Optimization of gaussian fuzzy membership functions and evaluation of the monotonicity property of fuzzy inference systems

In this paper, two issues relating to modeling of a monotonicity-preserving Fuzzy Inference System (FIS) are examined. The first is on designing or tuning of Gaussian Membership Functions (MFs) for a monotonic FIS. Designing Gaussian MFs for an FIS is difficult because of its spreading and curvature characteristics. In this study, the sufficient conditions are exploited, and the procedure of designing Gaussian MFs is formulated as a constrained optimization problem. The second issue is on the testing procedure for a monotonic FIS. As such, a testing procedure for a monotonic FIS model is proposed. Applicability of the proposed approach is demonstrated with a real world industrial application, i.e., Failure Mode and Effect Analysis. The results obtained are analysis and discussed. The outcomes show that the proposed approach is useful in designing a monotonicity-preserving FIS model.

On monotonic sufficient conditions of fuzzy inference systems and their applications

An important and difficult issue in designing a Fuzzy Inference System (FIS) is the specification of fuzzy sets and fuzzy rules. In this paper, two useful qualitative properties of the FIS model, i.e., the monotonicity and sub-additivity properties, are studied. The monotonic sufficient conditions of the FIS model with Gaussian membership functions are further analyzed. The aim is to incorporate the sufficient conditions into the FIS modeling process, which serves as a simple (which can be easily understood by domain users), easy-to-use (which can be easily applied to or can be a part of the FIS model), and yet reliable (which has a sound mathematical foundation) method to preserve the monotonicity property of the FIS model. Another aim of this paper is to demonstrate how these additional qualitative information can be exploited and extended to be part of the FIS designing procedure (i.e., for fuzzy sets and fuzzy rules design) via the sufficient conditions (which act as a set of useful governing equations for designing the FIS model). The proposed approach is able to avoid the "trial and error" procedure in obtaining a monotonic FIS model. To assess the applicability of the proposed approach, two practical problems are examined. The first is an FIS-based model for water level control, while the second is an FIS-based Risk Priority Number (RPN) model in Failure Mode and Effect Analysis (FMEA). To further illustrate the importance of the sufficient conditions as the governing equations, an analysis on the consequences of violating the sufficient conditions of the FIS-based RPN model is presented.

An evolutionary-based similarity reasoning scheme for monotonic multi-input fuzzy inference systems

In this paper, an Evolutionary-based Similarity Reasoning (ESR) scheme for preserving the monotonicity property of the multi-input Fuzzy Inference System (FIS) is proposed. Similarity reasoning (SR) is a useful solution for undertaking the incomplete rule base problem in FIS modeling. However, SR may not be a direct solution to designing monotonic multi-input FIS models, owing to the difficulty in getting a set of monotonically-ordered conclusions. The proposed ESR scheme, which is a synthesis of evolutionary computing, sufficient conditions, and SR, provides a useful solution to modeling and preserving the monotonicity property of multi-input FIS models. A case study on Failure Mode and Effect Analysis (FMEA) is used to demonstrate the effectiveness of the proposed ESR scheme in undertaking real world problems that require the monotonicity property of FIS models.

Sliding mode-like learning control for SISO complex systems with T-S fuzzy models

In this paper, a sliding mode-like learning control scheme is developed for a class of single input single output (SISO) complex systems. First, the Takagi-Sugeno (T-S) fuzzy modelling technique is employed to model the uncertain complex dynamical systems. Second, a sliding mode-like learning control is designed to drive the sliding variable to converge to the sliding surface, and the system states can then asymptotically converge to zero on the sliding surface. The advantages of this scheme are that: 1) the information about the uncertain system dynamics and the system model structure is not required for the design of the learning controller; 2) the closed-loop system behaves with a strong robustness with respect to uncertainties; 3) the control input is chattering-free. The sufficient conditions for the sliding mode-like learning control to stabilise the global fuzzy model are discussed in detail. A simulation example for the control of an inverted pendulum cart is presented to demonstrate the effectiveness of the proposed control scheme.

A robust learning control for SISO Nonlinear systems with T-S Fuzzy Model : C02-robust control

In this paper, a robust learning control is developed for a class of single input single output (SISO) nonlinear systems with T-S fuzzy model. It is seen that the proposed sliding mode learning control with the powerful Lipshitz-like condition can guarantee the stability, convergence and robustness of the closed-loop system without involving any assumptions on uncertain system dynamics. In addition, theconcept that the local system with the maximum membership function dominates the system dynamic behaviours helps to greatly simplify the control system design. It will be further seen that the continuous learning control ensures the advantage of chattering-free that may occur in conventional sliding mode systems. Simulation examples are presented to demonstrate the effectiveness of the proposed learning control through the comparison with the H-infinity control.

Enhancing an evolving tree-based text document visualization model with fuzzy c-Means clustering

An improved evolving model, i.e., Evolving Tree (ETree) with Fuzzy c-Means (FCM), is proposed for undertaking text document visualization problems in this study. ETree forms a hierarchical tree structure in which nodes (i.e., trunks) are allowed to grow and split into child nodes (i.e., leaves), and each node represents a cluster of documents. However, ETree adopts a relatively simple approach to split its nodes. Thus, FCM is adopted as an alternative to perform node splitting in ETree. An experimental study using articles from a flagship conference of Universiti Malaysia Sarawak (UNIMAS), i.e., Engineering Conference (ENCON), is conducted. The experimental results are analyzed and discussed, and the outcome shows that the proposed ETree-FCM model is effective for undertaking text document clustering and visualization problems.

