Showing papers in "IEEE Transactions on Fuzzy Systems in 2011"

A Fuzzy Association Rule-Based Classification Model for High-Dimensional Problems With Genetic Rule Selection and Lateral Tuning

Abstract: The inductive learning of fuzzy rule-based classification systems suffers from exponential growth of the fuzzy rule search space when the number of patterns and/or variables becomes high. This growth makes the learning process more difficult and, in most cases, it leads to problems of scalability (in terms of the time and memory consumed) and/or complexity (with respect to the number of rules obtained and the number of variables included in each rule). In this paper, we propose a fuzzy association rule-based classification method for high-dimensional problems, which is based on three stages to obtain an accurate and compact fuzzy rule-based classifier with a low computational cost. This method limits the order of the associations in the association rule extraction and considers the use of subgroup discovery, which is based on an improved weighted relative accuracy measure to preselect the most interesting rules before a genetic postprocessing process for rule selection and parameter tuning. The results that are obtained more than 26 real-world datasets of different sizes and with different numbers of variables demonstrate the effectiveness of the proposed approach.

Adaptive Output-Feedback Fuzzy Tracking Control for a Class of Nonlinear Systems

Abstract: This paper is concerned with the problem of adaptive fuzzy tracking control via output feedback for a class of uncertain single-input single-output (SISO) strict-feedback nonlinear systems. The dynamic feedback strategy begins with an input-driven filter. By utilizing fuzzy logic systems to approximate unknown and desired control input signals directly instead of the unknown nonlinear functions, an output-feedback fuzzy tracking controller is designed via a backstepping approach. It is shown that the proposed fuzzy adaptive output controller can guarantee that all the signals remain bounded and that the tracking error converges to a small neighborhood of the origin. Simulations results are presented to demonstrate the effectiveness of the proposed methods.

Model Approximation for Discrete-Time State-Delay Systems in the T–S Fuzzy Framework

Abstract: This paper is concerned with the problem of H∞ model approximation for discrete-time Takagi-Sugeno (T-S) fuzzy time-delay systems. For a given stable T- S fuzzy system, our attention is focused on the construction of a reduced-order model, which not only approximates the original system well in an H∞ performance but is also translated into a linear lower dimensional system. By applying the delay partitioning approach, a delay-dependent sufficient condition is proposed for the asymptotic stability with an H∞ error performance for the error system. Then, the H∞ model approximation problem is solved by using the projection approach, which casts the model approximation into a sequential minimization problem subject to linear matrix inequality (LMI) constraints by employing the cone complementary linearization algorithm. Moreover, by further extending the results, H∞ model approximation with special structures is obtained, i.e., delay-free model and zero-order model. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed methods.

A Kernel Fuzzy c-Means Clustering-Based Fuzzy Support Vector Machine Algorithm for Classification Problems With Outliers or Noises

Abstract: The support vector machine (SVM) has provided higher performance than traditional learning machines and has been widely applied in real-world classification problems and nonlinear function estimation problems. Unfortunately, the training process of the SVM is sensitive to the outliers or noises in the training set. In this paper, a common misunderstanding of Gaussian-function-based kernel fuzzy clustering is corrected, and a kernel fuzzy c-means clustering-based fuzzy SVM algorithm (KFCM-FSVM) is developed to deal with the classification problems with outliers or noises. In the KFCM-FSVM algorithm, we first use the FCM clustering to cluster each of two classes from the training set in the high-dimensional feature space. The farthest pair of clusters, where one cluster comes from the positive class and the other from the negative class, is then searched and forms one new training set with membership degrees. Finally, we adopt FSVM to induce the final classification results on this new training set. The computational complexity of the KFCM-FSVM algorithm is analyzed. A set of experiments is conducted on six benchmarking datasets and four artificial datasets for testing the generalization performance of the KFCM-FSVM algorithm. The results indicate that the KFCM-FSVM algorithm is robust for classification problems with outliers or noises.

