Fuzzy Systems

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Li-xin Wang - One of the best experts on this subject based on the ideXlab platform.

  • Analysis and design of hierarchical Fuzzy Systems
    IEEE Transactions on Fuzzy Systems, 1999
    Co-Authors: Li-xin Wang
    Abstract:

    In this letter, the hierarchical Fuzzy Systems are analyzed and designed. In the analysis part, we prove that the hierarchical Fuzzy Systems are universal approximators and analyze the sensitivity of the Fuzzy system output with respect to small perturbations in its inputs. In the design part, we derive a gradient descent algorithm for tuning the parameters of the hierarchical Fuzzy system to match the input-output pairs. The algorithm is simulated for two examples and the results show that the algorithm is effective and the hierarchical structure gives good approximation accuracy.

  • Universal approximation by hierarchical Fuzzy Systems
    Fuzzy Sets and Systems, 1998
    Co-Authors: Li-xin Wang
    Abstract:

    Abstract A serious problem limiting the applicability of standard Fuzzy controllers is the rule-explosion problem; that is, the number of rules increases exponentially with the number of input variables to the Fuzzy controller. A way to deal with this “curse of dimensionality” is to use the hierarchical Fuzzy Systems. A hierarchical Fuzzy system consists of a number of hierarchically connected low-dimensional Fuzzy Systems. It can be shown that the number of rules in the hierarchical Fuzzy system increases linearly with the number of input variables. In this paper, we prove that the hierarchical Fuzzy Systems are universal approximators; that is, they can approximate any nonlinear function on a compact set to arbitrary accuracy. Our proof is constructive, that is, we first construct a hierarchical Fuzzy system in a step-by-step manner, then prove that the constructed Fuzzy system satisfies an error bound, and finally show that the error bound can be made arbitrarily small.

  • Fuzzy Systems are universal approximators
    [1992 Proceedings] IEEE International Conference on Fuzzy Systems, 1992
    Co-Authors: Li-xin Wang
    Abstract:

    The author proves that Fuzzy Systems are universal approximators. The Stone-Weierstrass theorem is used to prove that Fuzzy Systems with product inference, centroid defuzzification, and a Gaussian membership function are capable of approximating any real continuous function on a compact set to arbitrary accuracy. This result can be viewed as an existence theorem of an optimal Fuzzy system for a wide variety of problems. >

Xiao-jun Zeng - One of the best experts on this subject based on the ideXlab platform.

  • An Evolving-Construction Scheme for Fuzzy Systems
    IEEE Transactions on Fuzzy Systems, 2010
    Co-Authors: Di Wang, Xiao-jun Zeng, John A. Keane
    Abstract:

    This paper proposes an evolving-construction scheme for Fuzzy Systems (ECSFS). ECSFS begins with a simple Fuzzy system and evolves its structure by adding more Fuzzy terms and rules to achieve a better accuracy in a "greedy'' way. An interesting feature of ECSFS is that it is able to automatically locate mathematically meaningful points, such as the extremum and inflexion points of the approximated function one by one, and then adds Fuzzy terms based on these points. Fuzzy Systems with such extreme points, like their Fuzzy terms, are more efficient than other Fuzzy Systems by using the same number of Fuzzy rules. As a result, ECSFS often achieves a better accuracy for Fuzzy-system identification compared with the previous methods when using the same number of Fuzzy rules. A number of simulation results are given to illustrate the advantages of the proposed scheme.

