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

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

Michael Hanss

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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Seifried, R., Burkhardt, M., and Held, A., 2013, “Trajectory Control of Serial and Parallel Flexible Manipulators Using Model Inversion,” Multibody Dynamics, J.-C. Samin, and P. Fisette, eds. (Computational Methods in Applied Sciences, Vol. 28), Springer, Berlin, pp. 53–75.
Rao, S., and Sawyer, J. P., 1995, “Fuzzy Finite Element Approach for Analysis of Imprecisely Defined Systems,” AIAA J., 33(12), pp. 2364–2370. 10.2514/3.12910
Moens, D., and Vandepitte, D., 2002, “Fuzzy Finite Element Method for Frequency Response Function Analysis of Uncertain Structures,” AIAA J., 40(1), pp. 126–136.
Moens, D., and Vandepitte, D., 2005, “A Survey of Non-Probabilistic Uncertainty Treatment in Finite Element Analysis (Special Issue on Computational Methods in Stochastic Mechanics and Reliability Analysis),” Comput. Methods Appl. Mech. Eng., 194(1216), pp. 1527–1555. 10.1016/j.cma.2004.03.019
Hanss, M., 2005, Applied Fuzzy Arithmetic—An Introduction with Engineering Applications, Springer, Berlin.
Zadeh, L. A., 1965, “Fuzzy Sets,” Inform. Control, 8(3), pp. 338–353. 10.1016/S0019-9958(65)90241-X
Walz, N.-P., Fischer, M., Hanss, M., and Eberhard, P., 2012, “Uncertainties in Multibody Systems—Potentials and Challenges,” Proceedings of the 4th International Conference on Uncertainty in Structural Dynamics, P. Sas, D. Moens, and S. Jonckheere, eds., Katholieke Universiteit Leuven, Leuven, Belgium, pp. 4643–4657.
Massa, F., Tison, T., and Lallemand, B., 2006, “A Fuzzy Procedure for the Static Design of Imprecise Structures,” Comput. Methods Appl. Mech. Eng., 195(912), pp. 925–941. 10.1016/j.cma.2005.02.015
Balu, A., and Rao, B., 2013, “Confidence Bounds on Design Variables Using High-Dimensional Model Representation-Based Inverse Reliability Analysis,” J. Struct. Eng., 139(6), pp. 985–996. 10.1061/(ASCE)ST.1943-541X.0000709
Haag, T., 2012, Forward and Inverse Fuzzy Arithmetic for Uncertainty Analysis With Applications to Structural Mechanics, Der Andere Verlag, Uelvesbüll, Germany.
Haag, T., Herrmann, J., and Hanss, M., 2010, “Identification Procedure for Epistemic Uncertainties Using Inverse Fuzzy Arithmetic,” Mech. Syst. Signal Process., 24(7), pp. 2021–2034. 10.1016/j.ymssp.2010.05.010
Kozlov, M., Tarasov, S., and Khachiyan, L., 1980, “The Polynomial Solvability of Convex Quadratic Programming,” USSR Computat. Math. Math. Phys., 20(5), pp. 223–228. 10.1016/0041-5553(80)90098-1
Li, G., Rosenthal, C., and Rabitz, H., 2001, “High Dimensional Model Representations,” J. Phys. Chemi. A, 105(33), pp. 7765–7777. 10.1021/jp010450t
Feuersänger, C., 2010, “Sparse Grid Methods for Higher Dimensional Approximation,” Ph.D. thesis, Institut für Numerische Simulation, Universität Bonn, Germany.
Sobol, I., 2001, “Global Sensitivity Indices for Nonlinear Mathematical Models and Their Monte Carlo Estimates,” Math. Comput. Simul., 55(1–3), pp. 271–280. 10.1016/S0378-4754(00)00270-6
Walz, N.-P., Burkhardt, M., Hanss, M., and Eberhard, P., 2015, “Sensitivity Computation for Uncertain Dynamical Systems Using High-Dimensional Model Representation and Hierarchical Grids,” Procedia IUTAM (Dynamical Analysis of Multibody Systems with Design Uncertainties), 13(1), pp. 127–137. 10.1016/j.piutam.2015.01.010
Bungartz, H.-J., and Griebel, M., 2004, “Sparse Grids,” Acta Numer., 13, pp. 147–269. 10.1017/S0962492904000182
Pflüger, D., 2010, Spatially Adaptive Sparse Grids for High-Dimensional Problems, Verlag Dr. Hut, Munich.
Adhikari, S., Chowdhury, R., and Friswell, M., 2011, “High Dimensional Model Representation Method for Fuzzy Structural Dynamics,” J. Sound Vib., 330(7), pp. 1516–1529. 10.1016/j.jsv.2010.10.010
Balu, A., and Rao, B., 2012, “High Dimensional Model Representation Based Formulations for Fuzzy Finite Element Analysis of Structures,” Finite Elem. Anal. Des., 50(1), pp. 217–230. 10.1016/j.finel.2011.09.012
Blajer, W., and Kołodziejczyk, K., 2004, “A Geometric Approach to Solving Problems of Control Constraints: Theory and a DAE Framework,” Multibody Syst. Dyn., 11(4), pp. 343–364. 10.1023/B:MUBO.0000040800.40045.51
Campbell, S. L., and Gear, C. W., 1995, “The Index of General Nonlinear DAEs,” Numer. Math., 72(2), pp. 173–196. 10.1007/s002110050165
Khalil, H. K., 2002, Nonlinear Systems, 3rd ed., Prentice Hall, Upper Saddle River, NJ.
Chen, D., and Paden, B., 1996, “Stable Inversion of Nonlinear Non-Minimum Systems,” Int. J. Control, 64(1), pp. 81–97. 10.1080/00207179608921618
Kailath, T., 1980, Linear Systems, Prentice-Hall, Englewood Cliffs, NJ.
Holzwarth, P., and Eberhard, P., 2015, “SVD-Based Improvements for Component Mode Synthesis in Elastic Multibody Systems,” Eur. J. Mech. A/Solids, 49(4), pp. 408–418. 10.1016/j.euromechsol.2014.08.009
Adhikari, S., 2006, “Damping Modelling Using Generalized Proportional Damping,” J. Sound Vib., 293(1–2), pp. 156–170. 10.1016/j.jsv.2005.09.034
Wallrapp, O., 1994, “Standardization of Flexible Body Modeling in Multibody System Codes, Part I: Definition of Standard Input Data,” Mech. Struct. Mach., 22(3), pp. 283–304. 10.1080/08905459408905214
Spong, M. W., Hutchinson, S., and Vidyasagar, M., 2006, Robot Modeling and Control, John Wiley & Sons, Hoboken, NJ.
Damaren, C. J., 2000, “Passivity and Noncollocation in the Control of Flexible Multibody Systems,” ASME J. Dyn. Syst. Meas. Control, 122(1), pp. 11–17. 10.1115/1.482423


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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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