Analytic Hierarchy Process (AHP) in Group Decision Making and its Optimization With an Allocation of Information Granularity

Abstract: In group decision making, one strives to reconcile differences of opinions (judgments) expressed by individual members of the group. Fuzzy-decision-making mechanisms bring a great deal of flexibility. By admitting membership degrees, we are offered flexibility to exploit different aggregation mechanisms and navigate a process of interaction among decision makers to achieve an increasing level of consistency within the group. While the studies reported so far exploit more or less sophisticated ways of adjusting/transforming initial judgments (preferences) of individuals, in this paper, we bring forward a concept of information granularity. Here, information granularity is viewed as an essential asset, which offers a decision maker a tangible level of flexibility using some initial preferences conveyed by each individual that can be adjusted with the intent to reach a higher level of consensus. Our study is concerned with an extension of the well-known analytic hierarchy process to the group decision-making scenario. More specifically, the admitted level of granularity gives rise to a granular matrix of pairwise comparisons. The granular entries represented, e.g., by intervals or fuzzy sets, supply a required flexibility using the fact that we select the most suitable numeric representative of the reciprocal matrix. The proposed concept of granular reciprocal matrices is used to optimize a performance index, which comes as an additive combination of two components. The first one expresses a level of consistency of the individual pairwise comparison matrices; by exploiting the admitted level of granularity, we aim at the minimization of the corresponding inconsistency index. The second part of the performance index quantifies a level of disagreement in terms of the individual preferences. The flexibility offered by the level of granularity is used to increase the level of consensus within the group. Given an implicit nature of relationships between the realizations of the granular pairwise matrices and the values of the performance index, we consider using particle swarm optimization as an optimization vehicle. Two scenarios of allocation of granularity among decision makers are considered, namely, a uniform allocation of granularity and nonuniform distribution of granularity, where the levels of allocated granularity are also subject to optimization. A number of numeric studies are provided to illustrate an essence of the method.

Asynchronous Output-Feedback Control of Networked Nonlinear Systems With Multiple Packet Dropouts: T–S Fuzzy Affine Model-Based Approach

Abstract: This paper investigates the problem of robust output-feedback control for a class of networked nonlinear systems with multiple packet dropouts. The nonlinear plant is represented by Takagi-Sugeno (T-S) fuzzy affine dynamic models with norm-bounded uncertainties, and stochastic variables that satisfy the Bernoulli random binary distribution are adopted to characterize the data-missing phenomenon. The objective is to design an admissible output-feedback controller that guarantees the stochastic stability of the resulting closed-loop system with a prescribed disturbance attenuation level. It is assumed that the plant premise variables, which are often the state variables or their functions, are not measurable so that the controller implementation with state-space partition may not be synchronous with the state trajectories of the plant. Based on a piecewise quadratic Lyapunov function combined with an S-procedure and some matrix inequality convexifying techniques, two different approaches to robust output-feedback controller design are developed for the underlying T-S fuzzy affine systems with unreliable communication links. The solutions to the problem are formulated in the form of linear matrix inequalities (LMIs). Finally, simulation examples are provided to illustrate the effectiveness of the proposed approaches.

Polynomial Fuzzy-Model-Based Control Systems: Stability Analysis Via Piecewise-Linear Membership Functions

Abstract: This paper presents the stability analysis of polynomial fuzzy-model-based (PFMB) control systems using the sum-of-squares (SOS) approach. The PFMB control system under consideration requires that the polynomial fuzzy model and polynomial fuzzy controller share neither the same premise membership functions nor the same number of fuzzy rules. This class of PFMB control systems offers a greater design flexibility to the polynomial fuzzy controller. However, due to the imperfectly matched membership functions, it usually produces more conservative stability conditions by following the traditional stability-analysis approach for the FMB control systems. To facilitate the stability analysis, piecewise-linear membership functions (PLMFs) are proposed, which offer a nice property that the grades of membership are governed by a finite number of sample points. Thus, it allows the PLMFs to be brought to the SOS-based stability conditions, which are applied to the PFMB control systems with the specified PLMFs rather than any shapes. The system stability can be examined by checking only the PFMB control system at the sample points. It is worth mentioning that the PLMFs, which are not necessarily implemented physically, are a mathematical tool to carry out the stability analysis. To verify the stability-analysis result, a simulation example is given to demonstrate the effectiveness of the proposed approach.

