Adaptive Control Algorithms in MIMO Linear Systems with Control Delay


In this paper we present two adaptive state control algorithms for the class of linear multi-input multi-output plants under external disturbances and the presence of control delay. Reference signal and external disturbance are considered as multi-harmonic signals with unknown frequencies, amplitudes, and initial phases. The algorithms are developed using the direct adaptive control method based on the internal model principle and do not require identification of disturbance parameters and/or the reference signal.

This is a preview of subscription content, log in to check access.


  1. 1.

    Johnson, C.D., Accommodation of External Disturbances in Linear Regulator and Servomechanism Problems, IEEE Trans. Autom. Control, 1971, vol. 16, no. 6, pp. 635–644.

    Article  Google Scholar 

  2. 2.

    Francis, D.A. and Wonham, W.M., The Internal Model Principle for Linear Multivariable Regulators, App. Math. Optim., 1975, vol. 2, pp. 170–194.

    MathSciNet  Article  Google Scholar 

  3. 3.

    Davison, E.J., The Robust Control of a Servomechanism Problem for Linear Time-Invariant Multivariable Systems, IEEE Trans. Autom. Control, 1976, vol. 21, pp. 25–34.

    MathSciNet  Article  Google Scholar 

  4. 4.

    Wonham, W.M., Linear Multivariable Control: A Geometric Approach, New York: Springer-Verlag, 1979. Translated under the title Lineinye mnogomernye sistemy upravleniya. Geometricheskii podkhod, Moscow: Nauka, 1980.

    Google Scholar 

  5. 5.

    Drozdov, V.N., Miroshnik, I.V., and Skorubskii, V.I., Sistemy avtomaticheskogo upravleniya s mikroEVM (Automated Control Systems with Microcomputers), Leningrad: Mashinostroenie, 1989.

    Google Scholar 

  6. 6.

    Bodson, M. and Douglas, S.C., Adaptive Algorithms for the Rejection of Sinusoidal Disturbances with Unknown Frequency, Automatica, 1997, vol. 33, no. 12, pp. 2213–2221.

    MathSciNet  Article  Google Scholar 

  7. 7.

    Nikiforov, V.O., Adaptive Servomechanism Controller with an Implicit Reference Model, Int. J. Control, 1997, vol. 68, no. 2, pp. 277–286.

    MathSciNet  Article  Google Scholar 

  8. 8.

    Nikiforov, V.O., Nonlinear Control System with Compensation for External Deterministic Disturbances, Izv. Ross. Akad. Nauk, Teor. Sist. Upravlen., 1997, no. 4, pp. 69–73.

  9. 9.

    Nikiforov, V.O., Adaptive Compensation for External Deterministic Disturbances, Mekhatronika, Avtomatiz., Upravl., 2003, no. 5, pp. 8–12.

  10. 10.

    Nikiforov, V.O., Adaptivnoe i robastnoe upravlenie s kompensatsiei vozmushchenii (Adaptive and Robust Control with Compensation for Disturbances), St. Petersburg: Nauka, 2003.

    Google Scholar 

  11. 11.

    Marino, R. and Tomei, P., Output Regulation for Linear Systems via Adaptive Internal Model, IEEE Trans. Autom. Control, 2003, vol. 48, no. 12, pp. 2199–2202.

    MathSciNet  Article  Google Scholar 

  12. 12.

    Byrnes, C.I. and Isidori, A., Nonlinear Internal Models for Output Regulation, IEEE Trans. Autom. Control, 2004, vol. 49, no. 12, pp. 2244–2247.

    MathSciNet  Article  Google Scholar 

  13. 13.

    Serrani, A., Isidori, A., and Marconi, L., Semi-Global Nonlinear Output Regulation with Adaptive Internal Model, IEEE Trans. Autom. Control, 2001, vol. 46, no. 8, pp. 1178–1194.

    Article  Google Scholar 

  14. 14.

    Nikiforov, V.O., Nonlinear Servocompensation of Unknown External Disturbances, Automatica, 2001, vol. 37, pp. 1647–1653.

    Article  Google Scholar 

  15. 15.

    Gerasimov, D.N., Pashenko, A.V., and Nikiforov, V.O., Improved Adaptive Compensation of Unmatched Multisinusoidal Disturbances in Uncertain Nonlinear Plants, Am. Control Conf. ASC, 2020.

  16. 16.

    Elliot, E. and Goodwin, G.C., Adaptive Implementation of the Internal Model Principle, Proc. 23d IEEE Conf. on Decision and Control, 1984, vol. 23, pp. 1292–1297.

