Nature-inspired optimal tuning of input membership functions of Takagi-Sugeno-Kang fuzzy models for Anti-lock Braking Systems

This paper suggests a synergy of fuzzy logic and nature-inspired optimization in terms of the nature-inspired optimal tuning of the input membership functions of a class of Takagi-Sugeno-Kang (TSK) fuzzy models dedicated to Anti-lock Braking Systems (ABSs). A set of TSK fuzzy models is proposed by a novel fuzzy modeling approach for ABSs. The fuzzy modeling approach starts with the derivation of a set of local state-space models of the nonlinear ABS process by the linearization of the first-principle process model at ten operating points. The TSK fuzzy model structure and the initial TSK fuzzy models are obtained by the modal equivalence principle in terms of placing the local state-space models in the rule consequents of the TSK fuzzy models. An operating point selection algorithm to guide modeling is proposed, formulated on the basis of ranking the operating points according to their importance factors, and inserted in the third step of the fuzzy modeling approach. The optimization problems are defined such that to minimize the objective functions expressed as the average of squared modeling errors over the time horizon, and the variables of these functions are a part of the parameters of the input membership functions. Two representative nature-inspired algorithms, namely a Simulated Annealing (SA) algorithm and a Particle Swarm Optimization (PSO) algorithm, are implemented to solve the optimization problems and to obtain optimal TSK fuzzy models. The validation and the comparison of SA and PSO and of the new TSK fuzzy models are carried out for an ABS laboratory equipment. The real-time experimental results highlight that the optimized TSK fuzzy models are simple and consistent with both training data and validation data and that these models outperform the initial TSK fuzzy models.     

A hybrid learning scheme combining EM and MASMOD algorithms forfuzzy local linearization modeling

Fuzzy local linearization (FLL) is a useful divide-and-conquer method for coping with complex problems such as modeling unknown nonlinear systems from data for state estimation and control. Based on a probabilistic interpretation of FLL, the paper proposes a hybrid learning scheme for FLL modeling, which uses a modified adaptive spline modeling (MASMOD) algorithm to construct the antecedent parts (membership functions) in the FLL model, and an expectation-maximization (EM) algorithm to parameterize the consequent parts (local linear models). The hybrid method not only has an approximation ability as good as most neuro-fuzzy network models, but also produces a parsimonious network structure (gain from MASMOD) and provides covariance information about the model error (gain from EM) which is valuable in applications such as state estimation and control. Numerical examples on nonlinear time-series analysis and nonlinear trajectory estimation using FLL models are presented to validate the derived algorithm.     

River flow forecasting through nonlinear local approximation in a fuzzy model

This study investigates the potential of nonlinear local function approximation in a Takagi–Sugeno (TS) fuzzy model for river flow forecasting. Generally, in a TS framework, the local approximation is performed by a linear model, while in this approach, linear function approximation is substituted using a nonlinear function approximation. The primary hypothesis herein is that the process being modeled (rainfall–runoff in this study) is highly nonlinear, and a linear approximation at the local domain might still leave a lot of unexplained variance by the model. In this study, subtractive clustering technique is used for domain partition, and neural network is used for function approximation. The modeling approach has been tested on two case studies: Kolar basin in India and Kentucky basin in USA. The results of fuzzy nonlinear local approximation (FNLLA) model are highly promising. The performance of the FNLLA is compared with that of a pure fuzzy inference system (FIS), and it is observed that both the models perform similar at 1-step-ahead forecasts. However, the FNLLA performs much better than FIS at higher lead times. It is also observed that FNLLA forecasts the river flow with lesser error compared to FIS. In the case of Kolar River, more than 40 % of the total data are forecasted with <2 % error by FNLLA at 1 h ahead, while the corresponding value for FIS is only 20 %. In the case of 3-h-ahead forecasts, these values are 25 % for FNLLA and 15 % for FIS. Performance of FNLLA in the case of Kentucky River basin was also better compared to FIS. It is also found that FNLLA simulates the peak flow better than FIS, which is certainly an improvement over the existing models.     