Integrated Fault Estimation and Accommodation Design for Discrete-Time Takagi–Sugeno Fuzzy Systems With Actuator Faults

Abstract: This paper addresses the problem of integrated robust fault estimation (FE) and accommodation for discrete-time Takagi-Sugeno (T-S) fuzzy systems. First, a multiconstrained reduced-order FE observer (RFEO) is proposed to achieve FE for discrete-time T-S fuzzy models with actuator faults. Based on the RFEO, a new fault estimator is constructed. Then, using the information of online FE, a new approach for fault accommodation based on fuzzy-dynamic output feedback is designed to compensate for the effect of faults by stabilizing the closed-loop systems. Moreover, the RFEO and the dynamic output feedback fault-tolerant controller are designed separately, such that their design parameters can be calculated readily. Simulation results are presented to illustrate our contributions.

Relaxed Stability Conditions for Continuous-Time T–S Fuzzy-Control Systems Via Augmented Multi-Indexed Matrix Approach

Abstract: This paper is concerned with the problem of developing an advanced strategy to reduce the conservatism in stability analysis and control synthesis of continuous-time Takagi-Sugeno (T-S) fuzzy systems. A novel augmented multi-indexed matrix approach is proposed to implement new right-hand-side slack variables technique for the homogenous polynomial setting. Combining with the Finsler lemma with homogenous-matrix Lagrange multipliers, convergent linear-matrix-inequality (LMI) relaxations for stability analysis are proposed by using the generalization of the Polya theorem for the case of positive polynomials with matrix-valued coefficients. A new type of state-feedback controller, namely, the homogeneous polynomially nonquadratic control law (HPNQCL), is developed to conceive less-conservative stabilization conditions. The obtained stability and stabilization conditions are further relaxed by using the proposed right-hand-side slack variables technique. Moreover, the advantages over the existing control schemes are certificated in theory. Three numerical examples are also provided to illustrate the effectiveness of the proposed methods.

TAIEX Forecasting Based on Fuzzy Time Series and Fuzzy Variation Groups

Abstract: In this paper, we present a new method to forecast the daily Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX) based on fuzzy time series and fuzzy variation groups, where the main input factor is the previous day's TAIEX, and the secondary factor is either the Dow Jones, the NASDAQ, the M 1b, or their combination. First, the proposed method fuzzifies the historical training data of the TAIEX into fuzzy sets to form fuzzy logical relationships. Second, it groups the fuzzy logical relationships into fuzzy logical relationship groups (FLRGs) based on the fuzzy variations of the secondary factor. Third, it evaluates the leverage of the fuzzy variations between the main factor and the secondary factor to construct fuzzy variation groups. Fourth, it gets the statistics of the fuzzy variations appearing in each fuzzy variation group. Fifth, it calculates the weights of the statistics of the fuzzy variations appearing in each fuzzy variation group, respectively. Finally, based on the weights of the statistics of the fuzzy variations appearing in the fuzzy variation groups and the FLRGs, it performs the forecasting of the daily TAIEX. Because the proposed method uses both fuzzy variation groups and FLRGs to analyze in detail the historical training data, it gets higher forecasting accuracy rates to forecast the TAIEX than the existing methods.

Multivariable Gaussian Evolving Fuzzy Modeling System

Abstract: This paper introduces a class of evolving fuzzy rule-based system as an approach for multivariable Gaussian adaptive fuzzy modeling. The system is an evolving Takagi-Sugeno (eTS) functional fuzzy model, whose rule base can be continuously updated using a new recursive clustering algorithm based on participatory learning. The fuzzy sets of the rule antecedents are multivariable Gaussian membership functions, which have been adopted to preserve information between input variable interactions. The parameters of the membership functions are estimated by the clustering algorithm. A weighted recursive least-squares algorithm updates the parameters of the rule consequents. Experiments considering time-series forecasting and nonlinear system identification are performed to evaluate the performance of the approach proposed. The multivariable Gaussian evolving fuzzy models are compared with alternative evolving fuzzy models and classic models with fixed structures. The results suggest that multivariable Gaussian evolving fuzzy modeling is a promising approach for adaptive system modeling.

