Mustafa Doğan, Janset Daşdemir, Berk Altıner

Adaptive output tracking of distributed parameter systems

In this paper, we consider the unknown trajectory tracking problem for stable distributed parameter systems. The main assumptions are that trajectory signals are generated by an unknown finite-dimensional exosystem that is a marginally stable system and the tracking error is available for measurement. In order to achieve perfect error regulation, a frequency estimator scheme is proposed to estimate unknown exosystem parameters, and the control law that is designed based on geometric output regulation theory is revisited. The success of the proposed method is demonstrated on a parabolic heat equation and a first-order hyperbolic partial differential equation.

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[1] Morris K, Levine W. Control of systems governed by partial differential equations. In: Levine W (editor). The control theory handbook 2nd ed. Boca Raton, Florida USA: CRC Press, 2010.
[2] Balas M. Feedback control of flexible systems. IEEE Transactions on Automatic Control 1978; 23 (4): 673-679. doi: 10.1109/TAC.1978.1101798
[3] Meurer T. Control of higher–dimensional PDEs: Flatness and backstepping designs. Berlin, Germany: Springer Science & Business Media, 2012.
[4] Francis BA, Wonham WM. The internal model principle for linear multivariable regulators. Applied mathematics and optimization 1975; 2 (2): 170-194. doi: 10.1007/BF01447855
[5] Isidori A, Byrnes CI. Output regulation of nonlinear systems. IEEE transactions on Automatic Control 1990; 35 (2): 131-140. doi: 10.1109/9.45168
[6] Pohjolainen S. Robust multivariable PI-controller for infinite dimensional systems. IEEE Transactions on Automatic Control 1982; 27 (1): 17-30. doi: 10.1109/TAC.1982.1102887
[7] Kobayashi T. Regulator problem for infinite-dimensional systems. Systems & Control Letters 1983; 3 (1): 31-39. doi: 10.1016/0167-6911(83)90035-X
[8] Byrnes CI, Laukó IG, Gilliam DS, Shubov VI. Output regulation for linear distributed parameter systems. IEEE Transactions on Automatic Control 2000; 45 (12): 2236-2252. doi: 10.1109/9.895561
[9] Natarajan V, Gilliam DS, Weiss G. The state feedback regulator problem for regular linear systems. IEEE Transactions on Automatic Control 2014; 59 (10): 2708-2723. doi: 10.1109/TAC.2014.2326491
[10] Deutscher J. Output regulation for linear distributed-parameter systems using finite-dimensional dual observers. Automatica 2011; 47 (11): 2468-2473. doi: 10.1016/j.automatica.2011.08.033
[11] Deutscher J. A backstepping approach to the output regulation of boundary controlled parabolic PDEs. Automatica 2015; 57: 56-64. doi: 10.1016/j.automatica.2015.04.008
[12] Deutscher J. Backstepping Design of Robust State Feedback Regulators for Linear 2×2 Hyperbolic Systems. IEEE Transactions on Automatic Control 2016; 62 (10): 5240-5247. doi: 10.1109/TAC.2016.2632423
[13] Paunonen L, Pohjolainen S. Robust controller design for infinite‐dimensional exosystems. International Journal of Robust and Nonlinear Control 2014; 24 (5): 825-858. doi: 10.1002/rnc.2920
[14] Hämäläinen T, Pohjolainen S. Robust regulation of distributed parameter systems with infinite-dimensional exosystems. SIAM Journal on Control and Optimization 2010; 48 (8): 4846-4873. doi: 10.1137/090757976
[15] Xu X, Dubljevic S. Output regulation problem for a class of regular hyperbolic systems. International Journal of Control 2016; 89 (1): 113-127. doi: 10.1080/00207179.2015.1060363
[16] Xu X, Dubljevic S. Output and error feedback regulator designs for linear infinite-dimensional systems. Automatica 2017; 83: 170-178. doi: 10.1016/j.automatica.2017.06.003
[17] Paunonen L, Phan D. Reduced order controller design for robust output regulation. IEEE Transactions on Automatic Control 2019; 65 (6): 2480-2493. doi: 10.1109/TAC.2019.2930185
[18] Guo W, Jin FF. Adaptive error feedback regulator design for 1 D heat equation. Automatica 2020; 113: 108810. doi: 10.1016/j.automatica.2020.108810
[19] Guo BZ, Meng T. Robust output feedback Control for an Euler-Bernoulli Beam Equation. In: IEEE 2020 American Control Conference (ACC); Denver, CO, USA; 2020. pp. 800-805.
[20] Wang X, Ji H, Sheng J. Output regulation problem for a class of SISO infinite dimensional systems via a finite dimensional dynamic control. Journal of Systems Science and Complexity, 2014; 27 (6): 1172-1191. doi: 10.1007/s11424- 014-2029-9
[21] Afshar S, Paunonen L. Adaptive robust output regulation control design. In: IEEE 2019 American Control Conference (ACC); Philadelphia, PA, USA; 2019. pp. 3038-3043.
[22] Curtain RF, Zwart H. An introduction to infinite-dimensional linear systems theory. New York, USA: Springer Science & Business Media, 1995.
[23] Luo Z H, Guo BZ, Morgül Ö. Stability and stabilization of infinite dimensional systems with applications. London, UK: Springer Science & Business Media, 2012.
[24] Aulisa E, Gilliam D. A practical guide to geometric regulation for distributed parameter systems. Boca Raton, Florida, USA: CRC Press, 2015.
[25] Obregon-Pulido G, Castillo-Toledo B, Loukianov A. A globally convergent estimator for n-frequencies. IEEE Transactions on Automatic Control 2002; 47 (5): 857-863. doi: 10.1109/TAC.2002.1000286
[26] Kailath T. Linear systems. Englewood Cliffs, NJ, USA: Prentice-Hall, 1980.
[27] Aksikas I, Fuxman A, Forbes JF, Winkin JJ. LQ control design of a class of hyperbolic PDE systems: Application to fixed-bed reactor. Automatica 2009; 45 (6): 1542-1548. doi: 10.1016/j.automatica.2009.02.017
[28] Xu X, Huang B, Dubljevic S. State feedback output regulation for a class of hyperbolic PDE systems. In: IEEE 2015 European Control Conference (ECC); Linz, Austria; 2015. pp. 521-526.