Identification of dynamical systems with a robust interval fuzzy model

In this paper we present a new method of interval fuzzy model identification. The method combines a fuzzy identification methodology with some ideas from linear programming theory. On a finite set of measured data, an optimality criterion that minimizes the maximal estimation error between the data and the proposed fuzzy model output is used. The idea is then extended to modelling the optimal lower and upper bound functions that define the band that contains all the measurement values. This results in a lower and an upper fuzzy model or a fuzzy model with a set of lower and upper parameters. The model is called the interval fuzzy model (INFUMO). The method can be used when describing a family of uncertain nonlinear functions or when the systems with uncertain physical parameters are observed. We believe that the fuzzy interval model can be very efficiently used, especially in fault detection and in robust control design.     

Interval fuzzy modeling applied to Wiener models with uncertainties.

This correspondence addresses the problem of interval fuzzy model identification and its use in the case of the robust Wiener model. The method combines a fuzzy identification methodology with some ideas from linear programming theory. On a finite set of measured data, an optimality criterion which minimizes the maximum estimation error between the data and the proposed fuzzy model output is used. The min-max optimization problem can then be seen as a linear programming problem that is solved to estimate the parameters of the fuzzy model in each fuzzy domain. This results in lower and upper fuzzy models that define the confidence interval of the observed data. The model is called the interval fuzzy model and is used to approximate the static nonlinearity in the case of the Wiener model with uncertainties. The resulting model has the potential to be used in the areas of robust control and fault detection.     

A robust design criterion for interpretable fuzzy models with uncertain data

We believe that nonlinear fuzzy filtering techniques may be turned out to give better robustness performance than the existing linear methods of estimation (H/sup 2/ and H/sup /spl infin// filtering techniques), because of the fact that not only linear parameters (consequents), but also the nonlinear parameters (membership functions) attempt to identify the uncertain behavior of the unknown system. However, the fuzzy identification methods must be robust to data uncertainties and modeling errors to ensure that the fuzzy approximation of unknown system's behavior is optimal in some sense. This study presents a deterministic approach to the robust design of fuzzy models in the presence of unknown but finite uncertainties in the identification data. We consider online identification of an interpretable fuzzy model, based on the robust solution of a regularized least-squares fuzzy parameters estimation problem. The aim is to resolve the difficulties associated with the robust fuzzy identification method due to lack of a priori knowledge about upper bounds on the data uncertainties. The study derives an optimal level of regularization that should be provided to ensure the robustness of fuzzy identification strategy by achieving an upper bound on the value of energy gain from data uncertainties and modeling errors to the estimation errors. A time-domain feedback analysis of the proposed identification approach is carried out with emphasis on stability, robustness, and steady-state issues. The simulation studies are provided to show the superiority of the proposed fuzzy estimation over the classical estimation methods.     

NOE TS fuzzy modelling of nonlinear dynamic systems with uncertainties using symbolic interval-valued data

An approach to Nonlinear Output Error (NOE) modelling using Takagi–Sugeno (TS) fuzzy model for a class of nonlinear dynamic systems having variability in their outputs is presented. Furthermore, the approach is compared and graphically illustrated with other alternate approaches on the basis of interval data and interval membership functions. Assuming the identification method can be repeated offline a number of times under similar conditions, multiple input–output time series can be obtained from the underlying system. These time series are pre-processed using the techniques of statistics and probability theory to generate the envelopes of response (curves outlining the upper and lower extremes of response) at each time instant. Two types of envelopes are described in this research: the max–min envelopes and the envelopes based on the confidence intervals provided by extended Chebyshev's inequality. By incorporating interval data in fuzzy modelling and using the theory of symbolic interval-valued data, a TS fuzzy model with interval antecedent and consequent parameters is obtained. This algorithm provides a model for predicting the expected response as well as envelopes. In order to validate the presented model, a simulation case study is devised in this paper. Moreover, it is demonstrated on the real data obtained from an electro-mechanical throttle valve.     

