Showing papers in "IEEE Transactions on Fuzzy Systems in 2006"

Interval Type-2 Fuzzy Logic Systems Made Simple

Abstract: To date, because of the computational complexity of using a general type-2 fuzzy set (T2 FS) in a T2 fuzzy logic system (FLS), most people only use an interval T2 FS, the result being an interval T2 FLS (IT2 FLS). Unfortunately, there is a heavy educational burden even to using an IT2 FLS. This burden has to do with first having to learn general T2 FS mathematics, and then specializing it to an IT2 FSs. In retrospect, we believe that requiring a person to use T2 FS mathematics represents a barrier to the use of an IT2 FLS. In this paper, we demonstrate that it is unnecessary to take the route from general T2 FS to IT2 FS, and that all of the results that are needed to implement an IT2 FLS can be obtained using T1 FS mathematics. As such, this paper is a novel tutorial that makes an IT2 FLS much more accessible to all readers of this journal. We can now develop an IT2 FLS in a much more straightforward way

A Survey on Analysis and Design of Model-Based Fuzzy Control Systems

Abstract: Fuzzy logic control was originally introduced and developed as a model free control design approach. However, it unfortunately suffers from criticism of lacking of systematic stability analysis and controller design though it has a great success in industry applications. In the past ten years or so, prevailing research efforts on fuzzy logic control have been devoted to model-based fuzzy control systems that guarantee not only stability but also performance of closed-loop fuzzy control systems. This paper presents a survey on recent developments (or state of the art) of analysis and design of model based fuzzy control systems. Attention will be focused on stability analysis and controller design based on the so-called Takagi-Sugeno fuzzy models or fuzzy dynamic models. Perspectives of model based fuzzy control in future are also discussed

A new version of 2-tuple fuzzy linguistic representation model for computing with words

Abstract: In this paper, we provide a new (proportional) 2-tuple fuzzy linguistic representation model for computing with words (CW), which is based on the concept of "symbolic proportion." This concept motivates us to represent the linguistic information by means of 2-tuples, which are composed by two proportional linguistic terms. For clarity and generality, we first study proportional 2-tuples under ordinal contexts. Then, under linguistic contexts and based on canonical characteristic values (CCVs) of linguistic labels, we define many aggregation operators to handle proportional 2-tuple linguistic information in a computational stage for CW without any loss of information. Our approach for this proportional 2-tuple fuzzy linguistic representation model deals with linguistic labels, which do not have to be symmetrically distributed around a medium label and without the traditional requirement of having "equal distance" between them. Moreover, this new model not only provides a space to allow a "continuous" interpolation of a sequence of ordered linguistic labels, but also provides an opportunity to describe the initial linguistic information by members of a "continuous" linguistic scale domain which does not necessarily require the ordered linguistic terms of a linguistic variable being equidistant. Meanwhile, under the assumption of equally informative (which is defined by a condition based on the concept of CCV), we show that our model reduces to Herrera and Mart/spl inodot//spl acute/nez's (translational) 2-tuple fuzzy linguistic representation model.

A new LMI-based approach to relaxed quadratic stabilization of T-S fuzzy control systems

Abstract: This paper proposes a new quadratic stabilization condition for Takagi-Sugeno (T-S) fuzzy control systems. The condition is represented in the form of linear matrix inequalities (LMIs) and is shown to be less conservative than some relaxed quadratic stabilization conditions published recently in the literature. A rigorous theoretic proof is given to show that the proposed condition can include previous results as special cases. In comparison with conventional conditions, the proposed condition is not only suitable for designing fuzzy state feedback controllers but also convenient for fuzzy static output feedback controller design. The latter design work is quite hard for T-S fuzzy control systems. Based on the LMI-based conditions derived, one can easily synthesize controllers for stabilizing T-S fuzzy control systems. Since only a set of LMIs is involved, the controller design is quite simple and numerically tractable. Finally, the validity and applicability of the proposed approach are successfully demonstrated in the control of a continuous-time nonlinear system.

