Mamdani Fuzzy System Research Articles

Cooperative Evolutionary Learning of Linguistic Fuzzy Rules and Parametric Aggregation Connectors for Mamdani Fuzzy Systems

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> There are two tasks in the design of linguistic fuzzy models for a concrete application: The derivation of the linguistic rule base and the setup of the inference system and the defuzzification method. Traditionally, the derivation of the linguistic rule base has been considered the most important task, but the use of the appropriate aggregation connectors in the inference system and the defuzzification interface can improve the fuzzy system behavior. In this paper, we take in consideration this idea, we propose an evolutionary learning method to learn a linguistic rule base and the parametric aggregation connectors of the inference and defuzzification in a single step. The aim of this methodology is to make possible a high level of positive synergy between the linguistic rule base and the aggregation connectors, improving the accuracy of the linguistic Mamdani fuzzy systems. Our proposal has shown good results solving three different applications. We introduce a statistical analysis of results for validating the model behavior on the applications used in the experimental study. We must remark that we present an experimental study with a double intention: a) to compare the behavior of the new approach in comparison with those ones that first learn the rule base and then adapt the connectors, and b) to analyze the rule bases obtained with fixed aggregation connectors and with the adaptive ones for showing the changes on the consequent rules, changes on labels that produce a better behavior of the linguistic model than the classic ones. </para>

PI-Fuzzy controllers for integral plants to ensure robust stability

New PI-fuzzy controllers comprising Takagi–Sugeno and Mamdani fuzzy systems to control a class of integral plants specific to servo systems are proposed in this paper. Linear PI controllers are designed in the first phase. They are tuned using the Extended Symmetrical Optimum method to ensure the desired overshoot and settling time with respect to step setpoint modifications and three types of load disturbance inputs. This type of PI controller design guarantees robust stability of the closed-loop system in response to parametric variations in a controlled plant. Following this, based on the results from the linear case and the modal equivalence principle, an original development method for PI-fuzzy controllers is proposed. Some experimental results are included to illustrate the effectiveness of the design process.

Fuzzy Anti-windup Schemes for NCTF Control of Point-to-point (PTP) Positioning Systems

The positioning systems generally need a controller to achieve high accuracy, fast response and robustness. In addition, ease of controller design and simplicity of controller structure are very important for practical application. For satisfying these requirements, NCTF (nominal characteristic trajectory following) controller has been proposed as a practical PTP positioning control. However, the effect of actuator saturation can not be completely compensated due to integrator windup because of plant parameter variations. This study presents a method to improve the NCTF controller for overcoming the problem of integrator windup by adopting fuzzy anti-windup schemes. Two fuzzy anti-windup schemes based on Mamdani and Takagi-Sugeno fuzzy system are developed and evaluated their effectiveness. The improved NCTF controller with the proposed fuzzy anti-windup schemes is evaluated through simulation using dynamic model of a rotary positioning system. The results show that the improved NCTF controller with Takagi-Sugeno-based fuzzy windup is the best scheme to compensate for the effect of integrator windup.

A Pareto-based multi-objective evolutionary approach to the identification of Mamdani fuzzy systems

In the last years, the numerous successful applications of fuzzy rule-based systems (FRBSs) to several different domains have produced a considerable interest in methods to generate FRBSs from data. Most of the methods proposed in the literature, however, focus on performance maximization and omit to consider FRBS comprehensibility. Only recently, the problem of finding the right trade-off between performance and comprehensibility, in spite of the original nature of fuzzy logic, has arisen a growing interest in methods which take both the aspects into account. In this paper, we propose a Pareto-based multi-objective evolutionary approach to generate a set of Mamdani fuzzy systems from numerical data. We adopt a variant of the well-known (2+2) Pareto Archived Evolutionary Strategy ((2+2)PAES), which adopts the one-point crossover and two appropriately defined mutation operators. (2+2)PAES determines an approximation of the optimal Pareto front by concurrently minimizing the root mean squared error and the complexity. Complexity is measured as sum of the conditions which compose the antecedents of the rules included in the FRBS. Thus, low values of complexity correspond to Mamdani fuzzy systems characterized by a low number of rules and a low number of input variables really used in each rule. This ensures a high comprehensibility of the systems. We tested our version of (2+2)PAES on three well-known regression benchmarks, namely the Box and Jenkins Gas Furnace, the Mackey-Glass chaotic time series and Lorenz attractor time series datasets. To show the good characteristics of our approach, we compare the Pareto fronts produced by the (2+2)PAES with the ones obtained by applying a heuristic approach based on SVD-QR decomposition and four different multi-objective evolutionary algorithms.

