二型Takagi-Sugeno-Kang模糊模型和不确定高斯混合模型的等价性

不确定的高斯混合模型和二型Takagi-Sugeno-Kang(TSK)模糊模型之间的对应关系被建立: 任何一个不确定的高斯混合模型都唯一对应着一个二型TSK模糊系统, 不确定的高斯混合模型的条件均值和二型TSK模糊 系统的输出是等价的. 基于此, 一种设计二型模糊系统的新方法被提出: 通过建立不确定的高斯混合模型确定二型TSK模糊系统, 即用概率统计的方法设计二型模糊系统. 仿真实验结果表明利用不确定高斯混合模型设计的二型模糊系统比其它模型具有更强的抗噪性和更快的速度.     

A Self-Evolving Interval Type-2 Fuzzy Neural Network With Online Structure and Parameter Learning

This paper proposes a self-evolving interval type-2 fuzzy neural network (SEIT2FNN) with online structure and parameter learning. The antecedent parts in each fuzzy rule of the SEIT2FNN are interval type-2 fuzzy sets and the fuzzy rules are of the Takagi–Sugeno–Kang (TSK) type. The initial rule base in the SEIT2FNN is empty, and the online clustering method is proposed to generate fuzzy rules that flexibly partition the input space. To avoid generating highly overlapping fuzzy sets in each input variable, an efficient fuzzy set reduction method is also proposed. This method independently determines whether a corresponding fuzzy set should be generated in each input variable when a new fuzzy rule is generated. For parameter learning, the consequent part parameters are tuned by the rule-ordered Kalman filter algorithm for high-accuracy learning performance. Detailed learning equations on applying the rule-ordered Kalman filter algorithm to the SEIT2FNN consequent part learning, with rules being generated online, are derived. The antecedent part parameters are learned by gradient descent algorithms. The SEIT2FNN is applied to simulations on nonlinear plant modeling, adaptive noise cancellation, and chaotic signal prediction. Comparisons with other type-1 and type-2 fuzzy systems in these examples verify the performance of the SEIT2FNN.      

Bayesian zero-order TSK fuzzy system modeling

Different from the existing TSK fuzzy system modeling methods, a novel zero-order TSK fuzzy modeling method called Bayesian zero-order TSK fuzzy system (B-ZTSK-FS) is proposed from the perspective of Bayesian inference in this paper. The proposed method B-ZTSK-FS constructs zero-order TSK fuzzy system by using the maximum a posteriori (MAP) framework to maximize the corresponding posteriori probability. First, a joint likelihood model about zero-order TSK fuzzy system is defined to derive a new objective function which can assure that both antecedents and consequents of fuzzy rules rather than only their antecedents of the most existing TSK fuzzy systems become interpretable. The defined likelihood model is composed of three aspects: clustering on the training set for antecedents of fuzzy rules, the least squares (LS) error for consequent parameters of fuzzy rules, and a Dirichlet prior distribution for fuzzy cluster memberships which is considered to not only automatically match the “sum-to-one” constraints on fuzzy cluster memberships, but also make the proposed method B-ZTSK-FS scalable for large-scale datasets by appropriately setting the Dirichlet index. This likelihood model indeed indicates that antecedent and consequent parameters of fuzzy rules can be linguistically interpreted and simultaneously optimized by the proposed method B-ZTSK-FS which is based on the MAP framework with the iterative sampling algorithm, which in fact implies that fuzziness and probability can co-jointly work for TSK fuzzy system modeling in a collaborative rather than repulsive way. Finally, experimental results on 28 synthetic and real-world datasets are reported to demonstrate the effectiveness of the proposed method B-ZTSK-FS in the sense of approximation accuracy, interpretability and scalability.     

