Nonlinear Dynamics

, Volume 92, Issue 3, pp 1317–1334 | Cite as

Fractional-order system identification for health monitoring

  • Kevin Leyden
  • Bill Goodwine
Original Paper


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.


Fractional calculus System identification Health monitoring Damage detection Frequency response 



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

Compliance with ethical standards

Conflicts of interest

The authors declare that they have no conflict of interest.


  1. 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)Google Scholar
  2. 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)Google Scholar
  3. 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)Google Scholar
  4. 4.
    Heymans, N., Bauwens, J.: Fractal rheological models and fractional differential equations for viscoelastic behavior. Rheol. Acta 33, 210 (1994)CrossRefGoogle Scholar
  5. 5.
    Mayes, J.: Reduction and approximation in large and infinite potential-driven flow networks. Ph.D. thesis, University of Notre Dame (2012)Google Scholar
  6. 6.
    Hartley, T.T., Lorenzo, C.F.: Fractional-order system identification based on continuous order-distributions. Signal Process. 83(11), 2287 (2003)CrossRefzbMATHGoogle Scholar
  7. 7.
    Das, S.: Functional Fractional Calculus. Springer, Berlin (2011)CrossRefzbMATHGoogle Scholar
  8. 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)Google Scholar
  9. 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)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Zhou, S., Cao, J., Chen, Y.: Genetic algorithm-based identification of fractional-order systems. Entropy 15(5), 1624 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Oustaloup, A.: La dérivation non entière. Hermes, Paris (1995)zbMATHGoogle Scholar
  12. 12.
    Ren, W., Beard, R.W., Atkins, E.M.: Information consensus in multivehicle cooperative control. IEEE Control Syst. Mag. 27, 71–82 (2007)CrossRefGoogle Scholar
  13. 13.
    Fax, J.A., Murray, R.M.: Information flow and cooperative control of vehicle formations. IEEE Trans. Autom. Control 49(9), 1465 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 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)Google Scholar
  15. 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)CrossRefGoogle Scholar
  16. 16.
    Murray, R.M.: Recent research in cooperative control of multivehicle systems. J. Dyn. Syst. Meas. Control 129(5), 571 (2007)CrossRefGoogle Scholar
  17. 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)CrossRefGoogle Scholar
  18. 18.
    McMickell, M.B., Goodwine, B.: Reduction and non-linear controllability of symmetric distributed systems. Int. J. Control 76(18), 1809 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    McMickell, M.B., Goodwine, B.: Motion planning for nonlinear symmetric distributed robotic formations. Int. J. Robot. Res. 26(10), 1025 (2007)CrossRefGoogle Scholar
  20. 20.
    Goodwine, B., Antsaklis, P.J.: Multi-agent compositional stability exploiting system symmetries. Automatica 49, 3158–3166 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 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)Google Scholar
  22. 22.
    Karsai, G., Sztipanovits, J.: Model-integrated development of cyber-physical systems. In: Software Technologies for Embedded and Ubiquitous Systems, pp. 46–54 (2008)Google Scholar
  23. 23.
    Lee, E.A.: CPS foundations. In: Design Automation Conference (DAC), 2010 47th ACM/IEEE, pp. 737–742. IEEE (2010)Google Scholar
  24. 24.
    Derler, P., Lee, E.A., Vincentelli, A.S.: Modeling cyber-physical systems. Proc. IEEE 100(1), 13 (2012)CrossRefGoogle Scholar
  25. 25.
    Reynders, E., Houbrechts, J., De Roeck, G.: Fully automated (operational) modal analysis. Mech. Syst. Signal Process. 29, 228 (2012)CrossRefGoogle Scholar
  26. 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)CrossRefGoogle Scholar
  27. 27.
    Chatzis, M.N., Chatzi, E.N.: A discontinuous unscented Kalman filter for non-smooth dynamic problems. Front. Built Environ. 3, 56 (2017)CrossRefGoogle Scholar
  28. 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)Google Scholar
  29. 29.
    Peeters, B., De Roeck, G.: Reference-based stochastic subspace identification for output-only modal analysis. Mech. Syst. Signal Process. 13(6), 855 (1999)CrossRefGoogle Scholar
  30. 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)CrossRefzbMATHGoogle Scholar
  31. 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)CrossRefGoogle Scholar
  32. 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)Google Scholar
  33. 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)Google Scholar
  34. 34.
    Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974)zbMATHGoogle Scholar
  35. 35.
    Baleanu, D., Machado, J.A.T., Luo, A.C.J. (eds.): Fractional Dynamics and Control. Springer, Berlin (2011)zbMATHGoogle Scholar
  36. 36.
    Ortigueira, M.: An introduction to the fractional continuous-time linear systems: the 21st century systems. IEEE Circuits Syst. Mag. 8(3), 19 (2008)CrossRefGoogle Scholar
  37. 37.
    Tenreiro Machado, J., Kiryakova, V., Mainardi, F.: Recent history of fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 16(3), 1140 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Silva, M.F., Tenreiro Machado, J., Lopes, A.: Fractional order control of a hexapod robot. Nonlinear Dyn. 38(1–4), 417 (2004)Google Scholar
  39. 39.
    Delavari, H., Lanusse, P., Sabatier, J.: Fractional order controller design for a flexible link manipulator robot. Asian J. Control 15, 783 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 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)Google Scholar
  42. 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)CrossRefGoogle Scholar
  43. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 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)CrossRefGoogle Scholar
  45. 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)CrossRefGoogle Scholar
  46. 46.
    Ortigueira, M.D., Machado, J.T.: What is a fractional derivative? J. Comput. Phys. 293, 4 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Aerospace and Mechanical EngineeringUniv. of Notre DameNotre DameUSA

Personalised recommendations