Adaptive Fuzzy Control for Synchronization of Nonlinear Teleoperators With Stochastic Time-Varying Communication Delays

Abstract: In this paper, adaptive fuzzy control is investigated for nonlinear teleoperators with time delays, which ensures synchronization of positions and velocities of the master and slave manipulators and does not rely on the use of the scattering transformation. Compared with the previous passivity framework, the communication delays are assumed to be stochastic time varying. By feedback linearization, the nonlinear dynamics of the teleoperation system is transformed into two subsystems: local master/slave position control with unmodeled dynamics and delayed motion synchronization. Then, based on linear matrix inequalities (LMI) and Markov jump linear systems, adaptive fuzzy-control strategies are developed for the nonlinear teleoperators with modeling uncertainties and external disturbances by using the approximation property of the fuzzy logic systems. It is proven that the master-slave teleoperation system is stochastically stable in mean square under specific LMI conditions, and all the signals of the resulting closed-loop system are uniformly bounded. Finally, the extensive simulations are performed to show the effectiveness of the proposed method.

Chaos Synchronization of Uncertain Fractional-Order Chaotic Systems With Time Delay Based on Adaptive Fuzzy Sliding Mode Control

Abstract: This paper proposes an adaptive fuzzy sliding mode control (AFSMC) to synchronize two different uncertain fractional-order time-delay chaotic systems, which are infinite dimensional in nature, and time delay is a source of instability. Because modeling the behavior of dynamical systems by fractional-order differential equations has more advantages than integer-order modeling, the adaptive time-delay fuzzy-logic system is constructed to approximate the unknown fractional-order time-delay-system functions. By using Lyapunov stability criterion, the free parameters of the adaptive fuzzy controller can be tuned online by output-feedback-control law and adaptive law. The sliding mode design procedure not only guarantees the stability and robustness of the proposed AFSMC, but it also guarantees that the external disturbance on the synchronization error can be attenuated. The simulation example is included to confirm validity and synchronization performance of the advocated design methodology.

Information Granularity in Fuzzy Binary GrC Model

Abstract: Zadeh's seminal work in theory of fuzzy-information granulation in human reasoning is inspired by the ways in which humans granulate information and reason with it. This has led to an interesting research topic: granular computing (GrC). Although many excellent research contributions have been made, there remains an important issue to be addressed: What is the essence of measuring a fuzzy-information granularity of a fuzzy-granular structure? What is needed to answer this question is an axiomatic constraint with a partial-order relation that is defined in terms of the size of each fuzzy-information granule from a fuzzy-binary granular structure. This viewpoint is demonstrated for fuzzy-binary granular structure, which is called the binary GrC model by Lin. We study this viewpoint from from five aspects in this study, which are fuzzy BINARY-granular-structure operators, partial-order relations, measures for fuzzy-information granularity, an axiomatic approach to fuzzy-information granularity, and fuzzy-information entropies.

Nonsynchronized-State Estimation of Multichannel Networked Nonlinear Systems With Multiple Packet Dropouts Via T–S Fuzzy-Affine Dynamic Models

Abstract: This paper investigates the problem of robust H∞ state estimation for a class of multichannel networked nonlinear systems with multiple packet dropouts. The nonlinear plant is represented by Takagi-Sugeno (T-S) fuzzy-affine dynamic models with norm-bounded uncertainties, and stochastic variables with general probability distributions are adopted to characterize the data missing phenomenon in output channels. The objective is to design an admissible state estimator guaranteeing the stochastic stability of the resulting estimation-error system with a prescribed H∞ disturbance attenuation level. It is assumed that the plant premise variables, which are often the state variables or their functions, are not measurable so that the estimator implementation with state-space partition may not be synchronized with the state trajectories of the plant. Based on a piecewise-quadratic Lyapunov function combined with S -procedure and some matrix-inequality-convexifying techniques, two different approaches are developed to robust filtering design for the underlying T-S fuzzy-affine systems with unreliable communication links. All the solutions to the problem are formulated in the form of linear-matrix inequalities (LMIs). Finally, simulation examples are provided to illustrate the effectiveness of the proposed approaches.

