This study presents an off-the-shelf error feedback controller that can completely rejects the sinusoidal disturbances whose magnitudes, phases, bias, frequencies and even the number of frequencies are all unknown. A closed-form solution is given under the assumptions that the plant has no uncertainty and has hyperbolic zero dynamics (i.e. there is no zero on the imaginary axis of the complex plane), and that the number of unknown frequencies has upper bound which is known. A non-minimum phase plant can be dealt with. The proposed controller is tested by a simulation, which verifies its effectiveness.

References

1. 1)
  - 18. Fedele, G., Ferrise, A.: ‘Biased sinusoidal disturbance compensation with unknown frequency’, IEEE Trans. Autom. Control, 2013, 58, (12), pp. 3207–3212 (doi: 10.1109/TAC.2013.2261656).
2. 2)
  - 26. Chen, C.T.: ‘Linear system theory and design’ (Oxford University Press, 1984, 2nd edn.).
3. 3)
  - 22. Marino, R., Tomei, P.: ‘An adaptive learning regulator for uncertain minimum phase systems with under modeled unknown exosystems’, Automatica, 2011, 47, (4), pp. 739–747 (doi: 10.1016/j.automatica.2011.01.019).
4. 4)
  - 13. Ding, Z., Liu, S.T.: ‘Asymptotic rejection of unknown sinusoidal disturbances in non-linear dynamic systems with delay’, IEE Proc. Control Theory Appl., 2010, 4, (6), pp. 1039–1046 (doi: 10.1049/iet-cta.2008.0264).
5. 5)
  - 1. Landau, I.D., Airimitoaie, T.B., Silva, A.C., Buche, G., Noë, M.: ‘Benchmark on adaptive regulation-rejection of unknown/time-varying multiple narrow band disturbances’, Eur. J. Control, 2013, 19, (4), pp. 237–252 (doi: 10.1016/j.ejcon.2013.05.007).
6. 6)
  - 9. Marino, R., Tomei, P.: ‘Output regulation for linear systems via adaptive internal model’, IEEE Trans. Autom. Control, 2003, 48, (12), pp. 2199–2202 (doi: 10.1109/TAC.2003.820143).
7. 7)
  - 16. Nikiforov, V.O.: ‘Adaptive non-linear tracking with complete compensation of unknown disturbances’, Eur. J. Control, 1998, 4, (2), pp. 132–139 (doi: 10.1016/S0947-3580(98)70107-4).
8. 8)
  - 29. Khalil, H.: ‘Nonlinear systems’ (Prentice-Hall, 2001, 3rd edn.).
9. 9)
  - M. Bodson , S.C. Douglas . Adaptive algorithms for the rejection of sinusoidal disturbances with unknown frequencies. Automatica , 10 , 2213 - 2221
10. 10)
  - B.D.O. Anderson . Exponential stability of linear equations arising in adaptive identification. IEEE Trans. Autom. Control , 1 , 83 - 88
11. 11)
  - 4. Kim, W., Kim, H., Chung, C.C., Tomizuka, M.: ‘Adaptive output regulation for the rejection of a periodic disturbance with an unknown frequency’, IEEE Trans. Control Syst. Tech., 2011, 19, (5), pp. 1296–1304 (doi: 10.1109/TCST.2010.2066276).
12. 12)
  - Z. Ding . Adaptive estimation and rejection of unknown sinusoidal disturbances in a class of non-minimum-phase nonlinear systems. IEE Proc.-Control Theory Appl. , 4 , 379 - 386
13. 13)
  - 27. Apostol, T.M.: ‘Resultants of cyclotomic polynomials’, Proc. Am. Math. Soc., 1970, 24, (3), pp. 457–462 (doi: 10.1090/S0002-9939-1970-0251010-X).
14. 14)
  - 7. Knobloch, H.W., Isidori, A., Flockerzi, D.: ‘Topics in control theory’ (Birkhauser, 1993).
15. 15)
  - 5. Kim, H., Shim, H., Jo, N.H.: ‘Adaptive add-on output regulator for rejection of sinusoidal disturbances and application to optical disc drives’, IEEE Trans. Ind. Electron., 2014, 61, (10), pp. 5490–5499 (doi: 10.1109/TIE.2014.2304920).
16. 16)
  - 20. Basturk, H.I., Krstic, M.: ‘State-derivative feedback for adaptive cancelation of unmatched disturbances in unknown strict-feedback lti systems’, Automatica, 2014, 50, (10), pp. 2539–2545 (doi: 10.1016/j.automatica.2014.08.002).
17. 17)
  - 19. Basturk, H.I., Krstic, M.: ‘Adaptive cancellation of matched unknown sinusoidal disturbances for LTI systems by state derivative feedback’, J. Dyn. Sys., Meas., Control, 2013, 135, (1), pp. 14501(1)–14501(6).
18. 18)
  - 3. Kim, H., Kim, W.H., Chung, C.C., Shim, H.: ‘Track following controller for optical disk drives based on adaptive output regulation’. Proc. 16th IFAC World Congress, 2005.
19. 19)
  - Q. Zhang . Adaptive observer for multiple-input-multiple-output (MIMO) linear time varying systems. IEEE Trans. Autom. Control , 3 , 525 - 529
20. 20)
  - 8. Huang, J.: ‘Nonlinear output regulation: theory and applications’ (SIAM, 2004).
21. 21)
  - 23. Marino, R., Santosuosso, G.: ‘Regulation of linear systems with unknown exosystems of uncertain order’, IEEE Trans. Autom. Control, 2007, 52, (2), pp. 352–359 (doi: 10.1109/TAC.2006.890376).
22. 22)
  - 28. Ioannou, P.A., Sun, J.: ‘Robust adaptive control’, (Prentice-Hall, 1996).
23. 23)
  - L.J. Brown , Q. Zhang . Periodic disturbance cancellation with uncertain frequency. Automatica , 631 - 637
24. 24)
  - 2. Serrani, A., Isidori, A., Marconi, L.: ‘Semiglobal nonlinear output regulation with adaptive internal model’, IEEE Trans. Autom. Control, 2001, 46, (8), pp. 1178–1194 (doi: 10.1109/9.940923).
25. 25)
  - B.A. Francis . The linear multivariable regulator problem. SIAM J. Control Opt. , 3 , 486 - 505
26. 26)
  - 24. Marino, R., Tomei, P.: ‘Disturbance cancellation for linear systems by adaptive internal models’, Automatica, 2013, 49, (5), pp. 1494–1500 (doi: 10.1016/j.automatica.2013.02.011).
27. 27)
  - 7. Nikiforov, V.O.: ‘Nonlinear servocompensation of unknown external disturbances’, Automatica, 2001, 37, pp. 1647–1653 (doi: 10.1016/S0005-1098(01)00117-0).
28. 28)
  - M. Bodson , J.S. Jensen , S.C. Douglas . Active noise control for periodic disturbances. IEEE Trans. Control Syst. Technol. , 1 , 200 - 205
29. 29)
  - Z. Ding . Asymptotic rejection of unknown sinusoidal disturbances in nonlinear systems. Automatica , 1 , 174 - 177
30. 30)
  - 21. Marino, R., Tomei, P.: ‘Output regulation for linear minimum phase systems with unknown order exosystem’, IEEE Trans. Autom. Control, 2007, 52, (10), pp. 2000–2005 (doi: 10.1109/TAC.2007.904288).

Linear systems with hyperbolic zero dynamics admit output regulator rejecting unknown number of unknown sinusoids

References

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