Fuzzy probabilistic approximation spaces and their information measures

Abstract: Rough set theory has proven to be an efficient tool for modeling and reasoning with uncertainty information. By introducing probability into fuzzy approximation space, a theory about fuzzy probabilistic approximation spaces is proposed in this paper, which combines three types of uncertainty: probability, fuzziness, and roughness into a rough set model. We introduce Shannon's entropy to measure information quantity implied in a Pawlak's approximation space, and then present a novel representation of Shannon's entropy with a relation matrix. Based on the modified formulas, some generalizations of the entropy are proposed to calculate the information in a fuzzy approximation space and a fuzzy probabilistic approximation space, respectively. As a result, uniform representations of approximation spaces and their information measures are formed with this work

Handling forecasting problems based on two-factors high-order fuzzy time series

Abstract: In our daily life, people often use forecasting techniques to predict weather, economy, population growth, stock, etc. However, in the real world, an event can be affected by many factors. Therefore, if we consider more factors for prediction, then we can get better forecasting results. In recent years, many researchers used fuzzy time series to handle prediction problems. In this paper, we present a new method to predict temperature and the Taiwan Futures Exchange (TAIFEX), based on the two-factors high-order fuzzy time series. The proposed method constructs two-factors high-order fuzzy logical relationships based on the historical data to increase the forecasting accuracy rate. The proposed method gets a higher forecasting accuracy rate than the existing methods.

Joint Propagation and Exploitation of Probabilistic and Possibilistic Information in Risk Assessment

Abstract: Random variability and imprecision are two distinct facets of the uncertainty affecting parameters that influence the assessment of risk. While random variability can be represented by probability distribution functions, imprecision (or partial ignorance) is better accounted for by possibility distributions (or families of probability distributions). Because practical situations of risk computation often involve both types of uncertainty, methods are needed to combine these two modes of uncertainty representation in the propagation step. A hybrid method is presented here, which jointly propagates probabilistic and possibilistic uncertainty. It produces results in the form of a random fuzzy interval. This paper focuses on how to properly summarize this kind of information; and how to address questions pertaining to the potential violation of some tolerance threshold. While exploitation procedures proposed previously entertain a confusion between variability and imprecision, thus yielding overly conservative results, a new approach is proposed, based on the theory of evidence, and is illustrated using synthetic examples

Nonsingular Terminal Sliding Mode Control of Robot Manipulators Using Fuzzy Wavelet Networks

Abstract: This paper presents an adaptive nonsingular terminal sliding mode (NTSM) tracking control design for robotic systems using fuzzy wavelet networks. Compared with linear hyperplane-based sliding control, terminal sliding mode controller can provide faster convergence and higher precision control. Therefore, a terminal sliding controller combined with the fuzzy wavelet network, which can accurately approximate unknown dynamics of robotic systems by using an adaptive learning algorithm, is an attractive control approach for robots. In addition, the proposed learning algorithm can on-line tune parameters of dilation and translation of fuzzy wavelet basis functions and hidden-to-output weights. Therefore, a robust control law is used to eliminate uncertainties including the inevitable approximation errors resulted from the finite number of fuzzy wavelet basis functions. The proposed controller requires no prior knowledge about the dynamics of the robot and no off-line learning phase. Moreover, both tracking performance and stability of the closed-loop robotic system can be guaranteed by Lyapunov theory. Finally, the effectiveness of the fuzzy wavelet network-based control approach is illustrated through comparative simulations on a six-link robot manipulator