Evolutionary-Group-Based Particle-Swarm-Optimized Fuzzy Controller With Application to Mobile-Robot Navigation in Unknown Environments

Abstract: This paper proposes an evolutionary-group-based particle-swarm-optimization (EGPSO) algorithm for fuzzy-controller (FC) design. The EGPSO uses a group-based framework to incorporate crossover and mutation operations into particle-swarm optimization. The EGPSO dynamically forms different groups to select parents in crossover operations, particle updates, and replacements. An adaptive velocity-mutated operation (AVMO) is incorporated to improve search ability. The EGPSO is applied to design all of the free parameters in a zero-order Takagi-Sugeno-Kang (TSK)-type FC. The objective of EGPSO is to improve fuzzy-control accuracy and design efficiency. Comparisons with different population-based optimizations of fuzzy-control problems demonstrate the superiority of EGPSO performance. In particular, the EGPSO-designed FC is applied to mobile-robot navigation in unknown environments. In this application, the robot learns to follow object boundaries through an EGPSO-designed FC. A simple learning environment is created to build this behavior without an exhaustive collection of input-output training pairs in advance. A behavior supervisor is proposed to combine the boundary-following behavior and the target-seeking behavior for navigation, and the problem of dead cycles is considered. Successful mobile-robot navigation in simulation and real environments verifies the EGPSO-designed FC-navigation approach.

A Fast and Scalable Multiobjective Genetic Fuzzy System for Linguistic Fuzzy Modeling in High-Dimensional Regression Problems

Abstract: Linguistic fuzzy modeling in high-dimensional regression problems poses the challenge of exponential-rule explosion when the number of variables and/or instances becomes high. One way to address this problem is by determining the used variables, the linguistic partitioning and the rule set together, in order to only evolve very simple, but still accurate models. However, evolving these components together is a difficult task, which involves a complex search space. In this study, we propose an effective multiobjective evolutionary algorithm that, based on embedded genetic database (DB) learning (involved variables, granularities, and slight fuzzy-partition displacements), allows the fast learning of simple and quite-accurate linguistic models. Some efficient mechanisms have been designed to ensure a very fast, but not premature, convergence in problems with a high number of variables. Further, since additional problems could arise for datasets with a large number of instances, we also propose a general mechanism for the estimation of the model error when using evolutionary algorithms, by only considering a reduced subset of the examples. By doing so, we can also apply a fast postprocessing stage for further refining the learned solutions. We tested our approach on 17 real-world datasets with different numbers of variables and instances. Three well-known methods based on embedded genetic DB learning have been executed as references. We compared the different approaches by applying nonparametric statistical tests for multiple comparisons. The results confirm the effectiveness of the proposed method not only in terms of scalability but in terms of the simplicity and generalizability of the obtained models as well.

LMI-Based Stability Analysis for Fuzzy-Model-Based Control Systems Using Artificial T–S Fuzzy Model

Abstract: This paper investigates the stability of fuzzy-model-based (FMB) control systems. An alternative stability-analysis approach using an artificial fuzzy system based on the Lyapunov stability theory is proposed. To facilitate the stability analysis, the continuous membership functions of the Takagi-Sugeno (T-S) fuzzy model are represented by the staircase ones. With the nice property of the staircase membership functions, it turns the set of infinite number of linear-matrix-inequality (LMI) based stability conditions into a finite one. Furthermore, the staircase membership functions carrying system information can be brought to the stability conditions to relax the stability conditions. The stability of the original FMB control systems is guaranteed by the satisfaction of the LMI-based stability conditions. The proposed stability analysis is applied to the FMB control systems of which the T-S fuzzy model and fuzzy controller do not share the same premise membership functions and, thus, is able to enhance the design flexibility of the fuzzy controller. A simulation example is given to illustrate the merits of the proposed approach.