    Google Scholar 

  17. 17.

    Palaniswami, M. and Goodwin, G.C., An Adaptive Implementation of the Internal Model Principle, Proc. 1987 Am. Control Conf., 1987, pp. 600–605.

  18. 18.

    Tao, G., Multivariable Adaptive Control: A Survey, Automatica, 2014, vol. 50. no. 11, pp. 2737–2764.

    MathSciNet  Article  Google Scholar 

  19. 19.

    Wang, L., Isidori, A., Liu, Z., and Su, H., Robust Output Regulation for Invertible Nonlinear MIMO Systems, Automatica, 2017, vol. 82, pp. 278–286.

    MathSciNet  Article  Google Scholar 

  20. 20.

    Pyrkin, A.A, Smyshlyaev, A., Bekiaris-Liberis, N., and Krstic, M., Rejection of Sinusoidal Disturbance of Unknown Frequency for Linear System with Input Delay, Am. Control Conf. Baltimore, 2010, pp. 5688–5693.

  21. 21.

    Pyrkin, A.A., Smyshlyaev, A., Bekiaris-Liberis, N., and Krstic, M., Output Control Algorithm for Unstable Plant with Input Delay and Cancellation of Unknown Biased Harmonic Disturbance, Time Delay System Conf., Prague, Czech Republic, 2010, pp. 39–44.

  22. 22.

    Pyrkin, A.A. and Bobtsov, A.A., Cancelation of Unknown Multiharmonic Disturbance for Nonlinear Plant with Input Delay, Int. J. Adaptive Control Signal Proc., 2012, vol. 26, no. 4, pp. 302–315.

    MathSciNet  Article  Google Scholar 

  23. 23.

    Wang, J., Vedyakov, A.A., Vediakova, A.O., Pyrkin, A.A., Bobtsov, A.A., and Shavetov, S.V., Output Adaptive Controller for a Class of MIMO Systems with Input Delay and Multisinusoidal Disturbance, IFAS-PapersOnLine, 2015, vol. 48, no. 11, pp. 892–899.

    Article  Google Scholar 

  24. 24.

    Pyrkin, A.A., Bobtsov, A.A., Nikiforov, V.O., Vedyakov, A.A., Kolyubin, S.A., and Borisov, O.I., Output Control Approach for Delayed Linear Systems with Adaptive Rejection of Multiharmonic Disturbance, IFAC Proc. Volumes, 2014, vol. 47. no. 3, pp. 12110–12115.

    Article  Google Scholar 

  25. 25.

    Pyrkin, A.A. and Bobtsov, A.A., Adaptive Controller for Linear System with Input Delay and Output Disturbance, IEEE Trans. Autom. Control, 2016, vol. 61, no. 12, pp. 4229–4234.

    MathSciNet  Article  Google Scholar 

  26. 26.

    Pyrkin, A.A., Bobtsov, A.A., Nikiforov, V.O., et al., Compensation of Polyharmonic Disturbance of State and Output of a Linear Plant with Delay in the Control Channel, Autom. Remote Control, 2015, vol. 76, no. 12, pp. 2124–2142.

    MathSciNet  Article  Google Scholar 

  27. 27.

    Narendra, K. and Annaswamy, A., Stable Adaptive Systems, New Jersey: Prentice Hall, 1989.

    Google Scholar 

  28. 28.

    Basturk, H.I. and Krstic, M., Adaptive Sinusoidal Disturbance Cancellation for Unknown LTI Systems Despite Input Delay, Automatica, 2015, vol. 58, 131–138.

    MathSciNet  Article  Google Scholar 

  29. 29.

    Gerasimov, D.N., Nikiforov, V.O., and Paramonov, A.V., Adaptive Disturbance Compensation in Delayed Linear Systems: Internal Model Approach, IEEE Conf. on Control Applications, 2015, pp. 1692–1696.

  30. 30.

    Annaswamy, A., Jang, J., and Lavretsky, E., Stability Margins for Adaptive Controllers in the Presence of Time-Delay, AIAA Guidance, Navigation, and Control Conf., Honolulu, 2008, AIAA 2008–6659.

  31. 31.

    Gerasimov, D.N., Paramonov, A.V., and Nikiforov, V.O., Algorithms of Adaptive Disturbance Compensation in Linear Systems with Arbitrary Input Delay, Int. J. Control, 2018.

  32. 32.

    Paramonov, A.V., Gerasimov, D.N., and Nikiforov, V.O., Fast Adaptive Compensation of Multi-Sinusoidal Disturbance in Linear MIMO Systems with Multiple Input Delays, Eur. Control Conf. ECC, 2018, pp. 2441–2446.