Fuzzy interpolative reasoning via scale and move transformations

Abstract: Interpolative reasoning does not only help reduce the complexity of fuzzy models but also makes inference in sparse rule-based systems possible. This paper presents an interpolative reasoning method by means of scale and move transformations. It can be used to interpolate fuzzy rules involving complex polygon, Gaussian or other bell-shaped fuzzy membership functions. The method works by first constructing a new inference rule via manipulating two given adjacent rules, and then by using scale and move transformations to convert the intermediate inference results into the final derived conclusions. This method has three advantages thanks to the proposed transformations: 1) it can handle interpolation of multiple antecedent variables with simple computation; 2) it guarantees the uniqueness as well as normality and convexity of the resulting interpolated fuzzy sets; and 3) it suggests a variety of definitions for representative values, providing a degree of freedom to meet different requirements. Comparative experimental studies are provided to demonstrate the potential of this method

Type-2 Fuzzistics for Symmetric Interval Type-2 Fuzzy Sets: Part 1, Forward Problems

Abstract: Interval type-2 fuzzy sets (T2 FS) play a central role in fuzzy sets as models for words and in engineering applications of T2 FSs. These fuzzy sets are characterized by their footprints of uncertainty (FOU), which in turn are characterized by their boundaries-upper and lower membership functions (MF). In this two-part paper, we focus on symmetric interval T2 FSs for which the centroid (which is an interval type-1 FS) provides a measure of its uncertainty. Intuitively, we anticipate that geometric properties about the FOU, such as its area and the center of gravities (centroids) of its upper and lower MFs, will be associated with the amount of uncertainty in such a T2 FS. The main purpose of this paper (Part 1) is to demonstrate that our intuition is correct and to quantify the centroid of a symmetric interval T2 FS, and consequently its uncertainty, with respect to such geometric properties. It is then possible, for the first time, to formulate and solve forward problems, i.e., to go from parametric interval T2 FS models to data with associated uncertainty bounds. We provide some solutions to such problems. These solutions are used in Part 2 to solve some inverse problems, i.e., to go from uncertain data to parametric interval T2 FS models (T2 fuzzistics)

Support-vector-based fuzzy neural network for pattern classification

Abstract: Fuzzy neural networks (FNNs) for pattern classification usually use the backpropagation or C-cluster type learning algorithms to learn the parameters of the fuzzy rules and membership functions from the training data. However, such kinds of learning algorithms usually cannot minimize the empirical risk (training error) and expected risk (testing error) simultaneously, and thus cannot reach a good classification performance in the testing phase. To tackle this drawback, a support-vector-based fuzzy neural network (SVFNN) is proposed for pattern classification in this paper. The SVFNN combines the superior classification power of support vector machine (SVM) in high dimensional data spaces and the efficient human-like reasoning of FNN in handling uncertainty information. A learning algorithm consisting of three learning phases is developed to construct the SVFNN and train its parameters. In the first phase, the fuzzy rules and membership functions are automatically determined by the clustering principle. In the second phase, the parameters of FNN are calculated by the SVM with the proposed adaptive fuzzy kernel function. In the third phase, the relevant fuzzy rules are selected by the proposed reducing fuzzy rule method. To investigate the effectiveness of the proposed SVFNN classification, it is applied to the Iris, Vehicle, Dna, Satimage, Ijcnn1 datasets from the UCI Repository, Statlog collection and IJCNN challenge 2001, respectively. Experimental results show that the proposed SVFNN for pattern classification can achieve good classification performance with drastically reduced number of fuzzy kernel functions.

Stability and stabilization of a class of fuzzy time-delay descriptor systems

Abstract: This paper studies a class of fuzzy time-delay descriptor systems in the extended Takagi-Sugeno (T-S) fuzzy model. Sufficient conditions are derived for the stability and stabilization in terms of linear matrix inequalities (LMIs). Illustrative examples are given to show the effectiveness and the advantages of the present results

Convergent results about the use of fuzzy simulation in fuzzy optimization problems

Abstract: We discuss the convergence of fuzzy simulation as it is employed in fuzzy optimization problems. Several convergence concepts for sequences of fuzzy variables are defined such as convergence in optimistic value. A new approach to approximating essentially bounded fuzzy variables with continuous possibility distributions is introduced. Applying the proposed approximation method to our previous work, we prove three convergence theorems about the use of fuzzy simulation in computing the credibility of a fuzzy event, finding the optimistic value of a return function, and calculating the expected value of a fuzzy variable