ROBUST ADAPTIVE FUZZY TRACKING CONTROL FOR A CLASS OF PERTURBED UNCERTAIN NONLINEAR SYSTEMS WITH UVCGF

In this paper, a robust adaptive fuzzy controller is presented for a wide class of perturbed uncertain nonlinear system with unknown virtual control gain function (UVCGF). The Mamdani fuzzy system is used to approximate unstructured uncertain functions in the system. The proposed algorithm, which incorporated Nussbaum-type gain into the decoupled backstepping approach, does not require a priori knowledge of the sign of UVCGF, and circumvents the controller-singularity problem gracefully in some existing literatures. It proved that the tracking error can be driven to a small residual set while keeping all signals in the closed-loop system semi-globally uniformly ultimately bounded (SGUUB). Simulation results are presented to validate the effectiveness of the proposed controller.

Fast inference using transition matrices: an extension to nonlinear operators

Fast inference using transition matrices (FITMs) was recently presented as a new fast algorithm for performing inferences in fuzzy systems. However, this algorithm only applies to the standard additive model (SAM) using the sum-product inference composition. In this paper, we show how this methodology carries over to more general Mamdani fuzzy systems, i.e., those using t-norms and t-conorms that satisfy a requirement of distributivity. In addition, FITM, as originally described, requires inputs to be fuzzy sets. We here extend this methodology to handle crisp inputs as well.

Mamdani fuzzy system: universal approximator to a class of random processes

The issue of fuzzy systems as universal approximators has drawn significant attention, but all results obtained are restricted to deterministic input-output (I/O) relationships. It should be noted that, in practice, many I/O systems, including fuzzy systems, operate in the environment which is essentially stochastic. In this paper, the Mamdani fuzzy systems are generalized as stochastic systems. By proving the Mamdani systems as universal approximators with L/sup 2/-norm, the approximation capability of the stochastic Mamdani systems to a class of random processes is systematically analyzed. In the mean square sense, such stochastic fuzzy systems are capable of approximating the prescribed random processes with arbitrary accuracy. Further, an efficient learning algorithm for the stochastic Mamdani systems is developed. Finally, a simulation example is employed to demonstrate our results.

TRAINING NEURAL NETWORKS AND NEURO-FUZZY SYSTEMS: A UNIFIED VIEW

Neural and neuro-fuzzy models are powerful nonlinear modelling tools. Different structures, with different properties, are widely used to capture static or dynamical nonlinear mappings. Static (non-recurrent) models share a common structure: a nonlinear stage, followed by a linear mapping. In this paper, the separability of linear and nonlinear parameters is exploited for completely supervised training algorithms. Examples of this unified view are presented, involving multilayer perceptrons, radial basis functions, wavelet networks, B-splines, Mamdani and TSK fuzzy systems.

Analyses for Lp(μ)-norm approximation capability of generalized Mamdani fuzzy systems

The connected piecewise linear functions are introduced to study the L p ( μ )-norm approximation capability of the generalized Mamdani fuzzy system. It is shown that such general fuzzy systems are the universal approximators to the p -integrable functions with L p ( μ )-norm. For the given degree of accuracy an upper bound of the fuzzy rule numbers needed in the fuzzy rule base of the system is estimated. Finally a simulation example is analyzed to demonstrate our conclusions.