A type-2 neural fuzzy system learned through type-1 fuzzy rules and its FPGA-based hardware implementation

This paper first proposes a type-2 neural fuzzy system (NFS) learned through its type-1 counterpart (T2NFS-T1) and then implements the built IT2NFS-T1 in a field-programmable gate array (FPGA) chip. The antecedent part of each fuzzy rule in the T2NFS-T1 uses interval type-2 fuzzy sets, while the consequent part uses a Takagi-Sugeno-Kang (TSK) type with interval combination weights. The T2NFS-T1 uses a simplified type-reduction operation to reduce system training time and hardware implementation cost. Given a training data set, a TSK type-1 NFS is first learned through structure and parameter learning. The built type-1 fuzzy logic system (FLS) is then extended to a type-2 FLS, where highly overlapped type-1 fuzzy sets are merged into interval type-2 fuzzy sets to reduce the total number of fuzzy sets. Finally, the rule consequent and antecedent parameters in the T2NFS-T1 are tuned using a hybrid of the gradient descent and rule-ordered recursive least square (RLS) algorithms. Simulation results and comparisons with various type-1 and type-2 FLSs verify the effectiveness and efficiency of the T2NFS-T1 for system modeling and prediction problems. A new hardware circuit using both parallel-processing and pipeline techniques is proposed to implement the learned T2NFS-T1 in an FPGA chip. The T2NFS-T1 chip reduces the hardware implementation cost in comparison to other type-2 fuzzy chips.     

A Recurrent Self-Evolving Interval Type-2 Fuzzy Neural Network for Dynamic System Processing

This paper proposes a recurrent self-evolving interval type-2 fuzzy neural network (RSEIT2FNN) for dynamic system processing. An RSEIT2FNN incorporates type-2 fuzzy sets in a recurrent neural fuzzy system in order to increase the noise resistance of a system. The antecedent parts in each recurrent fuzzy rule in the RSEIT2FNN are interval type-2 fuzzy sets, and the consequent part is of the Takagi-Sugeno-Kang (TSK) type with interval weights. The antecedent part of RSEIT2FNN forms a local internal feedback loop by feeding the rule firing strength of each rule back to itself. The TSK-type consequent part is a linear model of exogenous inputs. The RSEIT2FNN initially contains no rules; all rules are learned online via structure and parameter learning. The structure learning uses online type-2 fuzzy clustering. For the parameter learning, the consequent part parameters are tuned by a rule-ordered Kalman filter algorithm to improve learning performance. The antecedent type-2 fuzzy sets and internal feedback loop weights are learned by a gradient descent algorithm. The RSEIT2FNN is applied to simulations of dynamic system identifications and chaotic signal prediction under both noise-free and noisy conditions. Comparisons with type-1 recurrent fuzzy neural networks validate the performance of the RSEIT2FNN.     

Optimization of interval type-2 fuzzy logic controllers using evolutionary algorithms

A method for designing optimal interval type-2 fuzzy logic controllers using evolutionary algorithms is presented in this paper. Interval type-2 fuzzy controllers can outperform conventional type-1 fuzzy controllers when the problem has a high degree of uncertainty. However, designing interval type-2 fuzzy controllers is more difficult because there are more parameters involved. In this paper, interval type-2 fuzzy systems are approximated with the average of two type-1 fuzzy systems, which has been shown to give good results in control if the type-1 fuzzy systems can be obtained appropriately. An evolutionary algorithm is applied to find the optimal interval type-2 fuzzy system as mentioned above. The human evolutionary model is applied for optimizing the interval type-2 fuzzy controller for a particular non-linear plant and results are compared against an optimal type-1 fuzzy controller. A comparative study of simulation results of the type-2 and type-1 fuzzy controllers, under different noise levels, is also presented. Simulation results show that interval type-2 fuzzy controllers obtained with the evolutionary algorithm outperform type-1 fuzzy controllers.     

Online identification of evolving Takagi–Sugeno–Kang fuzzy models for crane systems

This paper suggests new evolving Takagi–Sugeno–Kang (TSK) fuzzy models dedicated to crane systems. A set of evolving TSK fuzzy models with different numbers of inputs are derived by the novel relatively simple and transparent implementation of an online identification algorithm. An input selection algorithm to guide modeling is proposed on the basis of ranking the inputs according to their important factors after the first step of the online identification algorithm. The online identification algorithm offers rule bases and parameters which continuously evolve by adding new rules with more summarization power and by modifying existing rules and parameters. The potentials of new data points are used with this regard. The algorithm is applied in the framework of the pendulum–crane system laboratory equipment. The evolving TSK fuzzy models are tested against the experimental data and a comparison with other TSK fuzzy models and modeling approaches is carried out. The comparison points out that the proposed evolving TSK fuzzy models are simple and consistent with both training data and testing data and that these models outperform other TSK fuzzy models.     