  • Approximation Capabilities of Hierarchical Fuzzy Systems
    IEEE Transactions on Fuzzy Systems, 2005
    Co-Authors: Xiao-jun Zeng, John A. Keane
    Abstract:

    Derived from practical application in location analysis and pricing, and based on the approach of hierarchical structure analysis of continuous functions, this paper investigates the approximation capabilities of hierarchical Fuzzy Systems. By first introducing the concept of the natural hierarchical structure, it is proved that continuous functions with natural hierarchical structure can be naturally and effectively approximated by hierarchical Fuzzy Systems to overcome the curse of dimensionality in both the number of rules and parameters. Then, based on Kolmogorov's theorem, it is shown that any continuous function can be represented as a superposition of functions with the natural hierarchical structure and can then be approximated by hierarchical Fuzzy Systems to achieve the universal approximation property. Further, the conditions under which the hierarchical Fuzzy approximation is superior to the standard Fuzzy approximation in overcoming the curse of dimensionality are analyzed

  • Decomposition property of Fuzzy Systems and its applications
    IEEE Transactions on Fuzzy Systems, 1996
    Co-Authors: Xiao-jun Zeng, M.g. Singh
    Abstract:

    This paper presents the decomposition property of Fuzzy Systems using a simple, constructive, decomposition procedure. That is, by properly dividing the input space into sub-input spaces, a general Fuzzy system is decomposed into several sub-Fuzzy Systems which are the simplest Fuzzy Systems in the sub-input spaces. Based on the decomposition property of Fuzzy Systems, the analysis of Fuzzy Systems can be divided into two steps: first, analyze the properties of the simplest Fuzzy Systems, and then, use the decomposition property to extend the results to general Fuzzy Systems. Using this idea, two applications of the decomposition property are given. The first is the application to the representation capability analysis of Fuzzy Systems. The second is the application to the analysis of a class of nonlinear control Systems. Then, based on the piecewise affine Fuzzy-system model, the existence condition and the design of a stable control for a class of single-input single-output (SISO) nonlinear Systems are presented.

  • Approximation accuracy analysis of Fuzzy Systems as function approximators
    IEEE Transactions on Fuzzy Systems, 1996
    Co-Authors: Xiao-jun Zeng, M.g. Singh
    Abstract:

    This paper establishes the approximation error bounds for various classes of Fuzzy Systems (i.e., Fuzzy Systems generated by different inferential and defuzzification methods). Based on these bounds, the approximation accuracy of various classes of Fuzzy Systems is analyzed and compared. It is seen that the class of Fuzzy Systems generated by the product inference and the center-average defuzzifier has better approximation accuracy and properties than the class of Fuzzy Systems generated by the min inference and the center-average defuzzifier, and the class of Fuzzy Systems defuzzified by the MoM defuzzifier. In addition, it is proved that Fuzzy Systems can represent any linear and multilinear function and explicit expressions of Fuzzy Systems generated by the MoM defuzzified method are given.

  • approximation theory of Fuzzy Systems spl minus siso case
    IEEE Transactions on Fuzzy Systems, 1994
    Co-Authors: Xiao-jun Zeng, M.g. Singh
    Abstract:

    In this paper, the approximation properties of MIMO Fuzzy Systems generated by the product inference are discussed. We first give an analysis of Fuzzy basic functions (FBF's) and present several properties of FBF's. Based on these properties of FBF's, we obtain several basic approximation properties of Fuzzy Systems: 1) basic approximation property which reveals the basic approximation mechanism of Fuzzy Systems; 2) uniform approximation bounds which give the uniform approximation bounds between the desired (control or decision) functions and Fuzzy Systems; 3) uniform convergent property which shows that Fuzzy Systems with defined approximation accuracy can always be obtained by dividing the input space into finer Fuzzy regions; and 4) universal approximation property which shows that Fuzzy Systems are universal approximators and extends some previous results on this aspect. The similarity between Fuzzy Systems and mathematical approximation is discussed and an idea to improve approximation accuracy is suggested based on uniform approximation bounds. >

Jurgen Adamy - One of the best experts on this subject based on the ideXlab platform.

  • Equilibria of continuous-time recurrent Fuzzy Systems
    Fuzzy Sets and Systems, 2006
    Co-Authors: Jurgen Adamy, A. Flemming
    Abstract:

    Abstract Unlike static Fuzzy Systems, recurrent Fuzzy Systems allow representing knowledge-based dynamic processes that can be stated in the form of “if … , then … ” rules, making it possible to model Systems that can only be described qualitatively. Further possible applications exist in the case where the dynamics of a system are quantitatively known, but only in certain mesh points. The interpolating character of the Fuzzy system between the dynamics of the mesh points yields a complete dynamic model. Based on discrete-time recurrent Fuzzy Systems this article presents first steps towards the theory of continuous-time recurrent Fuzzy Systems and provides criteria for the investigation of the dynamics of this class of Systems.

