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Fractional-order system identification for health monitoring

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Abstract

Fractional-order differential equations can describe the dynamics of robot formations and other high-order systems. These equations are useful models for such systems because of the flexibility afforded by including noninteger derivatives. A system’s fractional order may change in response to mechanical or operational damage, but the possibility of an order change is not typically considered in structural health monitoring or other system monitoring tools. Typically, the order is assumed to be an integer from the physics of the system, while behaviors are captured by parameters within the chosen model. In contrast, this work presents a procedure to identify the fractional order of a system’s dynamics across a variety of parameter changes; the inclusion of fractional orders allows order itself to measure dynamical shifts. This work presents the identification procedure, its mathematical foundations, and results from example systems representing two mobile robot formations. The fractional order changes in a manner consistent with the physical changes modeled by damage, suggesting that this procedure is widely applicable in health monitoring.

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References

  1. Goodwine, B.: Modeling a multi-robot system with fractional-order differential equations. In: Proceedings of the IEEE International Conference on Robotics and Automation, pp. 1763–1768 (2014)

  2. Goodwine, B.: Fractional-order dynamics in a random, approximately scale-free network of agents. In: Proceedings of the IEEE Conference on Control, Automation, Robotics and Vision, pp. 1581–1586 (2014)

  3. Leyden, K., Goodwine, B.: Using fractional-order differential equations for health monitoring of a system of cooperating robots. In: Proceedings of the IEEE International Conference on Robotics and Automation, pp. 366–371 (2016)

  4. Heymans, N., Bauwens, J.: Fractal rheological models and fractional differential equations for viscoelastic behavior. Rheol. Acta 33, 210 (1994)

    Article  Google Scholar 

  5. Mayes, J.: Reduction and approximation in large and infinite potential-driven flow networks. Ph.D. thesis, University of Notre Dame (2012)

  6. Hartley, T.T., Lorenzo, C.F.: Fractional-order system identification based on continuous order-distributions. Signal Process. 83(11), 2287 (2003)

    Article  MATH  Google Scholar 

  7. Das, S.: Functional Fractional Calculus. Springer, Berlin (2011)

    Book  MATH  Google Scholar 

  8. Liu, D.Y., Laleg-Kirati, T.M., Gibaru, O., Perruquetti, W.: Identification of fractional order systems using modulating functions method. In: American Control Conference (ACC), 2013, pp. 1679–1684. IEEE (2013)

  9. Narang, A., Shah, S.L., Chen, T.: Continuous-time model identification of fractional-order models with time delays. IET Control Theory Appl. 5(7), 900 (2011)

    Article  MathSciNet  Google Scholar 

  10. Zhou, S., Cao, J., Chen, Y.: Genetic algorithm-based identification of fractional-order systems. Entropy 15(5), 1624 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Oustaloup, A.: La dérivation non entière. Hermes, Paris (1995)

    MATH  Google Scholar 

  12. Ren, W., Beard, R.W., Atkins, E.M.: Information consensus in multivehicle cooperative control. IEEE Control Syst. Mag. 27, 71–82 (2007)

    Article  Google Scholar 

  13. Fax, J.A., Murray, R.M.: Information flow and cooperative control of vehicle formations. IEEE Trans. Autom. Control 49(9), 1465 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Leonard, N., Fiorelli, E.: Virtual leaders, artificial potentials, and coordinated control of groups. In: Proceedings of the 40th IEEE Conference on Decision and Control, pp. 2968–2973 (2001)

  15. Das, A.K., Fierro, R., Kumar, V., Ostrowski, J.P., Spletzer, J., Taylor, C.J.: A vision-based formation control framework. IEEE Trans. Robot. Autom. 18(5), 813 (2002)

    Article  Google Scholar 

  16. Murray, R.M.: Recent research in cooperative control of multivehicle systems. J. Dyn. Syst. Meas. Control 129(5), 571 (2007)

    Article  Google Scholar 

  17. Cao, Y., Yu, W., Ren, W., Chen, G.: An overview of recent progress in the study of distributed multi-agent coordination. IEEE Trans. Ind. Inform. 9(1), 427 (2013)

    Article  Google Scholar 

  18. McMickell, M.B., Goodwine, B.: Reduction and non-linear controllability of symmetric distributed systems. Int. J. Control 76(18), 1809 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. McMickell, M.B., Goodwine, B.: Motion planning for nonlinear symmetric distributed robotic formations. Int. J. Robot. Res. 26(10), 1025 (2007)

    Article  Google Scholar 

  20. Goodwine, B., Antsaklis, P.J.: Multi-agent compositional stability exploiting system symmetries. Automatica 49, 3158–3166 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. McMickell, M.B., Goodwine, B.: Reduction and non-linear controllability of symmetric distributed systems with drift. In: Proceedings of the IEEE International Conference on Robotics and Automation, pp. 3454–3460 (2002)

  22. Karsai, G., Sztipanovits, J.: Model-integrated development of cyber-physical systems. In: Software Technologies for Embedded and Ubiquitous Systems, pp. 46–54 (2008)