Robust Adaptive Fuzzy Control by Backstepping for a Class of MIMO Nonlinear Systems

Abstract: This paper presents a robust adaptive control method for a class of multi-input-multi-output (MIMO) nonlinear systems that are transformable to a parametric-strict-feedback form which has couplings among input channels and the appearance of parametric uncertainties in the input matrices. The proposed approach effectively combines the design techniques of robust adaptive control by backstepping and adaptive fuzzy-logic control in order to remove the matching-condition requirement and to provide boundedness of tracking errors, even under dominant model uncertainties and poor parameter adaptation. Unlike previous robust adaptive fuzzy controls of MIMO nonlinear systems, this research introduces the robustness terms explicitly in the controller structure to counteract the effects of model uncertainties and parameter-adaptation errors. Uniform boundedness of the MIMO nonlinear control system is proved, and simulation results further validate the effectiveness and performance of the proposed control method.

ANCFIS: A Neurofuzzy Architecture Employing Complex Fuzzy Sets

Abstract: Complex fuzzy sets (CFSs) are an extension of type-1 fuzzy sets in which the membership of an object to the set is a value from the unit disc of the complex plane. Although there has been considerable progress made in determining the properties of CFSs and complex fuzzy logic, there has yet to be any practical application of this concept. We present the adaptive neurocomplex-fuzzy-inferential system (ANCFIS), which is the first neurofuzzy system architecture to implement complex fuzzy rules (and, in particular, the signature property of rule interference). We have applied this neurofuzzy system to the domain of time-series forecasting, which is an important machine-learning problem. We find that ANCFIS performs well in one synthetic and five real-world forecasting problems and is also very parsimonious. Experimental comparisons show that ANCFIS is comparable with existing approaches on our five datasets. This work demonstrates the utility of complex fuzzy logic on real-world problems.

Adaptive Fuzzy Control With Guaranteed Convergence of Optimal Approximation Error

Abstract: With no a priori knowledge of plant boundary functions, a novel direct adaptive fuzzy controller (AFC) for a class of single-input single-output (SISO) uncertain affine nonlinear systems is developed in this paper. Based on the theory of fuzzy logic systems (FLSs) with variable universes of discourse (UDs), sufficient conditions that guarantee that the optimal fuzzy approximation error (FAE) is locally convergent are given. By the use of the output tracking error and its derivatives as input variables and by the selection of suitable adjusting parameters, a variable UD FLS with an optimal FAE local convergence is constructed, and its parameter adaptive law is derived by virtue of the Lyapunov stability theorem. Under the assumption that the optimal FAE is bounded, it is proved that the closed-loop system is asymptotically stable in the sense that all variables are uniformly ultimately bounded and that the tracking errors converge to zero. The proposed approach eliminates the influence of the FAE on the tracking errors by means of the inherent mechanism of the variable UD FLS. Thus, it has the potential to achieve high control performance without additional compensation under only a few fuzzy rules. Simulation studies demonstrate the superiority of the proposed AFC in terms of the settling time, tracking accuracy, smoothness of the control input, and robustness against external disturbances and parameter variations.

Weighted Fuzzy Rule Interpolation Based on GA-Based Weight-Learning Techniques

Abstract: In this paper, we propose a weighted fuzzy interpolative reasoning method for sparse fuzzy rule-based systems. It is based on a genetic algorithm (GA)-based weight-learning technique. The proposed method can deal with fuzzy rule interpolation with weighted antecedent variables. It also can deal with fuzzy rule interpolation based on polygonal membership functions and bell-shaped membership functions. We also propose a GA-based weight-learning algorithm to automatically learn the optimal weights of the antecedent variables of the fuzzy rules. Furthermore, we apply the proposed weighted fuzzy interpolative reasoning method and the proposed GA-based weight-learning algorithm to deal with the truck backer-upper control problem, the computer activity prediction problem, multivariate regression problems, and time series prediction problems. Based on statistical analysis techniques, the experimental results show that the proposed weighted fuzzy interpolative reasoning method by the use of the optimally learned weights that were obtained by the proposed GA-based weight-learning algorithm has statistically significantly smaller error rates than the existing methods.

