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Fuzzy neural network-based system identification of multi-storey shear buildings

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Abstract

This paper uses fuzzy neural network modelling for the identification of structural parameters of multi-storey shear buildings. First the identification has been done using response of the structure subject to ambient vibration with fuzzy initial condition. Then forced vibration with horizontal displacement in fuzzified form has been used to investigate the identification procedure. The model has been developed to handle the data in fuzzified form for multi-storey shear structure, and the procedure is tested for the identification of the stiffness parameters of simple example problems using the prior values of the design parameters.

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Acknowledgments

The authors would like to acknowledge funding from the Ministry of Earth Sciences, New Delhi, to carry out this investigation. The authors also thank the anonymous reviewers for their suggestions which have helped in improving the contents of this paper.

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Authors and Affiliations

  1. Department of Mathematics, National Institute of Technology, Rourkela, Rourkela, Odisha, 769008, India

    S. Chakraverty & Deepti Moyi Sahoo

Authors
  1. S. Chakraverty
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  2. Deepti Moyi Sahoo
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Correspondence to Deepti Moyi Sahoo.

Appendix: Fuzzy set theory and preliminaries

Appendix: Fuzzy set theory and preliminaries

The following Fuzzy arithmetic operations are used in this paper for defining fuzzified neural network.

  1. (A)

    Let X be an universal set. Then the fuzzy subset A of X is defined by its membership function

    $$\mu_{A} :\;X \to \left[ {0,1} \right]$$

    which assign a real number μ A (x) in the interval [0, 1], to each element x ∊ X, where the value of μ A (x) at x shows the grade of membership of x in A.

  2. (B)

    Given a fuzzy set A in X and any real number α ∊ [0, 1], then the α- cut or α-level or cut worthy set of A, denoted by A α is the crisp set

    $$A_{\alpha } = \left\{ {x \in X\left| {\mu_{A} (x) \ge \alpha } \right.} \right\}$$
  3. (C)

    A triangular fuzzy number (TFN) A can be defined as a triplet \(\left[ {\underline{a} ,ac,\bar{a}} \right]\). Its member ship function is defined as

    $$\mu_{A} (x) = \left\{ {\begin{array}{*{20}l} {0,} \hfill & {x < \underline{a} } \hfill \\ {\frac{{x - \underline{a} }}{{ac - \bar{a}}},} \hfill & {\underline{a} \le x \le ac} \hfill \\ {\frac{{\bar{a} - x}}{{\bar{a} - ac}},} \hfill & {ac \le x \le \bar{a}} \hfill \\ {0,} \hfill & {x > \bar{a}} \hfill \\ \end{array} } \right.$$

    Above TFN may be transformed to an interval form A α by α- cut as

    $$A_{\alpha } = \left[ {\underline{a}^{\left( \alpha \right)} ,\bar{a}^{(\alpha )} } \right] = \left[ {(ac - \underline{a} )\alpha + \underline{a} , - (\bar{a} - ac)\alpha + \bar{a}} \right]$$

    where a (α) and \(\overline{a}^{\left( \alpha \right)}\) are the lower and upper limits of the α-level set A α .

1.1 Operation of fuzzy numbers

We introduce arithmetic operations on fuzzy numbers and the results are expressed in terms of membership functions ∀x, y, z ∊ R as below

  1. 1.

    Addition: \(\mu_{A( + )B} (z) = \hbox{max} \{ \mu_{A} (x) \wedge \mu_{B} (y)|z = x + y\} .\)

  2. 2.

    Multiplication: \(\mu_{A( \cdot )B} (z) = \hbox{max} \{ \mu_{A} (x) \wedge \mu_{B} (y)|z = x \cdot y\} .\)

  3. 3.

    Nonlinear mapping: \(\mu_{f(Net)} (z) = \hbox{max} \left\{ {\mu_{Net} (x)|z = f(x)} \right\} .\)

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Chakraverty, S., Sahoo, D.M. Fuzzy neural network-based system identification of multi-storey shear buildings. Neural Comput & Applic 28, 597–612 (2017). https://doi.org/10.1007/s00521-015-2101-y

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