Necessary conditions on minimal system configuration for general MISO Mamdani fuzzy systems as universal approximators

Recent studies have shown that both Mamdani-type and Takagi-Sugeno-type fuzzy systems are universal approximators in that they can uniformly approximate continuous functions defined on compact domains with arbitrarily high approximation accuracy. In this paper, we investigate necessary conditions for general multiple-input single-output (MISO) Mamdani fuzzy systems as universal approximators with as minimal system configuration as possible. The general MISO fuzzy systems employ almost arbitrary continuous input fuzzy sets, arbitrary singleton output fuzzy sets, arbitrary fuzzy rules, product fuzzy logic AND, and the generalized defuzzifier containing the popular centroid defuzzifier as a special case. Our necessary conditions are developed under the practically sensible assumption that only a finite set of extrema of the multivariate continuous function to be approximated is available. We have first revealed a decomposition property of the general fuzzy systems: A r-input fuzzy system can always be decomposed to the sum of r simpler fuzzy systems where the first system has only one input variable, the second one two input variables, and the last one r input variables. Utilizing this property, we have derived some necessary conditions for the fuzzy systems to be universal approximators with minimal system configuration. The conditions expose the strength as well as limitation of the fuzzy approximation: (1) only a small number of fuzzy rules may be needed to uniformly approximate multivariate continuous functions that have a complicated formulation but a relatively small number of extrema; and (2) the number of fuzzy rules must be large in order to approximate highly oscillatory continuous functions. A numerical example is given to demonstrate our new results.

Comparison of necessary conditions for typical Takagi-Sugeno and Mamdani fuzzy systems as universal approximators

Both Takagi-Sugeno (TS) and Mamdani fuzzy systems are known to be universal approximators. We investigate whether one type of fuzzy approximators is more economical than the other. The TS fuzzy systems are the typical two-input single-output TS fuzzy systems. We first establish necessary conditions on minimal system configuration of the TS fuzzy systems as function approximators. We show that the number of the input fuzzy sets and fuzzy rules needed by the TS fuzzy systems depend on the number and locations of the extrema of the function to be approximated. The resulting conditions reveal the strength of the TS fuzzy approximators. The drawback, though, is that a large number of fuzzy rules must be employed to approximate periodic or highly oscillatory functions. We then compare these necessary conditions with the ones that we established for the general Mamdani fuzzy systems in our previous papers. Results of the comparison unveil that the minimal system configurations of the TS and Mamdani fuzzy systems are comparable. Finally, we prove that the minimal configuration of the TS fuzzy systems can be reduced and becomes smaller than that of the Mamdani fuzzy systems if nontrapezoidal or nontriangular input fuzzy sets are used. We believe that all the results in present paper hold for the TS fuzzy systems with more than two input variables but the proof seems to be mathematically difficult. Our new findings are valuable in designing more compact fuzzy systems, especially fuzzy controllers and models which are two most popular and successful applications of the fuzzy approximators.

Some remarks on the stability of Mamdani fuzzy control systems

This paper presents a passivity approach to the stability analysis of Mamdani fuzzy control systems. The main idea of the paper is that the fuzzy controller can be considered from an input-output viewpoint as a nonlinear dissipative operator. An input-output approach is provided by considering a dynamical dependency of inputs of the fuzzy controller. Then the passivity theory is applied in order to obtain conditions for absolute stability of the closed-loop control system.

Lyapunov-like stability theorems for continuous-time fuzzy control systems

The essential characteristics of fuzzy logic control are analysed in this paper. The locality of fuzzy control and the interpolation among fuzzy rules may be the most important characteristics of fuzzy control systems. Based on these two characteristics a new Lyapunov-like stability analysis approach is developed for both Mamdani and Takagi-Sugeno type fuzzy control systems. In this approach we apply the theory of generalized dynamical systems and contingent equations. Thus, the method can be used for both model-free and model-based fuzzy controller designs. Finally, an example is given to illustrate the application of the method.

An investigation of acidic phospholipids of myelin using CM cellulose chromatography and high-performance liquid chromatography

In this work we consider the application of an adaptive neural network control for a class of single input single output non linear systems. The method uses a neural network system of Radial Basis Function (RBF) type to approximate the feedback linearization law and a fuzzy inference system of Mamdani type to estimate the control signal error between the ideal unknown control signal and the actual control signal. The rule base of the Mamdani fuzzy system is constructed using simple expert reasoning. The parameters of the (RBF) controller are adapted and changed using the gradient descent law based on the estimated control error. The simulation is carried out on a three tanks system with the objective of controlling the level of one tank. The simulation results show that the proposed RBF-Mamdani scheme performs successful and robust control in comparison to the results obtained using a PI controller.