Development of a systematic methodology of fuzzy logic modeling

This paper proposes a systematic methodology of fuzzy logic modeling for complex system modeling. It has a unified parameterized reasoning formulation, an improved fuzzy clustering algorithm, and an efficient strategy of selecting significant system inputs and their membership functions. The reasoning mechanism introduces 4 parameters whose variation provides a continuous range of inference operation. As a result, we are no longer restricted to standard extremes in any step of reasoning. The fuzzy model itself can then adjust the reasoning process by optimizing the inference parameters based on input-output data. The fuzzy rules are generated through fuzzy c-means (FCM) clustering. Major bottlenecks are addressed and analytical solutions are suggested. We also address the classification process to extend the derived fuzzy partition to the entire output space. In order to select suitable input variables among a finite number of candidates (unlike traditional approaches) we suggest a new strategy through which dominant input parameters are assigned in one step and no iteration process is required. Furthermore, a clustering technique called fuzzy fine clustering is introduced to assign the input membership functions. In order to evaluate the proposed methodology, two examples-a nonlinear function and a gas furnace dynamic procedure-are investigated in detail. The significant improvement of the model is concluded compared to other fuzzy modeling approaches     

C-R模糊模型在连续搅拌反应釜中的应用

针对化工生产过程中强非线性对象的动态建模问题,本文提出了具有代表性的连续搅拌反应釜的非机理模糊建模方法。从系统的输入输出数据出发,根据辨识精度将系统空间划分为个子空间,然后由隶属函数将子空间模型联接成全局模型来表征系统的整体非线性特性。仿真结果表明,该模型输出能够很好地跟踪系统的参考输出,效果明显。     

Fuzzy bounded least-squares method for the identification of linearsystems

This paper presents the fuzzy bounded least-squares method which uses both linguistic information and numerical data to identify linear systems. This method introduces a new type of fuzzy system, i.e., a fuzzy interval system. The steps in the method are as follows: 1) to utilize all the available linguistic information to obtain a fuzzy interval system and then to use the fuzzy interval system to give the admissible model set (i.e., the set of all models which are acceptable and reasonable from the point of view of linguistic information); 2) to find a model in the admissible model set which best fits the available numerical data. It is shown that such a model can be obtained by a quadratic programming approach. By comparing this method with the least-squares method, it is proved that the model obtained by this method fits a real system better than the model obtained by the least-squares method. In addition, this method also checks the adequacy of linear models for modeling a given system during the identification process and can help one to decide whether it is necessary to use nonlinear models     

A novel data reduction method for Takagi–Sugeno fuzzy system design based on statistical design of experiment

This paper introduces a simple, systematic and effective method for designing Takagi–Sugeno (T–S) fuzzy systems utilizing a significantly smaller training data set versus existing methods. Creating proper training data is usually not an easy task and requires spending considerable time and resources. The proposed method first uses a three-level factorial design to partition the output space. Next the least square technique is used to estimate each of the partitioned output spaces. The membership functions are introduced with only three variables (min, max and number of membership functions). Fuzzy rules are generated with respect to the partitioned output surfaces and the membership functions. The proposed method is applied to two benchmark problems, controlling an inverted pendulum as well as modeling a nonlinear function. In the case of the inverted pendulum simulation results demonstrate significant improvement. In the case of nonlinear function modeling we demonstrated sufficient accuracy with only 9 training data, which represents 98% reduction in the number of training data compare to other method. Additionally, the proposed method offers extremely low computation time allowing it to be used with adaptive type systems.     

带结构风险最小化的最优区间回归模型辨识

针对来自模型结构、参数以及测量数据的不确定性等因素,传统的辨识方法获取的是确定性数学模型的点输出,其鲁棒性差,易受外界干扰.因此,采用区间输出比点输出更易于实际问题的研究.基于复杂系统的不确定性测量数据以及系统参数的不确定性,提出了最优区间回归模型辨识的一种新方法,该方法将逼近误差的L∞范数思想与结构风险最小化理论相结合,建立求解区间模型的最优化问题,应用线性规划独立求解区间模型的上界和下界模型.该方法在保证模型辨识精度的同时,其泛化性能得到进一步提高.实验分析表明,提出的方法对来自噪声以及参数不确定性的数据,可以从区间模型的辨识精度和泛化性能之间取其平衡.     