Design for Self-Organizing Fuzzy Neural Networks Based on Genetic Algorithms

Abstract: A novel hybrid learning algorithm based on a genetic algorithm to design a growing fuzzy neural network, named self-organizing fuzzy neural network based on genetic algorithms (SOFNNGA), to implement Takagi-Sugeno (TS) type fuzzy models is proposed in this paper. A new adding method based on geometric growing criterion and the epsiv-completeness of fuzzy rules is first used to generate the initial structure. Then a hybrid algorithm based on genetic algorithms, backpropagation, and recursive least squares estimation is used to adjust all parameters including the number of fuzzy rules. This has two steps: First, the linear parameter matrix is adjusted, and second, the centers and widths of all membership functions are modified. The GA is introduced to identify the least important neurons, i.e., the least important fuzzy rules. Simulations are presented to illustrate the performance of the proposed algorithm

Type-2 fuzzy hidden Markov models and their application to speech recognition

Abstract: This paper presents an extension of hidden Markov models (HMMs) based on the type-2 (T2) fuzzy set (FS) referred to as type-2 fuzzy HMMs (T2 FHMMs). Membership functions (MFs) of T2 FSs are three-dimensional, and this new third dimension offers additional degrees of freedom to evaluate the HMMs fuzziness. Therefore, T2 FHMMs are able to handle both random and fuzzy uncertainties existing universally in the sequential data. We derive the T2 fuzzy forward-backward algorithm and Viterbi algorithm using T2 FS operations. In order to investigate the effectiveness of T2 FHMMs, we apply them to phoneme classification and recognition on the TIMIT speech database. Experimental results show that T2 FHMMs can effectively handle noise and dialect uncertainties in speech signals besides a better classification performance than the classical HMMs.

LMI-based Integral fuzzy control of DC-DC converters

Abstract: In this paper, we propose a T-S fuzzy controller which combines the merits of: i) the capability for dealing with nonlinear systems; ii) the powerful LMI approach to obtain control gains; iii) the high performance of integral controllers; iv) the workable rigorous proof for exponential convergence of error signals; and v) the flexibility on tuning decay rate. The output regulation problems of a basic buck converter and a zero-voltage-transition (ZVT) buck converter are used as application examples to illustrate the control performance of the proposed methodology. First, we consider a general nonlinear system which can represent the large-signal models of the converters. After introducing an added integral state of output regulation error and taking coordinate translation on an equilibrium point, the resulting augmented system is represented into a Takagi-Sugeno (T-S) fuzzy model. Then, the concept of parallel distributed compensation is applied to design the control law whereby the control gains are obtained by solving linear matrix inequalities (LMIs). An interesting result is that the obtained control law is formed only by the linear state feedback signals weighted by grade functions. In addition, the robustness analysis is carried out when uncertainty and disturbance are taken into consideration. The performance of numerical simulations and practical experiments results is satisfactory.

Output Tracking Control for Fuzzy Systems Via Output Feedback Design

Abstract: Fuzzy observer-based control design is proposed to deal with the output tracking problem for nonlinear systems. For the purpose of tracking design, the new concept of virtual desired variables and, in turn the so-called generalized kinematics are introduced to simplify the design procedure. In light of this concept, the design procedure is split into two steps: i) Determine the virtual desired variables from the generalized kinematics; and ii) Determine the control gains just like solving linear matrix inequalities for stabilization problem. For immeasurable state variables, output feedback design is proposed. Here, we focus on a common feature held by many physical systems where their membership functions of fuzzy sets satisfy a Lipschitz-like property. Based on this setting, control gains and observer gains can be designed separately. Moreover, zero tracking error and estimation error are concluded. Three different types of systems, including nonlinear mass-spring systems, dc-dc converters, and induction motors are considered to demonstrate the design procedure. Their satisfactory simulation results verify the proposed approach