Adaptive Fuzzy Interpolation

Abstract: Fuzzy interpolative reasoning strengthens the power of fuzzy inference by the enhancement of the robustness of fuzzy systems and the reduction of the systems' complexity. However, after a series of interpolations, it is possible that multiple object values for a common variable are inferred, leading to inconsistency in interpolated results. Such inconsistencies may result from defective interpolated rules or incorrect interpolative transformations. This paper presents a novel approach for identification and correction of defective rules in interpolative transformations, thereby removing the inconsistencies. In particular, an assumption-based truth-maintenance system (ATMS) is used to record dependences between interpolations, and the underlying technique that the classical general diagnostic engine (GDE) employs for fault localization is adapted to isolate possible faulty interpolated rules and their associated interpolative transformations. From this, an algorithm is introduced to allow for the modification of the original linear interpolation to become first-order piecewise linear. The approach is applied to a realistic problem, which predicates the diarrheal disease rates in remote villages, to demonstrate the potential of this study.

Linguistic Summarization Using IF–THEN Rules and Interval Type-2 Fuzzy Sets

Abstract: Linguistic summarization (LS) is a data mining or knowledge discovery approach to extract patterns from databases. Many authors have used this technique to generate summaries like “Most senior workers have high salary,” which can be used to better understand and communicate about data; however, few of them have used it to generate IF-THEN rules like “IF X is large and Y is medium, THEN Z is small,” which not only facilitate understanding and communication of data but can also be used in decision-making. In this paper, an LS approach to generate IF-THEN rules for causal databases is proposed. Both type-1 and interval type-2 fuzzy sets are considered. Five quality measures-the degrees of truth, sufficient coverage, reliability, outlier, and simplicity-are defined. Among them, the degree of reliability is especially valuable for finding the most reliable and representative rules, and the degree of outlier can be used to identify outlier rules and data for close-up investigation. An improved parallel coordinates approach for visualizing the IF-THEN rules is also proposed. Experiments on two datasets demonstrate our LS and rule visualization approaches. Finally, the relationships between our LS approach and the Wang-Mendel (WM) method, perceptual reasoning, and granular computing are pointed out.

An Enhanced Type-Reduction Algorithm for Type-2 Fuzzy Sets

Abstract: Karnik and Mendel proposed an algorithm to compute the centroid of an interval type-2 fuzzy set efficiently. Based on this algorithm, Liu developed a centroid type-reduction strategy to carry out type reduction for type-2 fuzzy sets. A type-2 fuzzy set is decomposed into a collection of interval type-2 fuzzy sets by -cuts. Then, the Karnik-Mendel algorithm is called for each interval type-2 fuzzy set iteratively. However, the initialization of the switch point in each application of the Karnik-Mendel algorithm is not a good one. In this paper, we present an improvement to Liu's algorithm. We employ the previously obtained result to construct the starting values in the current application of the Karnik-Mendel algorithm. Convergence in each iteration, except the first one, can then speed up, and type reduction for type-2 fuzzy sets can be carried out faster. The efficiency of the improved algorithm is analyzed mathematically and demonstrated by experimental results.