  33. 33.

    Gerasimov, D.N., Miliushin, A.S., and Nikiforov, V.O., Algorithms of Adaptive Tracking of Unknown Multisinusoidal Signals in Linear Systems with Arbitrary Input Delay, Int. J. Adaptive Control Signal Proc., 2019, vol. 33, no. 6, pp. 900–912.

    MathSciNet  Article  Google Scholar 

  34. 34.

    Gerasimov, D.N., Miliushin, A.S., Paramonov, A.V., and Nikiforov, V.O., Algorithms of Adaptive Tracking of Unknown Multi-Sinusoidal Signals in MIMO Linear Systems with Multiple Input Delay, Proc. 2019 Am. Control Conf., 2019, pp. 3014–3019.

  35. 35.

    Nikiforov, V.O., Adaptive Servocompensation of Input Disturbances, IFAC Proc. Volumes, 1996, vol. 29, no. 1, pp. 5114–5119.

    Article  Google Scholar 

  36. 36.

    Nikiforov, V.O., Observers of External Deterministic Disturbances. I. Objects with Known Parameters, Autom. Remote Control, 2004, vol. 65, no. 10, pp. 1531–1541.

    MathSciNet  Article  Google Scholar 

  37. 37.

    Miroshnik, I.V., Nikiforov, V.O., and Fradkov, A.L., Nelineinoe i adaptivnoe upravlenie slozhnymi dinamicheskimi sistemami (Nonlinear and Adaptive Control for Complex Dynamical Systems), St. Petersburg: Nauka, 2000.

    Google Scholar 

  38. 38.

    Ljung, L., System Identification: Theory for the User, Englewood Cliffs: Prentice Hall, 1987. Translated under the title Identifikatsiya sistem. Teoriya dlya pol’zovatelya, Moscow: Nauka, 1991.

    Google Scholar 

  39. 39.

    Monopoli, R.V., Model Reference Adaptive Control with an Augmented Error Signal, IEEE Trans. Autom. Control, 1974, vol. 19, no. 5, pp. 474–484.

    Article  Google Scholar 

  40. 40.

    Nikiforov, V.O. and Fradkov, A.L., Adaptive Control Schemes with Extended Error, Autom. Remote Control, 1994, vol. 55, no. 9, pp. 1239–1255.

    MathSciNet  MATH  Google Scholar 

  41. 41.

    Morse, A.S., Global Stability of Parameter Adaptive Control Systems, IEEE Trans. Autom. Control, 1980, vol. 25, no. 3, pp. 433–439.

    MathSciNet  Article  Google Scholar 

  42. 42.

    Artstein, Z., Linear Systems with Delayed Controls: a Reduction, IEEE Trans. Autom. Control, 1982, vol. 27, no. 4, pp. 869–879.

    MathSciNet  Article  Google Scholar 

  43. 43.

    Engelborghs, K., Dambrine, M., and Roose, D., Limitations of a Class of Stabilization Methods for Delay Systems, IEEE Trans. Autom. Control, 2001, vol. 46, no. 2, pp. 336–339.

    MathSciNet  Article  Google Scholar 

  44. 44.

    Mondie, S. and Michiels, W., Finite Spectrum Assignment of Unstable Time-Delay Systems with a Safe Implementation, IEEE Trans. Autom. Control, 2003, vol. 48, no. 12, pp. 2207–2212.

    MathSciNet  Article  Google Scholar 

  45. 45.

    Zhong, Q.C., On Distributed Delay in Linear Control Laws—Part I: Discrete-Delay Implementations, IEEE Trans. Autom. Control, 2004, vol. 49, no. 11, pp. 2074–2080.

    Article  Google Scholar 

Download references


This work was supported by the Government of Russian Federation, grant no. 08-08.

Author information



Corresponding authors

Correspondence to V. O. Nikiforov or A. V. Paramonov or D. N. Gerasimov.

Additional information

This paper was recommended for publication by A.L. Fradkov, a member of the Editorial Board

Russian Text © The Author(s), 2020, published in Avtomatika i Telemekhanika, 2020, No. 6, pp. 153–172.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Nikiforov, V.O., Paramonov, A.V. & Gerasimov, D.N. Adaptive Control Algorithms in MIMO Linear Systems with Control Delay. Autom Remote Control 81, 1091–1106 (2020).

Download citation


  • adaptive tracking
  • MIMO system
  • disturbance compensation
  • control delay
  • internal model