Type-2 fuzzy modeling for desulphurization of steel process

This paper presents a new type-2 fuzzy logic system model for desulphurization process of a real steel industry in Canada. The type-2 fuzzy logic system permits us to model rule uncertainties where every membership value of an element has a second order membership value of its own. In this paper, we propose an indirect method to create second order membership grades that are amplitudes of type-2 secondary membership functions, where the primary memberships are extracted by implementation of fuzzy clustering approach. In this research, Gaussian Mixture Model (GMM) is used for the creation of second order membership grades. Furthermore, a reduction scheme is implemented which results in type-1 membership grades. In turn, this leads to a reduction of the complexity of the system. Two methods are used for the estimation of the membership functions: indirect and direct methods. In the indirect method, the system uses an interpolation scheme for the estimation of the most appropriate membership functions. In the direct method, the system is tuned by an inference algorithm for the optimization of the main parametric system. In this case, the parameters are: Schweizer and Sklar t-norm and s-norm, combination of FATI and FITA inference approaches, and Yager defuzzification. Finally, the system model is applied to the desulphurization process of a real steel industry in Canada. It is shown that the proposed type-2 fuzzy logic system is superior in comparison to multiple regression and type-1 fuzzy logic systems in terms of robustness, and error reduction.     

Interval type-2 neuro-fuzzy system with implication-based inference mechanism

Neuro-fuzzy systems have been proved to be an efficient tool for modelling real life systems. They are precise and have ability to generalise knowledge from presented data. Neuro-fuzzy systems use fuzzy sets – most commonly type-1 fuzzy sets. Type-2 fuzzy sets model uncertainties better than type-1 fuzzy sets because of their fuzzy membership function. Unfortunately computational complexity of type reduction in general type-2 systems is high enough to hinder their practical application. This burden can be alleviated by application of interval type-2 fuzzy sets. The paper presents an interval type-2 neuro-fuzzy system with interval type-2 fuzzy sets both in premises (Gaussian interval type-2 fuzzy sets with uncertain fuzziness) and consequences (trapezoid interval type-2 fuzzy set). The inference mechanism is based on the interval type-2 fuzzy Łukasiewicz, Reichenbach, Kleene-Dienes, or Brouwer–Gödel implications. The paper is accompanied by numerical examples. The system can elaborate models with lower error rate than type-1 neuro-fuzzy system with implication-based inference mechanism. The system outperforms some known type-2 neuro-fuzzy systems.     

Parameter Identification of Recurrent Fuzzy Systems With Fuzzy Finite-State Automata Representation

This paper presents the identification of nonlinear dynamical systems by recurrent fuzzy system (RFS) models. Two types of RFS models are discussed: the Takagi-Sugeno-Kang (TSK) type and the linguistic or Mamdani type. Both models are equivalent and the latter model may be represented by a fuzzy finite-state automaton (FFA). An identification procedure is proposed based on a standard general purpose genetic algorithm (GA). First, the TSK rule parameters are estimated and, in a second step, the TSK model is converted into an equivalent linguistic model. The parameter identification is evaluated in some benchmark problems for nonlinear system identification described in literature. The results show that RFS models achieve good numerical performance while keeping the interpretability of the actual system dynamics.     

Improving the interpretability of TSK fuzzy models by combiningglobal learning and local learning

The fuzzy inference system proposed by Takagi, Sugeno, and Kang, known as the TSK model in fuzzy system literature, provides a powerful tool for modeling complex nonlinear systems. Unlike conventional modeling where a single model is used to describe the global behavior of a system, TSK modeling is essentially a multimodel approach in which simple submodels (typically linear models) are combined to describe the global behavior of the system. Most existing learning algorithms for identifying the TSK model are based on minimizing the square of the residual between the overall outputs of the real system and the identified model. Although these algorithms can generate a TSK model with good global performance (i.e., the model is capable of approximating the given system with arbitrary accuracy, provided that sufficient rules are used and sufficient training data are available), they cannot guarantee the resulting model to have a good local performance. Often, the submodels in the TSK model may exhibit an erratic local behavior, which is difficult to interpret. Since one of the important motivations of using the TSK model (also other fuzzy models) is to gain insights into the model, it is important to investigate the interpretability issue of the TSK model. We propose a new learning algorithm that integrates global learning and local learning in a single algorithmic framework. This algorithm uses the idea of local weighed regression and local approximation in nonparametric statistics, but remains the component of global fitting in the existing learning algorithms. The algorithm is capable of adjusting its parameters based on the user's preference, generating models with good tradeoff in terms of global fitting and local interpretation. We illustrate the performance of the proposed algorithm using a motorcycle crash modeling example     

