New dynamic fuzzy structure and dynamic system identification

Fuzzy systems based on fuzzy if-then rules have been applied to various problems. The main application area has been fuzzy control problems. In many cases, such fuzzy systems can handle only a few input variables. This is because the number of fuzzy if-then rules exponentially increases as the number of input variables increases. In this paper, we try to design fuzzy classification systems based on fuzzy if-then rules for multidimensional pattern classification problems with many attributes. For designing such fuzzy classification systems, we compare two frameworks in the area of genetics-based machine learning: the Michigan approach and the Pittsburgh approach. The performance of fuzzy rule-based classification systems is also compared with that of various pattern classification methods. In computer simulations, we use a wine classification problem with 13 attributes, a cancer diagnosis problem with 9 attributes, and a credit approval problem with 14 attributes. © 1997 Scripta Technica, Inc. Electron Comm Jpn Pt 3, 80(12): 10–19, 1997

The interpolation functions of interval type-2 fuzzy systems and their universal approximation are investigated in this paper. Two types of fuzzification methods are designed to construct the antecedents and consequents of the type-2 inference rules. Then the properties of the fuzzy operator and the type-reduction algorithm are used to integrate all parts of the fuzzy system. Interpolation functions of interval type-2 fuzzy systems, which are proved to be universal approximators, are obtained based on three models, namely single input and single output, double inputs and single output, and multiple inputs and single output. The proposed approach is applied to approximate experiments of dynamic systems so as to evaluate the system performance. The system parameters are optimized by the QPSO algorithm. Experimental results for several data sets are given to show the approximation performances of the proposed interpolation functions are better than those of the interpolation function of the classical type-1 fuzzy system.

Fuzzy cognitive maps (FCMs) can structure virtual worlds that change with time. A FCM links causal events, actors, values, goals, and trends in a fuzzy feedback dynamical system. A fuzzy rule defines a fuzzy patch in the input -output state-space of a system. It links commonsense knowledge with state-space geometry. A FCM connects the fuzzy rules or causal flow paths that relate events. It can guide actors in a virtual world as the actors move through a web of cause and effect and react to events and to other actors. Experts draw FCM causal pictures of the virtual world. Complex FCMs can give virtual worlds with ?>new ?> or chaotic equilibrium behavior. Simple FCMs give virtual worlds with periodic behavior. They map input states to limit-cycle equilibria. A FCM limit cycle repeats a sequence of events or a chain of actions and responses. Limit cycles can control the steady-state rhythms and patterns in a virtual world. In nested FCMs each causal concept can control its own FCM or fuzzy function approximator. Appendix A shows how an additive fuzzy system can uniformly approximate any continuous (or bounded measurable) function on a compact domain to any degree of accuracy. This gives levels of fuzzy systems that can choose goals and causal webs as well as move objects and guide actors in the webs. FCM matrices sum to give a combined FCM virtual world for any number of knowledge sources. Adaptive FCMs change their fuzzy causal web as causal patterns change and as actors act and experts state their causal knowledge. Neural learning laws change the causal rules and the limit cycles. Actors learn new patterns and reinforce old ones. In complex FCMs the user can choose the dynamical structure of the virtual world from a spectrum that ranges from mildly to wildly nonlinear. We use an adaptive FCM to model an undersea virtual world of dolphins, fish, and sharks.

Almost all exact inversion methods provide inverse solutions for only one input variable of fuzzy systems. These methods have certain limitations on the fuzzy system structure such as monotonic rule bases, singleton rule consequents, and invertibility check. These requirements limit the modeling capabilities of the fuzzy systems and also may result in poor application performances. In this study, an exact analytical inversion method is presented for multi-input-single-output decomposable TS fuzzy systems with either singleton or linear consequents. In the proposed method, fuzzy system structures do not need to have monotonic rule bases, singleton rule consequents, or any invertibility conditions. Thus, more flexible fuzzy systems can be used in inverse model based applications. The proposed method provides a simple and systematic way to obtain unique inverse solutions of all input variables simultaneously with respect to any desired system output value. For this purpose, an inversion trajectory approach that guarantees the existence and uniqueness of the inverse solutions is introduced. The inversion trajectory consists of a set of paths defined on the specific edges of universe of discourses of the decomposed fuzzy subsystems. Using this approach, the inverse definition of the overall fuzzy system can easily be derived only by inverting the related decomposed fuzzy subsystems on this inversion trajectory and then combining their inverse definitions. In this way, the inverse definition of the overall fuzzy system is obtained as consisting of analytical solutions of linear and quadratic equations for singleton and linear consequent cases, respectively. Simulation studies are given for the inversion of two and three-input-single-output fuzzy systems, and the exactness and effectiveness of the proposed method are demonstrated.