一种线性化模糊内模自适应控制算法

针对非线性对象，提出一种线性化模糊内模自适应控制算法。该算法以一组模糊规则作为非线性对象内部模型，一条模糊规则表示一个局部线性系统；根据对象输入与输出测量值，利用TSK建模方法在线辨识局部模糊内部模型；同时依据辨识模型设计局部H2最优模糊控制规则，所有规则构成H2最优模糊控制器。仿真实验显示：该算法适用于非线性对象的控制，具有较好的鲁棒性和抗干扰能力。     

Adaptive fuzzy identification and predictive control for industrial processes

This paper proposes a method for adaptive identification and control for industrial applications. The learning of a T–S fuzzy model is performed from input/output data to approximate unknown nonlinear processes by a hierarchical genetic algorithm (HGA). The HGA approach is composed by five hierarchical levels where the following parameters of the T–S fuzzy system are learned: input variables and their respective time delays, antecedent fuzzy sets, consequent parameters, and fuzzy rules. In order to reduce the computational cost and increase the algorithm’s performance an initialization method is applied on HGA. To deal with nonlinear plants and time-varying processes, the T–S fuzzy model is adapted online to maintain the quality of the identification/control. The identification methodology is proposed for two application problems: (1) the design of data-driven soft sensors, and (2) the learning of a model for the Generalized predictive control (GPC) algorithm. The integration of the proposed adaptive identification method with the GPC results in an effective adaptive predictive fuzzy control methodology. To validate and demonstrate the performance and effectiveness of the proposed methodologies, they are applied on identification of a model for the estimation of the flour concentration in the effluent of a real-world wastewater treatment system; and on control of a simulated continuous stirred tank reactor (CSTR) and on a real experimental setup composed of two coupled DC motors. The results are presented, showing that the developed evolving T–S fuzzy model can identify the nonlinear systems satisfactorily and it can be used successfully as a prediction model of the process for the GPC controller.     

Fuzzy Regression Analysis by Support Vector Learning Approach

Support vector machines (SVMs) have been very successful in pattern classification and function approximation problems for crisp data. In this paper, we incorporate the concept of fuzzy set theory into the support vector regression machine. The parameters to be estimated in the SVM regression, such as the components within the weight vector and the bias term, are set to be the fuzzy numbers. This integration preserves the benefits of SVM regression model and fuzzy regression model and has been attempted to treat fuzzy nonlinear regression analysis. In contrast to previous fuzzy nonlinear regression models, the proposed algorithm is a model-free method in the sense that we do not have to assume the underlying model function. By using different kernel functions, we can construct different learning machines with arbitrary types of nonlinear regression functions. Moreover, the proposed method can achieve automatic accuracy control in the fuzzy regression analysis task. The upper bound on number of errors is controlled by the user-predefined parameters. Experimental results are then presented that indicate the performance of the proposed approach.     

A structure identification method of submodels for hierarchical fuzzy modeling using the multiple objective genetic algorithm

Fuzzy models describe nonlinear input‐output relationships with linguistic fuzzy rules. A hierarchical fuzzy modeling is promising for identification of fuzzy models of target systems that have many input variables. In the identification, (1) determination of a hierarchical structure of submodels, (2) selection of input variables of each submodel, (3) division of input and output space, (4) tuning of membership functions, and (5) determination of fuzzy inference method are carried out. This article presents a hierarchical fuzzy modeling method with an uneven division method of input space of each submodel. For selecting input variables of submodels, the multiple objective genetic algorithm (MOGA) is utilized. MOGA finds multiple models with different input variables and different numbers of fuzzy rules as compromising solutions. A human designer can choose desirable ones from these candidates. The proposed method is applied to acquisition of fuzzy rules from cyclists' pedaling data. In spite of a small number of data, the obtained model was able to give detailed suggestions to each cyclist. © 2002 Wiley Periodicals, Inc.     

Fuzzy rule-base driven orthogonal approximation

In this study, orthogonal approximation concept is applied to fuzzy systems. We propose a new useful model adapted from the well-known Sugeno type fuzzy system. The proposed fuzzy model is a generalization of the zero-order Sugeno fuzzy system model. Instead of linear functions in standard Sugeno model, we use nonlinear functions in the consequent part. The nonlinear functions are selected from a trigonometric orthogonal basis. Orthogonal function parameters are trained along with the Sugeno fuzzy system. The proposed model is demonstrated using three simulations—a nonlinear piecewise-continuous scalar function modeling and filtering, nonlinear dynamic system identification, and time series prediction. Finally some performance comparisons are carried out.     

