approximation techniques, an adaptive fault tolerant control (FTC) is designed to enhance the output performance of the vehicle suspension. Meanwhile, the command filtered scheme is introduced to solve the explosion of complexity problem in the traditional backstepping approach by avoiding virtual controller derivatives. In contrast to previous results, the proposed control can handle the fault tolerant problem and ensure the tracking error of vertical displacement converges into a small-predefined boundary by introducing the prescribed performance function. Moreover, the stability of the closed-loop system is analyzed according to the Lyapunov theory. Finally, comparative simulation examples and experimental studies are performed on the active pneumatic suspension test bench to verify the feasibility of the proposed scheme.

A Certain Class of Relatively Equi-Statistical Fuzzy Approximation Theorems

The aim of this paper is to introduce the notions of relatively deferred Nörlund uniform statistical convergence as well as relatively deferred Norlund point-wise statistical convergence through the dierence operator of fractional order of fuzzy-number-valued sequence of functions, and a type of convergence which lies between aforesaid notions, namely, relatively deferred Nörlund equi-statistical convergence. Also, we investigate the inclusion relations among these aforesaidnotions. As an application point of view, we establish a fuzzy approximation (Korovkin-type) theorem by using our new notion of relatively deferred Norlund equi-statistical convergence and intimate that this result is a non-trivial generalization of several well-established fuzzy Korovkin-type theorems which were presented in earlier works. Moreover, we estimate the fuzzy rate of the relatively deferred Nörlund equi-statistical convergence involving a non-zero scale function by using the fuzzy modulus of continuity.

Topological properties of multigranular rough sets on intuitionistic fuzzy approximation spaces

Two of the extensions of the basic rough set model introduced by Pawlak in 1982 is the rough sets on fuzzy approximation spaces introduced by De et al. in 2003 and its extension rough sets on intuitionistic fuzzy approximation spaces in 2006 by Tripathy. The model of Tripathy is dependent upon an intuitionistic fuzzy proximity relation. In an attempt to handle multiple granulations simultaneously two types of multigranulations called the optimistic and pessimistic multigranulation were introduced by Qian et al. in 2006 and 2010 respectively. The notion of multigranular rough sets over intuitionistic fuzzy approximation spaces were introduced by Tripathy et al. in 2018. Topological properties of rough sets, introduced by Pawlak have been studied extensively for different types of rough sets by Tripathy et al. Continuing this, here we introduce and study the concept of topological properties of multigranular rough sets on intuitionistic fuzzy approximation spaces.

Limited approximate bisimulations and the corresponding rough approximations

To measure the similarity of nodes in the neighboring subgraphs, Milner introduced the notion of k-limited bisimilarity. Recently, as a weaker version of k-limited bisimilarity, the notion of k-limited similarity was proposed and applied to graph pattern matching. (Bi)simulations have been widely used in comparing the behavior of fuzzy transition systems. In order to study the (bi)simulation semantics of labeled fuzzy transition systems in the residuated lattice-valued logic setting, we introduce an extension of labeled approximation spaces, called the quantitative fuzzy approximation spaces (QFASs), whose labels are equipped with a residuated lattice-valued equality relation. In a QFAS, we define a new notion of limited approximate similarity, to quantify to what extent one state is simulated by another in the neighboring subgraphs, and provide its properties. Based on the new notion, we give a definition of limited approximate simulation and discuss its properties. Then we introduce an ordered pair of relations, one on the state (vertex) set (limited approximate simulation) and one on the edge (transition) set induced by the relation on state set, called VE limited approximate simulation in this paper. We also present a new notion of limited approximate bisimilarity in a QFAS, to quantify to what extent two states are similar in the neighboring subgraphs, and give its properties. One main contribution of the paper is to give a condition for two states to be limited approximate bisimilar and investigate the degree of similarity between two states in a QFAS. Finally, we discuss the relationships between the rough approximations based on the underlying crisp relation induced by underlying labeled fuzzy relation and the rough approximations based on limited approximate bisimilarity.

Reducts in single valued neutrosophic ß-covering approximation spaces

The problem of reducts is an interesting issue in all rough set models. In this paper, we propose the concept of the reduct in a single valued neutrosophic β-covering approximation space (SVN-β-CAS). Moreover, reducts in SVN-β-CASs are investigated while adding and removing some objects of the universe, respectively. Firstly, the notion of the reduct in a SVN-β-CAS is presented. It can be seen as the generalization of the reduct in covering and fuzzy β-covering approximation spaces. Then, two new SVN-β-CASs are presented while adding and removing some objects of the original universe. Finally, some properties of reducts of SVN β-coverings are investigated while adding and removing some objects, respectively.