Application of type-2 neuro-fuzzy modeling in stock price prediction

We present an application of type-2 neuro-fuzzy modeling to stock price prediction based on a given set of training data. Type-2 fuzzy rules can be generated automatically by a self-constructing clustering method and the obtained type-2 fuzzy rules cab be refined by a hybrid learning algorithm. The given training data set is partitioned into clusters through input-similarity and output-similarity tests, and a type-2 TSK rule is derived from each cluster to form a fuzzy rule base. Then the antecedent and consequent parameters associated with the rules are refined by particle swarm optimization and least squares estimation. Experimental results, obtained by running on several datasets taken from TAIEX and NASDAQ, demonstrate the effectiveness of the type-2 neuro-fuzzy modeling approach in stock price prediction.     

Hybrid learning for interval type-2 fuzzy logic systems based on orthogonal least-squares and back-propagation methods

This paper presents a novel learning methodology based on a hybrid algorithm for interval type-2 fuzzy logic systems. Since only the back-propagation method has been proposed in the literature for the tuning of both the antecedent and the consequent parameters of type-2 fuzzy logic systems, a hybrid learning algorithm has been developed. The hybrid method uses a recursive orthogonal least-squares method for tuning the consequent parameters and the back-propagation method for tuning the antecedent parameters. Systems were tested for three types of inputs: (a) interval singleton, (b) interval type-1 non-singleton, and (c) interval type-2 non-singleton. Experiments were carried out on the application of hybrid interval type-2 fuzzy logic systems for prediction of the scale breaker entry temperature in a real hot strip mill for three different types of coil. The results proved the feasibility of the systems developed here for scale breaker entry temperature prediction. Comparison with type-1 fuzzy logic systems shows that hybrid learning interval type-2 fuzzy logic systems provide improved performance under the conditions tested.     

基于支持向量机的TSK模糊模型辨识与控制

研究非线性系统TSK模糊模型的辨识与控制,利用TSK模型,可以将线性控制理论应用于非线性系统控制。基于支持向量机和递推最小二乘法,辨识出TSK模糊模型,并且通过遗传算法优化隶属度函数参数,最小化辨识误差。针对TSK模型进行控制,控制器包括两个部分：权重最大子系统反馈控制及其监督控制,监督控制保证了系统的稳定性。辨识和控制仿真结果证明了算法的有效性。     

MPEG VBR video traffic modeling and classification using fuzzytechnique

We present an approach for MPEG variable bit rate (VBR) video modeling and classification using fuzzy techniques. We demonstrate that a type-2 fuzzy membership function, i.e., a Gaussian MF with uncertain variance, is most appropriate to model the log-value of I/P/B frame sizes in MPEG VBR video. The fuzzy c-means (FCM) method is used to obtain the mean and standard deviation (std) of T/P/B frame sizes when the frame category is unknown. We propose to use type-2 fuzzy logic classifiers (FLCs) to classify video traffic using compressed data. Five fuzzy classifiers and a Bayesian classifier are designed for video traffic classification, and the fuzzy classifiers are compared against the Bayesian classifier. Simulation results show that a type-2 fuzzy classifier in which the input is modeled as a type-2 fuzzy set and antecedent membership functions are modeled as type-2 fuzzy sets performs the best of the five classifiers when the testing video product is not included in the training products and a steepest descent algorithm is used to tune its parameters     

Optimal design of interval type 2 fuzzy controllers based on a simple tuning algorithm

Evolutionary algorithms are one of the most common choices reported in the literature for the tuning of fuzzy logic controllers based on either type-1 or type-2 fuzzy systems. An alternative to evolutionary algorithms is the simple tuning algorithm (STA-FLC), which is a methodology designed to improve the response of type-1 fuzzy logic controllers in a practical, intuitive and simple ways. This paper presents an extension of the simple tuning algorithm for fuzzy logic controllers based on the theory of type-2 fuzzy systems by using a parallel model implementation, it also includes a mechanism to calculate the feedback gain, new integral criteria parameters, and the effect of the AND/OR operator combinations on the fuzzy rules to improve the algorithm applicability and performance. All these improvements are demonstrated with experiments applied to different types of plants.     