Fuzzy identification of systems with unsupervised learning

The paper describes a mathematical tool to build a fuzzy model whose membership functions and consequent parameters rely on the estimates of a data set. The proposed method proved to be capable of approximating any real continuous function, also a strongly nonlinear one, on a compact set to arbitrary accuracy. Without resorting to domain experts, the algorithm constructs a model-free, complete function approximation system. Applications to the modeling of several functions among which classical nonlinear ones such as the Rosenbrock and the sine (x, y) functions are reported. The proposed algorithm can find applications in the development of fuzzy logic controllers (FLC).     

Traffic Flow Data Forecasting Based on Interval Type-2 Fuzzy Sets Theory

This paper proposes a long-term forecasting scheme and implementation method based on the interval type-2 fuzzy sets theory for traffic flow data. The type-2 fuzzy sets have advantages in modeling uncertainties because their membership functions are fuzzy. The scheme includes traffic flow data preprocessing module, type-2 fuzzification operation module and long-term traffic flow data forecasting output module, in which the Interval Approach acts as the core algorithm. The central limit theorem is adopted to convert point data of mass traffic flow in some time range into interval data of the same time range (also called confidence interval data) which is being used as the input of interval approach. The confidence interval data retain the uncertainty and randomness of traffic flow, meanwhile reduce the influence of noise from the detection data. The proposed scheme gets not only the traffic flow forecasting result but also can show the possible range of traffic flow variation with high precision using upper and lower limit forecasting result. The effectiveness of the proposed scheme is verified using the actual sample application.      

Robust Stability for Uncertain Delayed Fuzzy Hopfield Neural Networks With Markovian Jumping Parameters

This paper is concerned with the problem of the robust stability of nonlinear delayed Hopfield neural networks (HNNs) with Markovian jumping parameters by Takagi-Sugeno (T-S) fuzzy model. The nonlinear delayed HNNs are first established as a modified T-S fuzzy model in which the consequent parts are composed of a set of Markovian jumping HNNs with interval delays. Time delays here are assumed to be time-varying and belong to the given intervals. Based on Lyapunov-Krasovskii stability theory and linear matrix inequality approach, stability conditions are proposed in terms of the upper and lower bounds of the delays. Finally, numerical examples are used to illustrate the effectiveness of the proposed method.     

Fuzzy-logic-based process modeling using limited experimental data

Process modeling with limited experimental data is always a difficult task. It becomes even more difficult if the process is highly nonlinear and is characterized by multiple inputs and outputs. Under these circumstances, fuzzy logic may show its capabilities for model development. In this paper, an efficient fuzzy modeling methodology is introduced. The resulting fuzzy model consists of a number of fuzzy implications, each of which is of an IF–THEN form. The IF part consists of a set of logically related antecedents, while the THEN part contains a consequent expressed as a set of linear models. To ensure model simplicity and to accelerate the modeling process, an effective model-development route has been developed. To guarantee the model’s reliability, a t-test-based non-linearity analysis is proposed when each fuzzy implication is developed. The efficacy of the methodology is demonstrated by modeling two nonlinear industrial processes.     

Hybrid identification of fuzzy rule‐based models

In this study, we propose a hybrid identification algorithm for a class of fuzzy rule‐based systems. The rule‐based fuzzy modeling concerns structure optimization and parameter identification using the fuzzy inference methods and hybrid structure combined with two methods of optimization theories for nonlinear systems. Two types of inference methods of a fuzzy model concern a simplified and linear type of inference. The proposed hybrid optimal identification algorithm is carried out using a combination of genetic algorithms and an improved complex method. The genetic algorithms determine initial parameters of the membership function of the premise part of the fuzzy rules. In the sequel, the improved complex method (being in essence a powerful auto‐tuning algorithm) leads to fine‐tuning of the parameters of the respective membership functions. An aggregate performance index with a weighting factor is proposed in order to achieve a balance between performance of the fuzzy model obtained for the training and testing data. Numerical examples are included to evaluate the performance of the proposed model. They are also contrasted with the performance of the fuzzy models existing in the literature. © 2002 John Wiley & Sons, Inc.     

一种非线性模型的在线辨识方法

提出一种用于非线性模型在线辨识的模糊算法。该算法将非线性输入输出系统用时变线性系统模型来拟和。并把此非线性系统模型表示成模糊模型的形式，用在线调节模糊模型的方法来辨识时变线性模型的相关参数。在以往的模糊辨识方法中，均未给出在线调整非线性系统的模糊辨识算法。将递推模糊聚类方法与卡尔曼滤波法用于在线调整模糊模型参数，仿真算例表明了此算法的有效性与良好的实用价值。