  • sequential pattern recognition employing recurrent Fuzzy Systems
    Fuzzy Sets and Systems, 2004
    Co-Authors: Roland Kempf, Jurgen Adamy
    Abstract:

    Sequential pattern-recognition Systems check whether data strings, e.g., time series, exhibit certain pattern primitives in a specified order. As in the case of most other pattern-recognition methods, either conventional methods or Fuzzy Systems may be used here. This paper presents a sequential pattern-recognition system employing recurrent Fuzzy Systems that is employed as a monitoring system on continuous-casting Systems in the steel industry worldwide. Taking that application as a starting point, a general method for sequential pattern recognition in time series that uses recurrent Fuzzy Systems is described.

  • Equilibria of recurrent Fuzzy Systems
    Fuzzy Sets and Systems, 2003
    Co-Authors: Roland Kempf, Jurgen Adamy
    Abstract:

    Unlike static Fuzzy Systems, recurrent Fuzzy Systems are equipped with feedback loops and thus exhibit dynamic behaviors. The dynamics of a recurrent Fuzzy system is largely determined by its rule base. The dynamic behavior of a significant subclass of recurrent Fuzzy Systems may be immediately deduced from their rule base, without need for analyzing their mathematical description. Their equilibrium points may be readily identified and their stability behaviors investigated based on their rule base. The investigations involved lead to convergence theorems and other statements that preclude chaotic dynamics.

  • Regularity and chaos in recurrent Fuzzy Systems
    Fuzzy Sets and Systems, 2003
    Co-Authors: Jurgen Adamy, Roland Kempf
    Abstract:

    Abstract In this paper, we shall present a mathematical definition of recurrent Fuzzy Systems and begin to systematically investigate the underlying theory involved. Unlike static Fuzzy Systems, recurrent Fuzzy Systems are equipped with time-delayed feedback of their output and allow representing knowledge-based dynamic processes that may be stated in the form of “if …, then …” rules. We study their relationship to automata and show that they have an automaton-like behavior when appropriately designed. In other cases, recurrent Fuzzy system may exhibit chaotic behavior. We present sufficient conditions for the occurrence of chaos in recurrent Fuzzy Systems that can easily be checked solely on the basis of the qualitative, linguistically formulated models. We also discuss the extent to which state graphs may be used for describing the behaviors of recurrent Fuzzy Systems.

John A. Keane - One of the best experts on this subject based on the ideXlab platform.

  • An Evolving-Construction Scheme for Fuzzy Systems
    IEEE Transactions on Fuzzy Systems, 2010
    Co-Authors: Di Wang, Xiao-jun Zeng, John A. Keane
    Abstract:

    This paper proposes an evolving-construction scheme for Fuzzy Systems (ECSFS). ECSFS begins with a simple Fuzzy system and evolves its structure by adding more Fuzzy terms and rules to achieve a better accuracy in a "greedy'' way. An interesting feature of ECSFS is that it is able to automatically locate mathematically meaningful points, such as the extremum and inflexion points of the approximated function one by one, and then adds Fuzzy terms based on these points. Fuzzy Systems with such extreme points, like their Fuzzy terms, are more efficient than other Fuzzy Systems by using the same number of Fuzzy rules. As a result, ECSFS often achieves a better accuracy for Fuzzy-system identification compared with the previous methods when using the same number of Fuzzy rules. A number of simulation results are given to illustrate the advantages of the proposed scheme.