Predictive control based on local linear fuzzy models

This paper deals with predictive control based on fuzzy models. A novel algorithm (LOLIMOT) is proposed for the construction of Takagi-Sugeno fuzzy models. The rule consequents are optimized by a local orthogonal least-squares method that selects the significant regressors. The rule premises are optimized by a tree construction algorithm which partitions the input space in hyper-rectangles. A generalized predictive controller (GPC) and a dynamic matrix controller (DMC) are designed. Both controllers require the extraction of a linear model from the Takagi-Sugeno fuzzy model. For the GPC a new technique called local dynamic linearization is proposed that exploits the special structure of the local linear models. The DMC is based on the evaluation of a step response. The effectiveness of both the identification algorithm and the predictive controllers is shown by application to temperature control of an industrial-scale cross-flow heat exchanger.     

Nonlinear system modeling and robust predictive control based on RBF-ARX model

An integrated modeling and robust model predictive control (MPC) approach is proposed for a class of nonlinear systems with unknown steady state. First, the nonlinear system is identified off-line by RBF-ARX model possessing linear ARX model structure and state-dependent Gaussian RBF neural network type coefficients. On the basis of the RBF-ARX model, a combination of a local linearization model and a polytopic uncertain linear parameter-varying (LPV) model are built to approximate the present and the future system's nonlinear behavior, respectively. Subsequently, based on the approximate models, a min–max robust MPC algorithm with input constraint is designed for the output-tracking control of the nonlinear system with unknown steady state. The closed-loop stability of the MPC strategy is guaranteed by the use of parameter-dependent Lyapunov function and the feasibility of the linear matrix inequalities (LMIs). Simulation study to a NO_x decomposition process illustrates the effectiveness of the modeling and robust MPC approaches proposed in this paper.     

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     

A RBF-ARX model-based robust MPC for tracking control without steady state knowledge

A RBF-ARX modeling and robust model predictive control (MPC) approach to achieving output-tracking control of the nonlinear system with unknown steady-state knowledge is proposed. On the basis of the RBF-ARX model with considering the system time delay, a local linearization state-space model is obtained to represent the current behavior of the nonlinear system, and a polytopic uncertain linear parameter varying (LPV) state-space model is built to represent the future system’s nonlinear behavior. Based on the two models, a quasi-min–max MPC algorithm with constraint is designed for output-tracking control of the nonlinear system with unknown steady state knowledge. The optimization problem of the quasi-min–max MPC algorithm is finally converted to the convex linear matrix inequalities (LMIs) optimization problem. Closed-loop stability of the MPC strategy is guaranteed by the use of parameter-dependent Lyapunov function and feasibility of the LMIs. Two examples, i.e. the modeling and control of a continuously stirred tank reactor (CSTR) and a two tank system demonstrate the effectiveness of the RBF-ARX modeling and robust MPC approach.     

非线性布尔网络系统模糊建模与动态性能分析

针对非线性系统难以精确建模与动态性能分析的基本控制问题,基于模糊动态模型把布尔网络系统理论推广到非线性布尔网络系统,建立了模糊动态布尔网络控制系统的模型。引入模糊动态模型,对非线性布尔网络进行模糊建模,分别建立了非线性布尔网络系统的局部模型和全局模型。从系统的局部意义和全局意义上,对系统进行了能控性、能观性、稳定性等动态性能分析。最后,以多输入多输出的非线性布尔网络系统实例为具体研究对象,建立了系统的局部模型和全局模型,并对动态性能进行了仿真分析,得到了实验结果。实验结果表明,模糊动态布尔网络控制系统对非线性布尔网络系统的建模是有效的,动态性能分析是合理的,对模糊动态布尔网络控制系统的进一步分析有重要意义。     

Fuzzy functions based ARX model and new fuzzy basis function models for nonlinear system identification

