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Research Papers

Visualizing Non-Linear Control System Performance by Means of Multidimensional Scaling

[+] Author and Article Information
J. A. Tenreiro Machado

Institute of Engineering,
Polytechnic of Porto,
Department of Electrical Engineering,
Porto 4200-07, Portugal
e-mail: jtm@isep.ipp.pt

Contributed by the Design Engineering Division of ASME for publication in the Journal of Computational and Nonlinear Dynamics. Manuscript received November 21, 2011; final manuscript received May 6, 2013; published online July 18, 2013. Assoc. Editor: Hiroshi Yabuno.

J. Comput. Nonlinear Dynam 8(4), 041017 (Jul 18, 2013) (7 pages) Paper No: CND-11-1222; doi: 10.1115/1.4024540 History: Received November 21, 2011; Revised May 06, 2013

This paper addresses the use of multidimensional scaling in the evaluation of controller performance. Several nonlinear systems are analyzed based on the closed loop time response under the action of a reference step input signal. Three alternative performance indices, based on the time response, Fourier analysis, and mutual information, are tested. The numerical experiments demonstrate the feasibility of the proposed methodology and motivate its extension for other performance measures and new classes of nonlinearities.

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References

Atherton, D. P., 1975, Nonlinear Control Engineering, van Nostrand Reinhold, London.
J. E.Slotine, and Li, W., 1991, Applied Nonlinear Control, Prentice-Hall, Englewood Cliffs, NJ.
Cox, C. S., 1987, “Algorithms for Limit Cycle Prediction: A Tutorial Paper,” Int. J. Electr. Eng. Ed., 24(2), pp. 165–182.
FernandoB., and Duarte, J. T. M., 2009, “Fractional Describing Function of Systems With Coulomb Friction,” Nonlinear Dyn., 56(4), pp. 381–387. [CrossRef]
FernandoB., and Duarte, J. T. M., 2009, “Describing Function of Two Masses With Backlash,” Nonlinear Dyn., 56(4), pp. 409–413. [CrossRef]
Gelb, A., and Vander Velde, W. E., 1968, Multiple-Input Describing Functions and Non-Linear System Design, McGraw-Hill, New York.
Charles, L., and Phillips, R. D. H., 1996, Feedback Control Systems, Prentice-Hall, Englewood Cliffs, NJ.
Torgerson, W. S., 1958, Theory and Methods of Scaling, Wiley, New York.
Shepard, R. N., 1962, “The Analysis of Proximities: Multidimensional Scaling With an Unknown Distance Function,” Psychometrika, 27(I/II), pp. 219–246. [CrossRef]
Kruskal, J., 1964, “Multidimensional Scaling by Optimizing Goodness of Fit to a Nonmetric Hypothesis,” Psychometrika, 29(1), pp. 1–27. [CrossRef]
Kruskal, J. B., and Wish, M., 1978, Multidimensional Scaling, Sage Publications, Newbury Park.
Cox, T. F., and Cox, M. A. A., 2001, Multidimensional Scaling, Chapman and Hall, London.
Borg, I., and Groenen, P. J., 2005, Modern Multidimensional Scaling-Theory and Applications, Springer-Verlag, New York.
Martinez, W. L., and Martinez, A. R., 2005, Exploratory Data Analysis With MATLAB, Chapman and Hall, London.
de Leeuw, J., and Mair, P., 2009, “Multidimensional Scaling Using Majorization: Smacof in R,” J. Stat. Software, 31(3), pp. 1–30.
Machado, J. A. T., and Galhano, A. M. S., 2009, “Approximating Fractional Derivatives in the Perspective of System Control,” Nonlinear Dyn., 56(4), pp. 401–407. [CrossRef]
Machado, J., and Baleanu, D., 2011, “Characterization Approach to Modified Glassy Carbon Electrode-Nanofilm System Within Multidimensional Scaling,” J. Comput. Theor. Nanosci., 8(2), pp. 1–6. [CrossRef]
TenreiroMachado, J. A., and GonçaloM.Duarte, F. B. D., 2011, “Identifying Economic Periods and Crisis With the Multidimensional Scaling,” Nonlinear Dyn., 63(4), pp. 611–622. [CrossRef]
Chang, C.-H., and Chang, M.-K., 1994, “Analysis of Gain Margins and Phase Margins of a Nonlinear Reactor Control System,” IEEE Trans. Nucl. Sci., 41(4), pp. 1686–1691. [CrossRef]
Wu, B. F., Perng, J.-W., and Chin, H.-I., 2005, “Limit Cycle Analysis of Nonlinear Sampled-Data Systems by Gainphase Margin Approach,” J. Franklin Inst., 342(2), pp. 175–192. [CrossRef]
Cha, S.-H., 2008, “Taxonomy of Nominal Type Histogram Distance Measures,” Proc. of the American Conference on Applied Mathematics.
Shannon, C. E., 1948, “A Mathematical Theory of Communication,” Bell Syst. Tech. J., 27(7/10), pp. 379–423;623–656.
Jaynes, E. T., 1957, “Information Theory and Statistical Mechanics,” Phys. Rev., 106(4), pp. 620–630. [CrossRef]
Khinchin, A. I., 1957, Mathematical Foundations of Information Theory, Dover, New York.

Figures

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Fig. 1

Nonlinearities: Saturation, backlash and the combination of relay, and deadzone

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Fig. 2

Locus (K, θ, X) for the saturation, backlash, and combination of relay and deadzone

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Fig. 3

Two dimensional MDS map of the fifteen control systems for the measure based on signal time correlation and a unit step reference input

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Fig. 4

Three dimensional MDS map of the fifteen control systems for the measure based on signal time correlation and a unit step reference input

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Fig. 5

Shepard diagram for the two dimensional MDS map with fifteen control systems and the measure based on signal time correlation and a unit step reference input

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Fig. 6

Shepard diagram for the three dimensional MDS map with fifteen control systems and the measure based on signal time correlation and a unit step reference input

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Fig. 7

Stress versus number of dimensions of the MDS maps for the fifteen control systems and the measure based on signal time correlation and a unit step reference input

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Fig. 8

Three dimensional MDS map of the fifteen control systems for the measure based on frequency distance and a unit step reference input

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Fig. 9

Three dimensional MDS map of the fifteen control systems for the measure based on the mutual information and a unit step reference input

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Fig. 10

Three dimensional MDS map of the fifteen control systems for the measure based on signal time correlation and a double pulse reference input

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Fig. 11

Three dimensional MDS map of the fifteen control systems for the measure based on frequency distance and a double pulse reference input

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Fig. 12

Three dimensional MDS map of the fifteen control systems for the measure based on the mutual information and a double pulse reference input

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