  • Approximation Capabilities of Hierarchical Fuzzy Systems
    IEEE Transactions on Fuzzy Systems, 2005
    Co-Authors: Xiao-jun Zeng, John A. Keane
    Abstract:

    Derived from practical application in location analysis and pricing, and based on the approach of hierarchical structure analysis of continuous functions, this paper investigates the approximation capabilities of hierarchical Fuzzy Systems. By first introducing the concept of the natural hierarchical structure, it is proved that continuous functions with natural hierarchical structure can be naturally and effectively approximated by hierarchical Fuzzy Systems to overcome the curse of dimensionality in both the number of rules and parameters. Then, based on Kolmogorov's theorem, it is shown that any continuous function can be represented as a superposition of functions with the natural hierarchical structure and can then be approximated by hierarchical Fuzzy Systems to achieve the universal approximation property. Further, the conditions under which the hierarchical Fuzzy approximation is superior to the standard Fuzzy approximation in overcoming the curse of dimensionality are analyzed

  • FUZZ-IEEE - Separable Approximation Property of Hierarchical Fuzzy Systems
    The 14th IEEE International Conference on Fuzzy Systems 2005. FUZZ '05., 1
    Co-Authors: Xiao-jun Zeng, John A. Keane
    Abstract:

    This paper discusses the capabilities of standard hierarchical Fuzzy Systems to approximate continuous functions with natural hierarchical structure. The separable approximation property of hierarchical Fuzzy Systems is proved, that is, the construction of a hierarchical Fuzzy system with required approximation accuracy can be achieved by the separate construction of each sub-system with required approximation accuracy. This property provides a simple method to construct hierarchical Fuzzy Systems for function approximation. Based on the separable approximation property, it is further proved the structure approximation property of hierarchical Fuzzy Systems

Roland Kempf - One of the best experts on this subject based on the ideXlab platform.

  • sequential pattern recognition employing recurrent Fuzzy Systems
    Fuzzy Sets and Systems, 2004
    Co-Authors: Roland Kempf, Jurgen Adamy
    Abstract:

    Sequential pattern-recognition Systems check whether data strings, e.g., time series, exhibit certain pattern primitives in a specified order. As in the case of most other pattern-recognition methods, either conventional methods or Fuzzy Systems may be used here. This paper presents a sequential pattern-recognition system employing recurrent Fuzzy Systems that is employed as a monitoring system on continuous-casting Systems in the steel industry worldwide. Taking that application as a starting point, a general method for sequential pattern recognition in time series that uses recurrent Fuzzy Systems is described.

  • Equilibria of recurrent Fuzzy Systems
    Fuzzy Sets and Systems, 2003
    Co-Authors: Roland Kempf, Jurgen Adamy
    Abstract:

    Unlike static Fuzzy Systems, recurrent Fuzzy Systems are equipped with feedback loops and thus exhibit dynamic behaviors. The dynamics of a recurrent Fuzzy system is largely determined by its rule base. The dynamic behavior of a significant subclass of recurrent Fuzzy Systems may be immediately deduced from their rule base, without need for analyzing their mathematical description. Their equilibrium points may be readily identified and their stability behaviors investigated based on their rule base. The investigations involved lead to convergence theorems and other statements that preclude chaotic dynamics.

  • Regularity and chaos in recurrent Fuzzy Systems
    Fuzzy Sets and Systems, 2003
    Co-Authors: Jurgen Adamy, Roland Kempf
    Abstract:

    Abstract In this paper, we shall present a mathematical definition of recurrent Fuzzy Systems and begin to systematically investigate the underlying theory involved. Unlike static Fuzzy Systems, recurrent Fuzzy Systems are equipped with time-delayed feedback of their output and allow representing knowledge-based dynamic processes that may be stated in the form of “if …, then …” rules. We study their relationship to automata and show that they have an automaton-like behavior when appropriately designed. In other cases, recurrent Fuzzy system may exhibit chaotic behavior. We present sufficient conditions for the occurrence of chaos in recurrent Fuzzy Systems that can easily be checked solely on the basis of the qualitative, linguistically formulated models. We also discuss the extent to which state graphs may be used for describing the behaviors of recurrent Fuzzy Systems.