A new neural network-based type reduction algorithm for interval type-2 fuzzy logic systems

This paper introduces a new type reduction (TR) algorithm for interval type-2 fuzzy logic systems (IT2 FLSs). Flexibility and adaptiveness are the key features of the proposed non-parametric algorithm. Lower and upper firing strengths of rules as well as their consequent coefficients are fed into a neural network (NN). NN output is a crisp value that corresponds to the defuzzified output of IT2 FLSs. The NN type reducer is trained through minimization of an error-based cost function with the purpose of improving modelling and forecasting performance of IT2 FLS models. Simulation results indicate that application of the proposed TR algorithm greatly enhances modelling and forecasting performance of IT2 FLS models. This benefit is achieved in no cost, as the computational requirement of the proposed algorithm is less than or at most equivalent to traditional TR algorithms.

A new framework with similarity reasoning and monotone fuzzy rule relabeling for fuzzy inference systems

A complete and monotonically-ordered fuzzy rule base is necessary to maintain the monotonicity property of a Fuzzy Inference System (FIS). In this paper, a new monotone fuzzy rule relabeling technique to relabel a non-monotone fuzzy rule base provided by domain experts is proposed. Even though the Genetic Algorithm (GA)-based monotone fuzzy rule relabeling technique has been investigated in our previous work [7], the optimality of the approach could not be guaranteed. The new fuzzy rule relabeling technique adopts a simple brute force search, and it can produce an optimal result. We also formulate a new two-stage framework that encompasses a GA-based rule selection scheme, the optimization based-Similarity Reasoning (SR) scheme, and the proposed monotone fuzzy rule relabeling technique for preserving the monotonicity property of the FIS model. Applicability of the two-stage framework to a real world problem, i.e., failure mode and effect analysis, is further demonstrated. The results clearly demonstrate the usefulness of the proposed framework.

A new online updating framework for constructing monotonicity-preserving fuzzy inference systems

In this paper, a new online updating framework for constructing monotonicity-preserving Fuzzy Inference Systems (FISs) is proposed. The framework encompasses an optimization-based Similarity Reasoning (SR) scheme and a new monotone fuzzy rule relabeling technique. A complete and monotonically-ordered fuzzy rule base is necessary to maintain the monotonicity property of an FIS model. The proposed framework attempts to allow a monotonicity-preserving FIS model to be constructed when the fuzzy rules are incomplete and not monotonically-ordered. An online feature is introduced to allow the FIS model to be updated from time to time. We further investigate three useful measures, i.e., the belief, plausibility, and evidential mass measures, which are inspired from the Dempster- Shafer theory of evidence, to analyze the proposed framework and to give an insight for the inferred outcomes from the FIS model.

Interval-based and fuzzy set-based approaches to modeling of fuzzy inference systems with the local monotonicity property

Even though the importance of the local monotonicity property for function approximation problems is well established, there are relative few investigations addressing issues related to the fulfillment of the local monotonicity property in Fuzzy Inference System (FIS) modeling. We have previously conducted a preliminary study on the local monotonicity property of FIS models, with the assumption that the extrema point(s) (i.e., the maximum and/or minimum point(s)) is either known precisely or totally unknown. However, in some practical situations, the extrema point(s) can be known imprecisely (as an interval or a fuzzy set). In this paper, the imprecise information is exploited to construct an FIS model that fulfills the local monotonicity property. A procedure to estimate the extrema point(s) of a function is devised. Applicability of the findings to a datadriven modeling problem is further demonstrated.

Dynamic evolving neural-fuzzy inference system for rainfall-runoff (R-R) modelling

Dynamic Evolving Neural-Fuzzy Inference System (DENFIS) is a Takagi-Sugeno-type fuzzy inference system for online learning which can be applied for dynamic time series prediction. To the best of our knowledge, this is the first time that DENFIS has been used for rainfall-runoff (R-R) modeling. DENFIS model results were compared to the results obtained from the physically-based Storm Water Management Model (SWMM) and an Adaptive Network-based Fuzzy Inference System (ANFIS) which employs offline learning. Data from a small (5.6 km2) catchment in Singapore, comprising 11 separated storm events were analyzed. Rainfall was the only input used for the DENFIS and ANFIS models and the output was discharge at the present time. It is concluded that DENFIS results are better or at least comparable to SWMM, but similar to ANFIS. These results indicate a strong potential for DENFIS to be used in R-R modeling.

A new monotonicity index for fuzzy rule-based systems

A search in the literature reveals that mathematical conditions (usually sufficient conditions) for the Fuzzy Inference System (FIS) models to satisfy the monotonicity property have been developed. A monotonically-ordered fuzzy rule base is important to maintain the monotonicity property of an FIS. However, it may difficult to obtain a monotonically-ordered fuzzy rule base in practice. We have previously introduced the idea of fuzzy rule relabeling to tackle this problem. In this paper, we further propose a monotonicity index for the FIS system, which serves as a metric to indicate the degree of a fuzzy rule base fulfilling the monotonicity property. The index is useful to provide an indication whether a fuzzy rule base should (or should not) be used in practice, even with fuzzy rule relabeling. To illustrate the idea, the zero-order Sugeno FIS model is exemplified. We add noise as errors into the fuzzy rule base to formulate a set of non-monotone fuzzy rules. As such, the metric also acts as a measure of noise in the fuzzy rule base. The results show that the proposed metric is useful to indicate the degree of a fuzzy rule base fulfilling the monotonicity property.