Stability analysis of recurrent type-2 TSK fuzzy systems with nonlinear consequent part

Neural Computing and Applications - A necessary condition for stability of a class of recurrent type-2 TSK fuzzy systems is presented. In this system, the antecedent part is indeed represented by...     

Self-adaptive interval type-2 neural fuzzy network control for PMLSM drives

This paper proposes a self-adaptive interval type-2 neural fuzzy network (SAIT2NFN) control system for the high-precision motion control of permanent magnet linear synchronous motor (PMLSM) drives. The antecedent parts in the SAIT2NFN use interval type-2 fuzzy sets to handle uncertainties in PMLSM drives, including payload variation, external disturbance, and sense noise. The SAIT2NFN is firstly trained to model the inverse dynamics of PMLSM through concurrent structure and parameter learning. The fuzzy rules in the SAIT2NFN can be generated automatically by using online clustering algorithm to obtain a suitable-sized network structure, and a back propagation is proposed to adjust all network parameters. Then, a robust SAIT2NFN inverse control system that consists of the SAIT2NFN and an error-feedback controller is proposed to control the PMLSM drive in a changing environment. Moreover, the Kalman filtering algorithm with a dead zone is derived using Lyapunov stability theorem for online fine-tuning all network parameters to guarantee the convergence of the SAIT2NFN. Experimental results show that the proposed SAIT2NFN control system achieves the best tracking performance in comparison with type-1 NFN control systems.     

Reinforcement Interval Type-2 Fuzzy Controller Design by Online Rule Generation and Q-Value-Aided Ant Colony Optimization

This paper proposes a new reinforcement-learning method using online rule generation and Q-value-aided ant colony optimization (ORGQACO) for fuzzy controller design. The fuzzy controller is based on an interval type-2 fuzzy system (IT2FS). The antecedent part in the designed IT2FS uses interval type-2 fuzzy sets to improve controller robustness to noise. There are initially no fuzzy rules in the IT2FS. The ORGQACO concurrently designs both the structure and parameters of an IT2FS. We propose an online interval type-2 rule generation method for the evolution of system structure and flexible partitioning of the input space. Consequent part parameters in an IT2FS are designed using Q-values and the reinforcement local-global ant colony optimization algorithm. This algorithm selects the consequent part from a set of candidate actions according to ant pheromone trails and Q-values, both of which are updated using reinforcement signals. The ORGQACO design method is applied to the following three control problems: (1) truck-backing control; (2) magnetic-levitation control; and (3) chaotic-system control. The ORGQACO is compared with other reinforcement-learning methods to verify its efficiency and effectiveness. Comparisons with type-1 fuzzy systems verify the noise robustness property of using an IT2FS.     

Hierarchical type-2 neuro-fuzzy BSP model

This paper presents a novel hybrid interval type-2 neuro-fuzzy inference system, with automatic learning of all its parameters, to handle uncertainty. This new model, called hierarchical type-2 neuro-fuzzy BSP model (T2-HNFB), combines the paradigms of the type-2 fuzzy inference systems and neural networks with recursive partitioning techniques (binary space partitioning - BSP). The model is able to automatically create and expand its own structure, to reduce limitations on the number of inputs and to extract fuzzy linguistic rules from a dataset, as well as to efficiently model and manipulate most types of uncertainty existing in real situations. In addition, it provides an interval for its output, which can be regarded as a measure of uncertainty and constitutes important information for real applications. In this context, this model overcomes the limitations of the conventional type-2 and type-1 fuzzy inference systems. Experimental results show that the results provided by the T2-HNFB model are close to and in several cases better than the best results supplied by the other models used for comparison.