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Robust adaptive stabilization of linear time-invariant dynamic systems by using fractional-order holds and multirate sampling controls. (English) Zbl 1190.93092
Summary: This paper presents a strategy for designing a robust discrete-time adaptive controller for stabilizing Linear Time-Invariant (LTI) continuous-time dynamic systems. Such systems may be unstable and noninversely stable in the worst case. A reduced-order model is considered to design the adaptive controller. The control design is based on the discretization of the system with the use of a multirate sampling device with fast-sampled control signal. A suitable on-line adaptation of the multirate gains guarantees the stability of the inverse of the discretized estimated model, which is used to parameterize the adaptive controller. A dead zone is included in the parameters estimation algorithm for robustness purposes under the presence of unmodeled dynamics in the controlled dynamic system. The adaptive controller guarantees the boundedness of the system measured signal for all time. Some examples illustrate the efficacy of this control strategy.
MSC:
93D21 Adaptive or robust stabilization
93C55 Discrete-time control/observation systems
93C05 Linear systems in control theory
93B11 System structure simplification
93C40 Adaptive control/observation systems
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References:
[1] P. A. Ioannou and J. Sun, Robust Adaptive Control, Prentice-Hall, Englewood Cliffs, NJ, USA, 1996. · Zbl 0839.93002
[2] S. Alonso-Quesada and M. De la Sen, “Robust adaptive control of discrete nominally stabilizable plants,” Applied Mathematics and Computation, vol. 150, no. 2, pp. 555-583, 2004. · Zbl 1041.93029 · doi:10.1016/S0096-3003(03)00291-1
[3] S. Alonso-Quesada and M. De la Sen, “Robust adaptive stabilizer for linear systems with imperfectly known point delays using a multi-estimation model,” Dynamics of Continuous, Discrete & Impulsive Systems. Series B, vol. 15, no. 5, pp. 683-708, 2008. · Zbl 1155.93036
[4] M. De la Sen and A. Ibeas, “On the global asymptotic stability of switched linear time-varying systems with constant point delays,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 231710, 31 pages, 2008. · Zbl 1166.34040 · doi:10.1155/2008/231710 · eudml:130387
[5] M. De la Sen and A. Ibeas, “Stability results of a class of hybrid systems under switched continuous-time and discrete-time control,” Discrete Dynamics in Nature and Society, vol. 2009, Article ID 315713, 28 pages, 2009. · Zbl 1193.34014 · doi:10.1155/2009/315713 · eudml:233183
[6] A. Ibeas, M. De la Sen, and S. Alonso-Quesada, “Stable multi-estimation model for single-input single-output discrete adaptive control systems,” International Journal of Systems Science, vol. 35, no. 8, pp. 479-501, 2004. · Zbl 1064.93025 · doi:10.1080/00207720412331280918
[7] M. De la Sen and S. Alonso-Quesada, “Model matching via multirate sampling with fast sampled input guaranteeing the stability of the plant zeros: extensions to adaptive control,” IET Control Theory & Applications, vol. 1, no. 1, pp. 210-225, 2007. · doi:10.1049/iet-cta:20050133
[8] S. Liang and M. Ishitobi, “Properties of zeros of discretised system using multirate input and hold,” IEE Proceedings: Control Theory and Applications, vol. 151, no. 2, pp. 180-184, 2004. · doi:10.1049/ip-cta:20040038
[9] S. Liang, M. Ishitobi, and Q. Zhu, “Improvement of stability of zeros in discrete-time multivariable systems using fractional-order hold,” International Journal of Control, vol. 76, no. 17, pp. 1699-1711, 2003. · Zbl 1047.93033 · doi:10.1080/00207170310001631945
[10] B. Iri\vcanin and S. Stević, “On some rational difference equations,” Ars Combinatoria, vol. 92, pp. 67-72, 2009. · Zbl 1224.39014
[11] S. Stević, “On a generalized max-type difference equation from automatic control theory,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 3-4, pp. 1841-1849, 2010. · Zbl 1194.39007 · doi:10.1016/j.na.2009.09.025
[12] K. S. Narendra and A. M. Annaswamy, Stable Adaptive Systems, Prentice-Hall, Englewood Cliffs, NJ, USA, 1989. · Zbl 0758.93039
[13] P. A. Ioannou and Datta, “Robust adaptive control: a unified approach,” Proceedings of the IEEE, vol. 79, no. 12, pp. 1736-1768, 1991. · doi:10.1109/5.119551
[14] R. H. Middleton, G. C. Goodwin, D. J. Hill, and D. Q. Mayne, “Design issues in adaptive control,” IEEE Transactions on Automatic Control, vol. 33, no. 1, pp. 50-58, 1988. · Zbl 0637.93040 · doi:10.1109/9.360
[15] S. Alonso-Quesada and M. De la Sen, “Robust adaptive control with multiple estimation models for stabilization of a class of non-inversely stable time-varying plants,” Asian Journal of Control, vol. 6, no. 1, pp. 59-73, 2004.
[16] K. G. Arvanitis, “An algorithm for adaptive pole placement control of linear systems based on generalized sampled-data hold functions,” Journal of the Franklin Institute, vol. 336, no. 3, pp. 503-521, 1999. · Zbl 0964.93063 · doi:10.1016/S0016-0032(98)00044-1
[17] M. J. Błachuta, “On approximate pulse transfer functions,” IEEE Transactions on Automatic Control, vol. 44, no. 11, pp. 2062-2067, 1999. · Zbl 0955.93029 · doi:10.1109/9.802916
[18] L. Chen and L. Chen, “Permanence of a discrete periodic Volterra model with mutual interference,” Discrete Dynamics in Nature and Society, vol. 20089, Article ID 205481, 9 pages, 2009. · Zbl 1178.39004 · doi:10.1155/2009/205481 · eudml:229133
[19] Y.-H. Fan and L.-L. Wang, “Permanence for a discrete model with feedback control and delay,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 945109, 8 pages, 2008. · Zbl 1163.39011 · doi:10.1142/S1793524508000369
[20] C. Wei and L. Chen, “A delayed epidemic model with pulse vaccination,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 746951, 12 pages, 2008. · Zbl 1149.92329 · doi:10.1155/2008/746951 · eudml:129368
[21] H. R. Karimi, M. Zapateiro, and N. Luo, “New delay-dependent stability criteria for uncertain neutral systems with mixed time-varying delays and nonlinear perturbations,” Mathematical Problems in Engineering, vol. 2009, Article ID 759248, 22 pages, 2009. · Zbl 1182.34097 · doi:10.1155/2009/759248 · eudml:45898
[22] G. Tao and P. A. Ioannou, “Model reference adaptive control for plants with unknown relative degree,” IEEE Transactions on Automatic Control, vol. 38, no. 6, pp. 976-982, 1993. · Zbl 0786.93063 · doi:10.1109/9.222314
[23] G. C. Goodwin and K. S. Sin, Adaptive Filtering Prediction and Control, Prentice-Hall, Englewood Cliffs, NJ, USA, 1984. · Zbl 0653.93001
[24] C. A. Desoer and M. Vidyasagar, Feedback Systems: Input-Output Properties, Academic Press, New York, NY, USA, 1975. · Zbl 0327.93009
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