  23. Lee, E.A.: CPS foundations. In: Design Automation Conference (DAC), 2010 47th ACM/IEEE, pp. 737–742. IEEE (2010)

  24. Derler, P., Lee, E.A., Vincentelli, A.S.: Modeling cyber-physical systems. Proc. IEEE 100(1), 13 (2012)

    Article  Google Scholar 

  25. Reynders, E., Houbrechts, J., De Roeck, G.: Fully automated (operational) modal analysis. Mech. Syst. Signal Process. 29, 228 (2012)

    Article  Google Scholar 

  26. Rainieri, C., Fabbrocino, G.: Development and validation of an automated operational modal analysis algorithm for vibration-based monitoring and tensile load estimation. Mech. Syst. Signal Process. 60, 512 (2015)

    Article  Google Scholar 

  27. Chatzis, M.N., Chatzi, E.N.: A discontinuous unscented Kalman filter for non-smooth dynamic problems. Front. Built Environ. 3, 56 (2017)

    Article  Google Scholar 

  28. Shirdel, A.H., Björk, K.M., Toivonen, H.T.: Identification of linear switching system with unknown dimensions. In: 47th Hawaii International Conference on System Sciences (HICSS), 2014, pp. 1344–1352. IEEE (2014)

  29. Peeters, B., De Roeck, G.: Reference-based stochastic subspace identification for output-only modal analysis. Mech. Syst. Signal Process. 13(6), 855 (1999)

    Article  Google Scholar 

  30. Juang, J.N., Pappa, R.S.: An eigensystem realization algorithm for modal parameter identification and model reduction. J. Guid. Control. Dyn. 8(5), 620 (1985)

    Article  MATH  Google Scholar 

  31. Chatzi, E.N., Smyth, A.W.: The unscented Kalman filter and particle filter methods for nonlinear structural system identification with non-collocated heterogeneous sensing. Struct. Control Health Monit. 16(1), 99 (2009)

    Article  Google Scholar 

  32. Jamaludin, I., Wahab, N., Khalid, N., Sahlan, S., Ibrahim, Z., Rahmat, M.F.: N4SID and MOESP subspace identification methods. In: IEEE 9th International Colloquium on Signal Processing and Its Applications (CSPA), 2013, pp. 140–145. IEEE (2013)

  33. Chen, C.W., Juang, J.N., Lee, G.: Frequency domain state-space system identification. In: American Control Conference, 1993, pp. 3057–3061. IEEE (1993)

  34. Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974)

    MATH  Google Scholar 

  35. Baleanu, D., Machado, J.A.T., Luo, A.C.J. (eds.): Fractional Dynamics and Control. Springer, Berlin (2011)

    MATH  Google Scholar 

  36. Ortigueira, M.: An introduction to the fractional continuous-time linear systems: the 21st century systems. IEEE Circuits Syst. Mag. 8(3), 19 (2008)

    Article  Google Scholar 

  37. Tenreiro Machado, J., Kiryakova, V., Mainardi, F.: Recent history of fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 16(3), 1140 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  38. Silva, M.F., Tenreiro Machado, J., Lopes, A.: Fractional order control of a hexapod robot. Nonlinear Dyn. 38(1–4), 417 (2004)

  39. Delavari, H., Lanusse, P., Sabatier, J.: Fractional order controller design for a flexible link manipulator robot. Asian J. Control 15, 783 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  40. Chen, Y., Moore, K.L.: Analytical stability bound for a class of delayed fractional-order dynamic systems. Nonlinear Dyn. 29(1–4), 191 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  41. Zhao, C., Xue, D., Chen, Y.: A fractional order PID tuning algorithm for a class of fractional order plants. In: Proceedings of the IEEE International Conference on Mechatronics and Automation, pp. 216–221 (2005)

  42. Monje, C.A., Vinagre, B.M., Feliu, V., Chen, Y.: Tuning and auto-tuning of fractional order controllers for industry applications. Control Eng. Pract. 16(7), 798 (2008)

    Article  Google Scholar 

  43. Cao, Y., Ren, W.: Distributed formation control for fractional-order systems: dynamic interaction and absolute/relative damping. Syst. Control Lett. 59(34), 233 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  44. Cao, Y., Li, Y., Ren, W., Chen, Y.Q.: Distributed coordination of networked fractional-order systems. IEEE Trans. Syst. Man Cybern. Part B Cybern. 40(2), 362 (2010)

    Article  Google Scholar 

  45. Goodwine, B., Leyden, K.: Recent results in fractional-order modeling in multi-agent systems and linear friction welding. IFAC PapersOnLine 48(1), 380 (2015)

    Article  Google Scholar 

  46. Ortigueira, M.D., Machado, J.T.: What is a fractional derivative? J. Comput. Phys. 293, 4 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors gratefully acknowledge many interesting and fruitful discussions with Fabio Semperlotti and Mihir Sen.

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Authors and Affiliations

  1. Aerospace and Mechanical Engineering, Univ. of Notre Dame, 365 Fitzpatrick Hall, Notre Dame, IN, 46556, USA

    Kevin Leyden & Bill Goodwine

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  1. Kevin Leyden
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  2. Bill Goodwine
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Correspondence to Kevin Leyden.

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The authors declare that they have no conflict of interest.

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The partial support of the US National Science Foundation under the RI Grant No. IIS-1527393 is gratefully acknowledged.

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Leyden, K., Goodwine, B. Fractional-order system identification for health monitoring. Nonlinear Dyn 92, 1317–1334 (2018). https://doi.org/10.1007/s11071-018-4128-y

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