Fuzzy system identification via chaotic ant swarm

A novel dynamic fuzzy logic system (DFLS) structure is presented, and its properties investigated. Based on the DFLS, an identification scheme is developed using the Lyapunov synthesis approach with a projection algorithm modification, which is shown to be globally stable for a large class of nonlinear dynamic systems in the sense that all system parameters and variables are uniformly bounded. The application of DFLS-based identifier to practical systems is illustrated by identification of a flexible link manipulator with highly nonlinear joint friction. The effectiveness of the identification scheme in dealing with complex systems is demonstrated. The significance of human expertise in improving system performance is illustrated in a simulation experiment. ©1997 by John Wiley & Sons, Inc.

Zero-order TSK-type fuzzy system learning using a two-phase swarm intelligence algorithm

We report on experiments designed to highlight the strengths and weaknesses of an autonomous rule acquisition algorithm for the fuzzy controller of a simulated mobile robot. The algorithm is a Pittsburgh-style fuzzy classifier system. The fuzzy classifier system paradigm is an elegant and versatile combination of evolutionary and lifetime reinforcement learning based on an underlying fuzzy logic structure. It possesses a powerful potential to be a general-purpose linguistically interpretable problem-solver for continuous real-valued domains. We have tested the performance and robustness of many controllers using this approach, a sample of which, are presented here. We find that, although the robot controllers can often be quite robust to environmental changes after learning, they also sometimes display critical weaknesses in certain scenarios. We also compare performance of this algorithm with that of a hand-coded fuzzy controller. We find that it is more difficult than one would initially expect to derive a set of hand-coded rules that are as versatile as the best of those acquired by the fuzzy classifier system. It is worth noting that the task to be learned here is itself quite simple, there are far more complex problems within the mobile robot domain, thus providing a forward impetus to the work.

<abstract> <p>The asymptotic behavior of solutions of fuzzy control systems is a component of the study of fuzzy control theory. The study of stability for T-S (Takagi-Sugeno) fuzzy systems, which process qualitative data through linguistic expressions, is the subject of this paper. Asymptotic stability is conservative in many real-world applications due to measurement noise and other disruptions. The ultimate limit, which indicates that the mistakes stay in a specific area close to the origin after a long enough amount of time, is a crucial characteristic that is frequently defined for such systems. We are interested with the problem of the state feedback controller for T-S fuzzy models with uncertainties where the global exponential ultimate boundedness of solutions is studied for certain fuzzy control systems. We use common quadratic Lyapunov function and parallel distributed compensation controller techniques to study the asymptotic behavior of the solutions of fuzzy control system in presence of perturbations. An example demonstrating the validity of the main result is discussed.</p> </abstract>

This paper investigates the robust control for dynamical systems subject to uncertainty. The uncertainty is assumed to be (possibly fast) time varying and bounded. The bound is unknown but lies within a prescribed fuzzy set (hence the fuzzy dynamical system). We propose an approach for the robust control design which is implemented in two steps. First, a class of robust controls is proposed based on tunable parameters. The proposed controls are deterministic and are not conventionally IF-THEN rules based. By the Lyapunov minimax approach, we prove that the proposed controls are able to guarantee deterministic system performance, namely, uniform boundedness and ultimate uniform boundedness. Second, optima seeking from the proposed controls is considered to improve fuzzy system performance. We formulate the optima-seeking problem as a two-player (one leader and one follower) Stackelberg game by developing two cost functions, each of which is in charge of one tunable parameter (i.e., the player). Each cost function consists of an average fuzzy system performance index and the associated player's control effort. We show that the solution of the optimal design problem (i.e., the optima of the tunable parameters), which is called the Stackelberg strategy, always exists and how to obtain the backwards-induction outcome is provided. Simulation results on the walking control of a biped robot model are presented for demonstration.