Distributivity and conditional distributivity of a uninorm and a continuous t-conorm

Abstract: The open problem recalled by Klement in the Linz2000 closing session, related to distributivity and conditional distributivity of a uninorm and a continuous t-conorm, is solved for the most usual known classes of uninorms. From the obtained results, it is deduced that distributivity and conditional distributivity are equivalent for these cases. It is remarkable that solutions appear involving not only strict t-conorms but also ordinal sums of the maximum with a strict t-conorm. Conversely, the distributivity of a t-conorm over a uninorm is also studied leading only to already known solutions. Moreover, the dual case of distributivity and conditional distributivity involving uninorms and continuous t-norms is also solved, proving again the equivalence of both kinds of distributivities

Robust $H_{\infty}$ Fuzzy Control Approach for a Class of Markovian Jump Nonlinear Systems

Abstract: This paper addresses the problem of stabilizing a class of nonlinear systems subject to Markovian jump parameters using a robust stochastic fuzzy controller with Hinfin performance. The class of jump nonlinear systems considered is described by a fuzzy model composed of two levels: A crisp level which represents the jumps and a fuzzy level which represents the system nonlinearities. Considering the approximation error between the fuzzy model and the jump nonlinear system as norm-bounded uncertainties, we develop a systematic technique based on a coupled Lyapunov function to obtain a robust stochastic fuzzy controller which guarantees the L2 gain of the closed-loop system in respect to external inputs to be equal to or less than a prescribed value. A simulation example on an industrial power plant operating in a cogeneration scheme is presented to illustrate the effectiveness of the proposed stabilizing controller in reducing oscillations as well as maintaining a desired operation condition in the presence of fluctuations in the local load

Observability and decentralized control of fuzzy discrete-event systems

Abstract: Fuzzy discrete-event systems as a generalization of (crisp) discrete-event systems have been introduced in order that it is possible to effectively represent uncertainty, imprecision, and vagueness arising from the dynamic of systems. A fuzzy discrete-event system has been modeled by a fuzzy automaton; its behavior is described in terms of the fuzzy language generated by the automaton. In this paper, we are concerned with the supervisory control problem for fuzzy discrete-event systems with partial observation. Observability, normality, and co-observability of crisp languages are extended to fuzzy languages. It is shown that the observability, together with controllability, of the desired fuzzy language is a necessary and sufficient condition for the existence of a partially observable fuzzy supervisor. When a decentralized solution is desired, it is proved that there exist local fuzzy supervisors if and only if the fuzzy language to be synthesized is controllable and co-observable. Moreover, the infimal controllable and observable fuzzy superlanguage, and the supremal controllable and normal fuzzy sublanguage are also discussed. Simple examples are provided to illustrate the theoretical development

Implicative Fuzzy Associative Memories

Abstract: Associative neural memories are models of biological phenomena that allow for the storage of pattern associations and the retrieval of the desired output pattern upon presentation of a possibly noisy or incomplete version of an input pattern. In this paper, we introduce implicative fuzzy associative memories (IFAMs), a class of associative neural memories based on fuzzy set theory. An IFAM consists of a network of completely interconnected Pedrycz logic neurons with threshold whose connection weights are determined by the minimum of implications of presynaptic and postsynaptic activations. We present a series of results for autoassociative models including one pass convergence, unlimited storage capacity and tolerance with respect to eroded patterns. Finally, we present some results on fixed points and discuss the relationship between implicative fuzzy associative memories and morphological associative memories

Soft transition from probabilistic to possibilistic fuzzy clustering

Abstract: In the fuzzy clustering literature, two main types of membership are usually considered: A relative type, termed probabilistic, and an absolute or possibilistic type, indicating the strength of the attribution to any cluster independent from the rest. There are works addressing the unification of the two schemes. Here, we focus on providing a model for the transition from one schema to the other, to exploit the dual information given by the two schemes, and to add flexibility for the interpretation of results. We apply an uncertainty model based on interval values to memberships in the clustering framework, obtaining a framework that we term graded possibility. We outline a basic example of graded possibilistic clustering algorithm and add some practical remarks about its implementation. The experimental demonstrations presented highlight the different properties attainable through appropriate implementation of a suitable graded possibilistic model. An interesting application is found in automated segmentation of diagnostic medical images, where the model provides an interactive visualization tool for this task