In this study, auto regressive with exogenous input (ARX) modeling is improved with fuzzy functions concept (FF-ARX). Fuzzy function with least squares estimation (FF-LSE) method has been recently developed and widely used with a small improvement with respect to least squares estimation method (LSE). FF-LSE is structured with only inputs and their membership values. This proposed model aims to increase the capability of the FF-LSE by widening the regression matrix with lagged input–output values. In addition, by using same idea, we proposed also two new fuzzy basis function models. In the first, basis of the fuzzy system and lagged input–output values are structured together in the regression matrix and named as “L-FBF”. Secondly, instead of using basis function, the membership values of the lagged input–output values are used in the regression matrix by using Gaussian membership functions, called “M-FBF”. Therefore, the power of the fuzzy basis function is also enhanced. For the corresponding models, antecedent part parameters for the input vectors are determined with fuzzy c-means (FCM) clustering algorithm. The consequent parameters of the all models are estimated with the LSE. The proposed models are utilized and compared for the identification of nonlinear benchmark problems.     

RBF-ARX model-based MPC strategies with application to a water tank system

A hybrid pseudo-linear RBF-ARX model that combines Gaussian radial basis function (RBF) networks and linear ARX model structure is utilized for representing the dynamic behavior of a class of smooth nonlinear and non-stationary systems. This model is locally linear at each working point and globally nonlinear within whole working range. Based on the structural characteristics of the RBF-ARX model, three receding horizon predictive control (RBF-ARX-MPC) strategies are designed: (1) the RBF-ARX-MPC algorithm based on single-point linearization (MPC-SPL); (2) the RBF-ARX-MPC algorithm based on multi-point linearization (MPC-MPL); and (3) the RBF-ARX-MPC algorithm based on globally nonlinear optimization (MPC-GNO). In the MPC-SPL, the future multi-step-ahead predictive output of the system is obtained based on the local linearization of the RBF-ARX model at only current working-point, while in the MPC-MPL the future long-term output prediction is obtained according to the future local characteristics from previous online optimization results of the RBF-ARX model based MPC. In the MPC-GNO, the globally nonlinear characteristics of the RBF-ARX model are fully used for online getting control variables of the MPC. Real-time control experiments for the three type MPCs are carried out on a water tank system, which are also compared with a classical PID control and a traditional linear ARX model-based MPC. The results verify that the modeling method and the model-based predictive control strategies are realizable and effective for the nonlinear and unstable system. Moreover, it is also shown that the MPC-GNO can obtain better control performance but need more computation time compared to the other MPCs, which makes it possible to be applied into some slowly varying processes.     

Direct fuzzy model‐reference adaptive control

Intelligent systems may be viewed as a framework for solving the problems of nonlinear system control. The intelligence of the system in the nonlinear or changing environment is used to recognize in which environment the system currently resides and to service it appropriately. This paper presents a general methodology of adaptive control based on multiple models in fuzzy form to deal with plants with unknown parameters which depend on known plant variables. We introduce a novel model‐reference fuzzy adaptive control system which is based on the fuzzy basis function expansion. The generality of the proposed algorithm is substantiated by the Stone‐Weierstrass theorem which indicates that any continuous function can be approximated by fuzzy basis function expansion. In the sense of adaptive control this implies the adaptive law with fuzzified adaptive parameters which are obtained using Lyapunov stability criterion. The combination of adaptive control theory based on models obtained by fuzzy basis function expansion results in fuzzy direct model‐reference adaptive control which provides higher adaptation ability than basic adaptive‐control systems. The proposed control algorithm is the extension of direct model‐reference fuzzy adaptive‐control to nonlinear plants. The direct fuzzy adaptive controller directly adjusts the parameter of the fuzzy controller to achieve approximate asymptotic tracking of the model‐reference input. The main advantage of the proposed approach is simplicity together with high performance, and it has been shown that the closed‐loop system using the direct fuzzy adaptive controller is globally stable and the tracking error converges to the residual set which depends on fuzzification properties. The proposed approach can be implemented on a wide range of industrial processes. In the paper the foundation of the proposed algorithm are given and some simulation examples are shown and discussed. © 2002 Wiley Periodicals, Inc.     