Sub-optimal control of fuzzy linear dynamical systems under granular differentiability concept

In this paper, a new method for fuzzy inference system optimization is presented. The proposed method performs type-1, interval type-2 and general type-2 fuzzy inference systems design using a hierarchical genetic algorithm as optimization method. This method is an improvement of a fuzzy system optimization approach presented in previous works where only the optimization of type-1 and interval type-2 fuzzy inference systems was performed using a human recognition application. The human recognition is performed using three biometric measures namely iris, ear, and voice, where the main idea is to perform the combination of responses in the modular neural networks using an optimized fuzzy inference system to improve the final results without and with noisy conditions. The main contribution of the proposed method is an optimization able to select type of fuzzy logic, granulation of each fuzzy variable and fuzzy rules selection to design optimal fuzzy inference systems applied in combining modular neural networks responses. The results obtained show the effectiveness of the proposed method to design structures of fuzzy inference systems. Statistical comparisons are performed with previous results, where better results can be observed using the proposed method. The design of optimal structures of fuzzy inference systems include among other parameters; type of fuzzy logic (type-1, interval and general type-2 fuzzy logic), type of inference model (Mamdani model or Sugeno model), and consequents of the fuzzy if–then rules.

In this paper, an interval type-2 Takagi-Sugeno fuzzy classification system (IT2T-SFCS) learned by particle swarm optimization (PSO) and support vector machine (SVM) for antecedent and consequent parameters optimization is proposed. The IT2T-SFCS is constructed by fuzzy if-then rules whose antecedents are interval type-2 fuzzy sets and consequents are linear state equations. The antecedents of IT2T-SFCS use the fuzzy iterative self-organizing data analysis technique (ISODATA) and PSO to learn and calculate the optimal centers and the uncertain widths of the Gaussian membership functions. Consequent parameters in IT2T-SFCS are learned through SVM for the purpose of achieving higher generalization ability. The proposed IT2T-SFCS is able to directly handle uncertainties, minimize the effects of uncertainties and get the better generalization performance, which inherits the benefits of interval type-2 T-S fuzzy system and SVM. For demonstration, IT2T-SFCS is used as a classifier in gender recognition. The experimental results show that the performance of the proposed IT2T-SFCS is superior to that of the previous mainstream classifiers.

40 - Techniques and Applications of Genetic Algorithm-Based Methods for Designing Compact Fuzzy Classification Systems

Interpretability improvements to find the balance interpretability-accuracy in fuzzy modeling: an overview.- Regaining comprehensibility of approximative fuzzy models via the use of linguistic hedges.- Identifying flexible structured premises for mining concise fuzzy knowledge.- A multiobjective genetic learning process for joint feature selection and granularity and contexts learning in fuzzy rule-based classification systems.- Extracting linguistic fuzzy models from numerical data-AFRELI algorithm.- Constrained optimization of fuzzy decision trees.- A new method for inducing a set of interpretable fuzzy partitions and fuzzy inference systems from data.- A Feature Ranking Algorithm for Fuzzy Modelling Problems.- Interpretability in multidimensional classification.- Interpretable semi-mechanistic fuzzy models by clustering, OLS and FIS model reduction.- Trade-off between approximation accuracy and complexity: TS controller design via HOSVD based complexity minimization.- Simplification and reduction of fuzzy rules.- Effect of rule representation in rule base reduction.- Singular value-based fuzzy reduction with relaxed normalization condition.- Interpretability, complexity, and modular structure of fuzzy systems.- Hierarchical genetic fuzzy systems: accuracy, interpretability and design autonomy.- About the trade-off between accuracy and interpretability of Takagi-Sugeno models in the context of nonlinear time series forecasting.- Accurate, transparent and compact fuzzy models by multi-objective evolutionary algorithms.- Transparent fuzzy systems in modeling and control.- Uniform fuzzy partitions with cardinal splines and wavelets: getting interpretable linguistic fuzzy models.- Relating the theory of partitions in MV-logic to the design of interpretable fuzzy systems.- A formal model of interpretability of linguistic variables.- Expressing relevance and interpretability of rule-based systems.- Conciseness of fuzzy models.- Exact trade-off between approximation accuracy and interpretability: solving the saturation problem for certain FRBSs.- Interpretability improvement of RBF-based neurofuzzy systems using regularized learning.- Extracting fuzzy classification rules from fuzzy clusters on the basis of separating hyperplanes.