Scalable TSK Fuzzy Modeling for Very Large Datasets Using Minimal-Enclosing-Ball Approximation

Abstract: In order to overcome the difficulty in Takagi-Sugeno-Kang (TSK) fuzzy modeling for large datasets, scalable TSK (STSK) fuzzy-model training is investigated in this study based on the core-set-based minimal-enclosing-ball (MEB) approximation technique. The specified L2-norm penalty-based -insensitive criterion is first proposed for TSK-model training, and it is found that such TSK fuzzy-model training can be equivalently expressed as a center-constrained MEB problem. With this finding, an STSK fuzzy-model-training algorithm, which is called STSK, for large or very large datasets is then proposed by using the core-set-based MEB-approximation technique. The proposed algorithm has two distinctive advantages over classical TSK fuzzy-model training algorithms: The maximum space complexity for training is not reliant on the size of the training dataset, and the maximum time complexity for training is linear with the size of the training dataset, as confirmed by extensive experiments on both synthetic and real-world regression datasets.

On the Continuity of Type-1 and Interval Type-2 Fuzzy Logic Systems

Abstract: This paper studies the continuity of the input-output mappings of fuzzy logic systems (FLSs), including both type-1 (T1) and interval type-2 (IT2) FLSs. We show that a T1 FLS being an universal approximator is equivalent to saying that a T1 FLS has a continuous input-output mapping. We also derive the condition under which a T1 FLS is discontinuous. For IT2 FLSs, we consider six type-reduction and defuzzification methods (the Karnik-Mendel method, the uncertainty bound method, the Wu-Tan method, the Nie-Tan method, the Du-Ying method, and the Begian-Melek-Mendel method) and derive the conditions under which continuous and discontinuous input-output mappings can be obtained. Guidelines for designing continuous IT2 FLSs are also given. This paper is to date the most comprehensive study on the continuity of FLSs. Our results will be very useful in the selection of the parameters of the membership functions to achieve a desired continuity (e.g., for most traditional modeling and control applications) or discontinuity (e.g., for hybrid and switched systems modeling and control).

A New Fuzzy Lyapunov Function for Relaxed Stability Condition of Continuous-Time Takagi–Sugeno Fuzzy Systems

Abstract: This paper presents a new fuzzy Lyapunov function (FLF) for the stability analysis of continuous-time Takagi-Sugeno (T-S) fuzzy systems. Unlike conventional FLFs, the proposed one depends not only on the fuzzy weighting functions of the T-S fuzzy systems but on their first-order time derivatives as well. Based on the proposed FLF, a sufficient stability condition is derived in the form of linear matrix inequalities, depending on the upper bounds on the second-order time derivative of the fuzzy weighting functions, as well as the first-order ones. Finally, some examples demonstrate that the proposed condition can provide less conservative results than the previous ones in the literature.

Dynamic Output Feedback Predictive Control for Nonlinear Systems Represented by a Takagi–Sugeno Model

Abstract: This paper addresses the output feedback predictive control for a Takagi-Sugeno (T-S) fuzzy system with bounded noise. The controller optimizes an infinite-horizon objective function respecting the input and state constraints. The control law is parameterized as a dynamic output feedback that is dependent on the membership functions, and the closed-loop stability is specified by the notion of quadratic boundedness. Online algorithms that guarantee the recursive feasibility of the convex optimization problem and the convergence of the augmented state to a neighborhood of the equilibrium point are proposed in this paper. A numerical example is given to illustrate the effectiveness of the proposed output feedback controllers.

Prediction Interval Construction and Optimization for Adaptive Neurofuzzy Inference Systems

Abstract: The performance of an adaptive neurofuzzy inference system (ANFIS) significantly drops when uncertainty exists in the data or system operation. Prediction intervals (PIs) can quantify the uncertainty associated with ANFIS point predictions. This paper first presents a methodology to adapt the delta technique for the construction of PIs for outcomes of the ANFIS models. As the ANFIS models are linear in their consequent part, the ANFIS-based PIs are computationally less expensive than neural network (NN)-based PIs. Second, this paper proposes a method to optimize ANFIS-based PIs. A new PI-based cost function is developed for the training of the ANFIS models. A simulated annealing-based algorithm is applied to minimize the new nonlinear cost function and adjust the premise and consequent parameters of the ANFIS model. Using three real-world case studies, it is shown that ANFIS-based PIs are computationally less expensive than NN-based PIs. The application of the proposed optimization algorithm leads to better quality PIs than optimized NN-based PIs.