Adaptive fuzzy fault‐tolerant control for non‐linear systems under actuator and sensor faults: the practical fixed‐time stability

This study investigates the adaptive fault-tolerate control for strict-feedback non-linear system, in which actuator and sensor faults may happen simultaneously. Moreover, the additive and multiplicative faults are considered, which cover the bias, drift, loss of accuracy, and loss of effectiveness faults. Under the multiple sensor faults, a modified backstepping technique involving in the faulted states is proposed by utilising the fuzzy approximation to the unknown non-linear functions. Later, the adaptive fuzzy fault-tolerant control strategy is designed to ensure the practical fixed-time stability of the closed-loop system. Meanwhile, the convergence time is independent of the initial states of the system, and the control performance can be ensured for arbitrary bounded initial states in this study. Finally, the Buck converter circuit is employed to demonstrate the effectiveness of the proposed fault-tolerant control scheme.

Disturbance-Observer-Based Fuzzy Control for a Robot Manipulator Using an EMG-Driven Neuromusculoskeletal Model

Robot manipulators have been extensively used in complex environments to complete diverse tasks. The teleoperation control based on human-like adaptivity in the robot manipulator is a growing and challenging field. This paper developed a disturbance-observer-based fuzzy control framework for a robot manipulator using an electromyography- (EMG-) driven neuromusculoskeletal (NMS) model. The motion intention (desired torque) was estimated by the EMG-driven NMS model with EMG signals and joint angles from the user. The desired torque was transmitted into the desired velocity for the robot manipulator system through an admittance filter. In the robot manipulator system, a fuzzy logic system, utilizing an integral Lyapunov function, was applied for robot manipulator systems subject to model uncertainties and external disturbances. To compensate for the external disturbances, fuzzy approximation errors, and nonlinear dynamics, a disturbance observer was integrated into the controller. The developed control algorithm was validated with a 2-DOFs robot manipulator in simulation. The results indicate the proposed control framework is effective and crucial for the applications in robot manipulator control.

Observer-Based Composite Adaptive Type-2 Fuzzy Control for PEMFC Air Supply Systems

Polymer electrolyte membrane fuel cell (PEMFC) air supply systems are usually affected negatively by model uncertainties, external disturbance, and unmeasured variables. In this article, we propose a composite adaptive type-2 fuzzy controller based on a high-gain observer and a disturbance observer for oxygen excess ratio (OER) of PEMFC air supply systems. First, the derivatives of system output, which are unavailable due to limited sensors, are estimated via the high-gain observer. Then, interval type-2 fuzzy logic systems (IT2 FLSs) are adopted to approximate the unknown system dynamics and the disturbance observer is designed to estimate compound disturbance including unknown external disturbance and fuzzy approximation error. Finally, in order to improve the tracking performance, two composite adaptive updating laws are constructed by utilizing the estimated tracking error and the modeling error. Theoretical analysis shows that the system tracking error is uniformly ultimately bounded by Lyapunov stability theory. Numerical simulations and hardware-in-loop experiments are presented to demonstrate the effectiveness and superiority of the proposed controller.

Picture Fuzzy Rough Set and Rough Picture Fuzzy Set on Two Different Universes and Their Applications

The major concern of this article is to propose the notion of picture fuzzy rough sets (PFRSs) over two different universes which depend on δ , ζ , ϑ -cut of picture fuzzy relation ℛ on two different universes (i.e., by combining picture fuzzy sets (PFSs) with rough sets (RSs)). Then, we discuss several interesting properties and related results on the PFRSs. Furthermore, we define some notions related to PFRSs such as (Type-I/Type-II) graded PFRSs, the degree α and β with respect to ℛ δ , ζ , ϑ on PFRSs, and (Type-I/Type-II) generalized PFRSs based on the degree α and β with respect to ℛ δ , ζ , ϑ and investigate the basic properties of above notions. Finally, an approach based on the rough picture fuzzy approximation operators on two different universes in decision-making problem is introduced, and we give an example to show the validity of this approach.

Attribute reduction with fuzzy rough self-information measures

The fuzzy rough set is one of the most effective methods for dealing with the fuzziness and uncertainty of data. However, in most cases this model only considers the information provided by the lower approximation of a decision when it is used to attribute reduction. In a realistic environment, the uncertainty of information is related to lower approximation as well as upper approximation. In this study, we construct four kinds of uncertainty measures by combining fuzzy rough approximations with the concept of self-information. These uncertainty measures can be employed to evaluate the classification ability of attribute subsets. The relationships between these measures are discussed in detail. It is proven that the fourth measure, called relative decision self-information, is better for attribute reduction than the other measures because it considers both the lower and upper approximations of a fuzzy decision. The proposed measures are generalizations of classical measures based on fuzzy rough sets. Finally, we have designed a greedy algorithm for attribute reduction. We validate the effectiveness of the proposed method by comparing the experimental results for efficiency and accuracy with those of three other algorithms using fundamental data.