Structure and stability analysis of general Mamdani fuzzy dynamic models

Mamdani fuzzy models have always been used as black‐box models. Their structures in relation to the conventional model structures are unknown. Moreover, there exist no theoretical methods for rigorously judging model stability and validity. I attempt to provide solutions to these issues for a general class of fuzzy models. They use arbitrary continuous input fuzzy sets, arbitrary fuzzy rules, arbitrary inference methods, Zadeh or product fuzzy logic AND operator, singleton output fuzzy sets, and the centroid defuzzifier. I first show that the fuzzy models belong to the NARX (nonlinear autoregressive with the extra input) model structure, which is one of the most important and widely used structures in classical modeling. I then divide the NARX model structure into three nonlinear types and investigate how the settings of the fuzzy model components, especially input fuzzy sets, dictate the relations between the fuzzy models and these types. I have found that the fuzzy models become type‐2 models if and only if the input fuzzy sets are linear or piecewise linear (e.g., trapezoidal or triangular), becoming type 3 if and only if at least one input fuzzy set is nonlinear. I have also developed an algorithm to transfer type‐2 fuzzy models into type‐1 models as far as their input–output relationships are concerned, which have some important properties not shared by the type‐2 models. Furthermore, a necessary and sufficient condition has been derived for a part of the general fuzzy models to be linear ARX models. I have established a necessary and sufficient condition for judging local stability of type‐1 and type‐2 fuzzy models. It can be used for model validation and control system design. Three numeric examples are provided. Our new findings provide a theoretical foundation for Mamdani fuzzy modeling and make it more consistent with the conventional modeling theory. © 2005 Wiley Periodicals, Inc. Int J Int Syst 20: 103–125, 2005.     

Fuzzy basis functions, universal approximation, and orthogonalleast-squares learning

Fuzzy systems are represented as series expansions of fuzzy basis functions which are algebraic superpositions of fuzzy membership functions. Using the Stone-Weierstrass theorem, it is proved that linear combinations of the fuzzy basis functions are capable of uniformly approximating any real continuous function on a compact set to arbitrary accuracy. Based on the fuzzy basis function representations, an orthogonal least-squares (OLS) learning algorithm is developed for designing fuzzy systems based on given input-output pairs; then, the OLS algorithm is used to select significant fuzzy basis functions which are used to construct the final fuzzy system. The fuzzy basis function expansion is used to approximate a controller for the nonlinear ball and beam system, and the simulation results show that the control performance is improved by incorporating some common-sense fuzzy control rules.     

Immunity-based hybrid learning methods for approximator structure and parameter adjustment

From the point of view of information processing the immune system is a highly parallel and distributed intelligent system which has learning, memory, and associative retrieval capabilities. In this paper we present two immunity-based hybrid learning approaches for function approximation (or regression) problems that involve adjusting the structure and parameters of spatially localized models (e.g., radial basis function networks). The number and centers of the receptive fields for local models are specified by immunity-based structure adaptation algorithms, while the parameters of the local models, which enter in a linear fashion, are tuned separately using a least-squares method. The effectiveness of the procedure is demonstrated through a nonlinear function approximation problem and a nonlinear dynamical system modeling problem.     

Generalized neurofuzzy network modeling algorithms usingBezier-Bernstein polynomial functions and additive decomposition

This paper introduces a new neurofuzzy model construction algorithm for nonlinear dynamic systems based upon basis functions that are Bezier-Bernstein polynomial functions. This paper is generalized in that it copes with n-dimensional inputs by utilising an additive decomposition construction to overcome the curse of dimensionality associated with high n. This new construction algorithm also introduces univariate Bezier-Bernstein polynomial functions for the completeness of the generalized procedure. Like the B-spline expansion based neurofuzzy systems, Bezier-Bernstein polynomial function based neurofuzzy networks hold desirable properties such as nonnegativity of the basis functions, unity of support, and interpretability of basis function as fuzzy membership functions, moreover with the additional advantages of structural parsimony and Delaunay input space partition, essentially overcoming the curse of dimensionality associated with conventional fuzzy and RBF networks. This new modeling network is based on additive decomposition approach together with two separate basis function formation approaches for both univariate and bivariate Bezier-Bernstein polynomial functions used in model construction. The overall network weights are then learnt using conventional least squares methods. Numerical examples are included to demonstrate the effectiveness of this new data based modeling approach.     