Reliable $rm H_infty $ Fuzzy Control for Continuous-Time Nonlinear Systems With Actuator Failures

Abstract: This paper is concerned with the design of reliable Hinfin fuzzy controllers for continuous-time nonlinear systems with actuator failures. The Takagi and Sugeno fuzzy model is employed to represent a nonlinear system. The objective is to find a stabilizing state-feedback fuzzy controller such that the nominal Hinfin performance is optimized while satisfying a prescribed Hinfin performance constraint in the actuator failure cases. Based on the linear matrix inequality (LMI) techniques, two efficient methods for the design of a suboptimal reliable Hinfin fuzzy controller are proposed. Different Lyapunov functions are used during the design for the nominal and actuator failure cases, which lead to a less conservative controller design. In the first method, a single Lyapunov function is used for the actuator failure cases. The second method adopts a parameter-dependent Lyapunov function for the actuator failure cases, which further reduces the conservatism of the design. Finally, numerical simulations on the chaotic Rossler system are given to illustrate the effectiveness of the proposed design methods

Robust tracking control of an electrically driven robot: adaptive fuzzy logic approach

Abstract: This paper is concerned with the robust tracking control of an electrically driven robot with the model uncertainties in the robot dynamics and the motor dynamics. The motors driving the joints of the robot are assumed to be equipped with only the joint position and the current measurement devices. Adaptive fuzzy logic and adaptive backstepping method are employed to provide the solution to the control problem. The suggested method does not require the measurement of the velocity nor the acceleration. Simulation results from a two-link electrically driven robot show the satisfactory performance of the proposed control scheme even in the presence of internal model uncertainties in both the robot and motor dynamics and external disturbances

Model reference output feedback fuzzy tracking control design for nonlinear discrete-time systems with time-delay

Abstract: In this study, a model reference fuzzy tracking control design for nonlinear discrete-time systems with time-delay is introduced. First, the Takagi and Sugeno (TS) fuzzy model is employed to approximate a nonlinear discrete-time system with time-delay. Next, based on the fuzzy model, a fuzzy observer-based fuzzy controller is developed to reduce the tracking error as small as possible for all bounded reference inputs. The advantage of proposed tracking control design is that only a simple fuzzy observer-based controller is used in our approach without feedback linearization technique and complicated adaptive scheme. By the proposed method, the fuzzy tracking control design problem is parameterized in terms of a linear matrix inequality problem (LMIP). The LMIP can be efficiently solved using the convex optimization techniques. Simulation example is given to illustrate the design procedures and tracking performance of the proposed method.

On the properties of OWA operator weights functions with constant level of orness

Abstract: The result of aggregation performed by the ordered weighted averaging (OWA) operator heavily depends upon the weighting vector used. A number of methods have been presented for obtaining the associated weights. In this paper, we present analytic forms of OWA operator weighting functions, each of which has properties of rank-based weights and a constant level of orness, irrespective of the number of objectives considered. These analytic forms provide significant advantages for generating the OWA weights over previously reported methods. First, the OWA weights can be efficiently generated by using proposed weighting functions without solving a complicated mathematical program. Moreover, convex combinations of these specific OWA operators can be used to generate the OWA operators with any predefined values of orness once specific values of orness are a priori stated by the decision maker. Those weights have a property of constant level of orness as well. Finally, the OWA weights generated at a predefined value of orness make almost no numerical difference with maximum entropy OWA weights in terms of dispersion

Fuzzy H/sub /spl infin// output feedback control design for singularly perturbed systems with pole placement constraints: an LMI approach

