An integral and MRAC-based approach to the adaptive stabilisation of a class of linear time-delay systems with unknown parameters

Ramírez Jerónimo, Luis Felipe ; Saldivar, Belem  ; Aguilar-Ibañez, Carlos ; Acosta, José Ángel

International Journal of Applied Mathematics and Computer Science, Tome 34 (2024) no. 1, pp. 81-91.

Résumé

The design of a novel strategy based on the model reference adaptive control method for the stabilisation of a second-order linear time-delay system with unknown parameters is presented. The proposed approach is developed under the assumption that only one state of the system is available, and the sign of the control gain is known. First, the integral operator is applied to obtain a new representation of the original system, where the whole state is known. The use of the integral operator decomposes the control problem into two subproblems that are solved by using the model reference adaptive control method and the backstepping procedure. The effectiveness of the proposed approach is illustrated through an academic example and a practical application case regarding a chemical reactor recycle system.

Détail

Keywords: MRAC, time delay system, backstepping procedure, integral operator
Mots-clés : MRAC, układ z opóźnieniem, operator całkowy

Bibliographie

[1] Aguilar-Ibañez, C., García-Canseco, E. and Suárez-Castañón, M.S. (2013). Control adaptable basado en inmersión e invarianza (I&I) de una clase de sistemas lineales de segundo orden con parámetros desconocidos, Congreso Nacional de Control Automático (CNCA), Ensenada, Baja California Sur, Mexico, pp. 56-61.

[2] Aguilar-Ibañez, C., Saldivar, B., Jiménez-Lizarraga, M., García-Canseco, E. and Garrido, R. (2021). Parametric uncertain second-order linear system output-adaptive stabilization: An integral and MRCA based approach, European Journal of Control 57: 76-81.

[3] Ali, A.K. and Mahmoud, M.M. (2022). Improved design of nonlinear control systems with time delay, International Journal of Robotics and Control Systems 2(2): 317-331.

[4] Åström, K.J. and Wittenmark, B. (2008). Adaptive Control, 2nd Edn, Dover Publications, Inc., Mineola.

[5] Bekcheva, M., Mounier, H. and Greco, L. (2017). Control of differentially flat linear delay systems with constraints, IFAC-PapersOnLine 50(1): 13348-13353.

[6] Bresch-Pietri, D., Chauvin, J. and Petit, N. (2012). Adaptive control scheme for uncertain time-delay systems, Automatica 48(8): 1536-1552.

[7] Evesque, S., Annaswamy, A., Niculescu, S. and Dowling, A. (2003). Adaptive control of a class of time-delay systems, Journal of Dynamic Systems, Measurement and Control 125(2): 186-193.

[8] Fang, M. and Park, J. (2013). A multiple integral approach to stability of neutral time-delay systems, Applied Mathematics and Computation 224: 714-718.

[9] Gu, K., Kharitonov, V.L. and Chen, J. (2003). Stability of Time-Delay Systems, Birkhäuser, New York.

[10] Hua, C., Sun, Z. and Chen, Z. (2022). Stabilization for time-delay nonlinear systems with unknown time-varying control coefficients, Journal of the Franklin Institute 359(16): 8895-8909.

[11] Jia, X., Xu, S., Lu, J., Li, Y., Chu, Y. and Zhang, Z. (2018). Adaptive control for uncertain nonlinear time-delay systems in a lower-triangular form, Journal of the Franklin Institute 355(9): 3911-3925.

[12] Mathiyalagan, K. and Sangeetha, G. (2019). Finite-time stabilization of nonlinear time delay systems using LQR based sliding mode control, Journal of the Franklin Institute 356(7): 3948-3964.

[13] Mera, M., Castanos, F. and Poznyak, A. (2014). Quantised and sampled output feedback for nonlinear systems, International Journal of Control 87(12): 2475-2487.

[14] Mirkin, B. and Gutman, P.-O. (2010). Lyapunov-based adaptive output-feedback control of MIMO nonlinear plants with unknown, time-varying state delays, IFAC Proceedings Volumes 43(2): 33-38.

[15] Mirkin, B.M. and Gutman, P.-O. (2005). Output feedback model reference adaptive control for multi-input-multi-output plants with state delay, Systems & Control Letters 54(10): 961-972.

[16] Mirkin, B., Mirkin, E.L. and Gutman, P.-O. (2008). Model reference adaptive control of nonlinear plant with dead time, 2008 47th IEEE Conference on Decision and Control, Cancún, Quintana Roo, Mexico, pp. 1920-1924.

[17] Nguyen, K.D. (2018). A predictor-based model reference adaptive controller for time-delay systems, IEEE Transactions on Automatic Control 63(12): 4375-4382.

[18] Niculescu, S.-I. and Annaswamy, A.M. (2003). An adaptive smith-controller for time-delay systems with relative degree n* ≤ 2, Systems & Control Letters 49(5): 347-358.

[19] Orlov, Y., Belkoura, L., Richard, J.-P. and Dambrine, M. (2003). Adaptive identification of linear time-delay systems, International Journal of Robust and Nonlinear Control: IFAC-Affiliated Journal 13(9): 857-872.

[20] Orlov, Y., Kolmanovsky, I. and Gomez, O. (2009). Adaptive identification of linear time-delay systems: From theory toward application to engine transient fuel identification, International Journal of Adaptive Control and Signal Processing 23(2): 150-165.

[21] Ortega, R., Hsu, L. and Astolfi, A. (2003). Immersion and invariance adaptive control of linear multivariable systems, Systems & Control Letters 49(1): 37-47.

[22] Phoojaruenchanachai, S., Uahchinkul, K. and Prempraneerach, Y. (1998). Robust stabilisation of a state delayed system, IEEE Proceedings-Control Theory and Applications 145(1): 87-91.

[23] Yao, D., Li, J. and Wu, J. (2019). Adaptive control design with assigned tracking accuracy for a class of nonlinearly parameterized input-delayed systems, Mathematical Problems in Engineering 2019, Article ID: 8602719.

[24] Yuan, S., Zhang, L. and Baldi, S. (2019). Adaptive stabilization of impulsive switched linear time-delay systems: A piecewise dynamic gain approach, Automatica 103: 322-329.

[25] Yuan, S., Zhang, L., Zhang, F., Wan, Y. and Baldi, S. (2018). Robust adaptive stabilization of switched higher-order planar nonlinear systems with unknown time-varying delays, IEEE International Conference on Systems, Man, and Cybernetics (SMC), Miyazaki, Japan, pp. 3249-3254.