Nonlinear state estimation using fuzzy local linear models

A local modelling-based approach to nonlinear state estimation using a Sugeno fuzzy inference framework is presented. Four new fuzzy Kalman filters are proposed on this basis, and simulation results presented suggest potential improvements when compared with conventional extended Kalman filtering.     

Recurrent neuro-fuzzy networks for nonlinear process modeling

A type of recurrent neuro-fuzzy network is proposed in this paper to build long-term prediction models for nonlinear processes. The process operation is partitioned into several fuzzy operating regions. Within each region, a local linear model is used to model the process. The global model output is obtained through the centre of gravity defuzzification which is essentially the interpolation of local model outputs. This modeling strategy utilizes both process knowledge and process input/output data. Process knowledge is used to initially divide the process operation into several fuzzy operating regions and to set up the initial fuzzification layer weights. Process I/O data are used to train the network. Network weights are such trained so that the long-term prediction errors are minimized. Through training, membership functions of fuzzy operating regions are refined and local models are learnt. Based on the recurrent neuro-fuzzy network model, a novel type of nonlinear model-based long range predictive controller can be developed and it consists of several local linear model-based predictive controllers. Local controllers are constructed based on the corresponding local linear models and their outputs are combined to form a global control action by using their membership functions. This control strategy has the advantage that control actions can be calculated analytically avoiding the time consuming nonlinear programming procedures required in conventional nonlinear model-based predictive control. The techniques have been successfully applied to the modeling and control of a neutralization process.     

An Analytical Study on Structure, Stability and Design of General Nonlinear Takagi–Sugeno Fuzzy Control Systems

In this paper, we first study analytical structure of general nonlinear Takagi-Sugeno (TS, for short) fuzzy controllers, then establish a condition for analytically determining asymptotic stability of the fuzzy control systems at the equilibrium point, and finally use the stability condition in design of the control systems that are at least locally stable. The general TS fuzzy controllers use arbitrary input fuzzy sets, any types of fuzzy logic AND, TS fuzzy rules with linear consequent and the generalized defuzzifier which contains the popular centroid defuzzifier as a special case. We have mathematically proved that the general TS fuzzy controllers are nonlinear controllers with variable gains continuously changing with controllers’ input variables. Using Lyapunov’s linearization method, we have established a necessary and sufficient condition for analytically determining local asymptotic stability of TS fuzzy control systems, each of which is made up of a fuzzy controller of the general class and a nonlinear plant. We show that the condition can be used in practice even when the plant model is not explicitly known. We have utilized the stability condition to design, with or without plant model, general TS fuzzy control systems that are at least locally stable. Three numerical examples are given to illustrate in detail how to use our new results. Our results offer four important practical advantages: (1) our stability condition, being a necessary and sufficient one, is the tightest possible stability condition, (2) the condition is simple and easy to use partially because it only needs the fuzzy controller structure around the equilibrium point, (3) the condition can be used for determining system local stability and designing fuzzy control systems that are stable at least around the equilibrium point even when the explicit plant models are unavailable, and (4) the condition covers a very broad range of nonlinear TS fuzzy control systems, for which a meaningful global stability condition seems impossible to establish.     

On the interpretation and identification of dynamic Takagi-Sugenofuzzy models

Dynamic Takagi-Sugeno fuzzy models are not always easy to interpret, in particular when they are identified from experimental data. It is shown that there exists a close relationship between dynamic Takagi-Sugeno fuzzy models and dynamic linearization when using affine local model structures, which suggests that a solution to the multiobjective identification problem exists. However, it is also shown that the affine local model structure is a highly sensitive parametrization when applied in transient operating regimes. Due to the multiobjective nature of the identification problem studied here, special considerations must be made during model structure selection, experiment design, and identification in order to meet both objectives. Some guidelines for experiment design are suggested and some robust nonlinear identification algorithms are studied. These include constrained and regularized identification and locally weighted identification. Their usefulness in the present context is illustrated by examples