Abstract: This paper examines the problem of designing an H/sub /spl infin// output feedback controller with pole placement constraints for singular perturbed Takagi-Sugeno (TS) fuzzy models. We propose a fuzzy H/sub /spl infin// output feedback controller that not only guarantees the /spl Lscr//sub 2/-gain of the mapping from the exogenous input noise to the regulated output to be less than some prescribed value, but also ensures closed-loop poles of each subsystem are in a prespecified linear matrix inequality (LMI) region. In order to alleviate the numerical stiffness caused by the singular perturbation /spl epsiv/, the design technique is formulated in terms of a family of /spl epsiv/-independent linear matrix inequalities. The proposed approach can be applied both standard and nonstandard singularly perturbed nonlinear systems. A numerical example is provided to illustrate the design developed in this paper.

Ranking type-2 fuzzy numbers

Abstract: Type-2 fuzzy sets are a generalization of the ordinary fuzzy sets in which each type-2 fuzzy set is characterized by a fuzzy membership function. In this paper, we consider the problem of ranking a set of type-2 fuzzy numbers. We adopt a statistical viewpoint and interpret each type-2 fuzzy number as an ensemble of ordinary fuzzy numbers. This enables us to define a type-2 fuzzy rank and a type-2 rank uncertainty for each intuitionistic fuzzy number. We show the reasonableness of the results obtained by examining several test cases

Mixed Feedforward/Feedback Based Adaptive Fuzzy Control for a Class of MIMO Nonlinear Systems

Abstract: This paper proposes a mixed feedforward/feedback (FFB) based adaptive fuzzy controller design for a class of multiple-input-multiple-output (MIMO) uncertain nonlinear systems. By integrating both feedforward and feedback compensation, we introduce the FFB-based fuzzy controller composed of a feedforward fuzzy compensator and a robust error-feedback compensator. To achieve a forward compensation of uncertainties, the feedforward fuzzy compensator takes the desired commands as premise variables of fuzzy rules and adaptively adjusts the consequent part from an error measure. Meanwhile, the feedback controller part is constructed based on Hinfin control techniques and nonlinear damping design. Then, the attenuation of both disturbances and estimated fuzzy parametric errors is guaranteed from a linear matrix inequality (LMI)-based gain design. The main advantages are: i) a simpler architecture for implementation is provided; and ii) the typical boundedness of assumption on fuzzy universal approximation errors is not required. Finally, an inverted pendulum system and a two-link robot are taken as application examples to show the expected performance

Deriving Analytical Input–Output Relationship for Fuzzy Controllers Using Arbitrary Input Fuzzy Sets and Zadeh Fuzzy AND Operator

Abstract: A fuzzy controller uses either Zadeh or product fuzzy AND operator, with the former being more frequently used than the latter. We have recently published a novel technique for deriving analytical input-output relation for the fuzzy controllers that use Zadeh AND operator and arbitrary trapezoidal input fuzzy sets, including triangular ones as special cases. In this paper, we have developed a general technique based on that technique to cover arbitrary types of input fuzzy sets. Moreover, we have established some necessary and sufficient conditions to characterize general relationship between shape of input fuzzy sets and shape of input space divisions, an important and integral issue because analytical relationship differs in different regions of input space. The new technique and the shape relations are applicable to any type of fuzzy controllers (e.g., Mamdani type or Takagi-Sugeno type). The analytical structures that we have derived provide an unprecedented opportunity to insightfully and rigorously examine the advantages and shortcomings of different design choices available for various components of the fuzzy controllers. We have focused on type selection for input fuzzy sets of Mamdani fuzzy controllers. Our preliminary analysis indicates that the fuzzy controllers using trapezoidal fuzzy sets may be understood (and possibly analyzed and designed) more sensibly and easily in the context of conventional control theory than the fuzzy controllers using any other types of fuzzy sets. Our proposition is that trapezoidal fuzzy sets should be the first choice and used most of time. Possible implication for automatic learning of input fuzzy sets via neural networks or genetic algorithms is briefly discussed