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Research Papers

A Comprehensive Fuzzy Uncertainty Analysis of a Controlled Nonlinear System With Unstable Internal Dynamics

[+] Author and Article Information
Nico-Philipp Walz

Institute of Engineering and Computational Mechanics, University of Stuttgart,
70569 Stuttgart, Germany
e-mail: nico-philipp.walz@itm.uni-stuttgart.de

Markus Burkhardt

Institute of Engineering and Computational Mechanics, University of Stuttgart,
70569 Stuttgart, Germany
e-mail: markus.burkhardt@itm.uni-stuttgart.de

Peter Eberhard

Professor
Institute of Engineering and Computational Mechanics, University of Stuttgart,
70569 Stuttgart, Germany
e-mail: peter.eberhard@itm.uni-stuttgart.de

Michael Hanss

Professor
Institute of Engineering and Computational Mechanics, University of Stuttgart,
70569 Stuttgart, Germany
e-mail: michael.hanss@itm.uni-stuttgart.de

Manuscript received October 25, 2014; final manuscript received June 8, 2015; published online October 2, 2015. Assoc. Editor: Alba Sofi.

ASME J. Risk Uncertainty Part B 1(4), 041008 (Oct 02, 2015) (12 pages) Paper No: RISK-14-1077; doi: 10.1115/1.4030810 History: Received October 25, 2014; Accepted June 10, 2015; Online October 02, 2015

Fuzzy uncertainty analyses disclose a deeper insight and provide a better understanding of complex systems with highly interdependent parameters. In contrast to probability theory, fuzzy arithmetic is concerned with epistemic uncertainties, which originate from a lack of knowledge or from idealizing assumptions in the modeling process. Direct fuzzy arithmetic can be used to illustrate how parameter uncertainties propagate through a system. In contrast, inverse fuzzy arithmetic can be used to identify admissible parameter uncertainties that obey defined error bounds. In addition, fuzzy arithmetic is capable of providing global sensitivity analyses. Therefore, an improved formulation for inverse analyses as well as a new concept for the computation of global sensitivities is presented. These tools are used here to assess the model-based feed-forward control of a nonlinear system with unstable internal dynamics.

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References

Figures

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Fig. 1

Experimental setup of the parallel manipulator with the three active and highlighted strain gauges

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Fig. 2

Triangular fuzzy number; the dots indicate the membership degree μ

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Fig. 3

Parallel manipulator with flexible links (left) and mode shapes of the long link (right)

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Fig. 4

Total influence measures K ˜iT of the drive positions s1 and s2 with respect to the model parameters of Table 2

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Fig. 5

Total influence measures K ˜iT of the driving forces u1 and u2 with respect to the model parameters of Table 2

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Fig. 6

Fuzzy coordinates of the end-effector for fuzzy-valued initial conditions

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Fig. 7

Curvature at the second strain gauge (see Fig. 1) for fuzzy-valued initial conditions

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Fig. 8

End-effector position in the x−y-plane with fuzzy-valued initial conditions with α-cut bounds computed in Cartesian and natural coordinates; the upper plot shows the overall motion and the lower plot shows a detailed view of the highlighted region (black rectangle)

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Fig. 9

Deviations of end-effector position from nominal trajectory; the dashed line displays the specified tolerance

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