**AUTHORS:**Chiou-Jye Huang, Kai-Hung Lu, Hsin-Chuan Chen

**Download as PDF**

**ABSTRACT:**
This paper presents a novel feedback linearization control of nonlinear systems with uncertainties for
the tracking and almost disturbance decoupling and develops an Acquired Immunity Deficiency Syndrome
control strategy. The main contribution of this study is to construct a controller, under appropriate conditions,
such that the resulting closed-loop system is valid for any initial condition and bounded tracking signal with the
following characteristics: input-to-state stability with respect to disturbance inputs and almost disturbance
decoupling. In order to demonstrate the applicability, this paper develops the feedback linearization design for
the control of a mathematical HIV/AIDS model system to improve the viral load. The performances of drug
treatment based on our proposed novel nonlinear geometric feedback control approach are better than some
existing approaches, i.e., the healthy CD+ T cell population can be kept in original cells per cubic millimeter and
the viral load is reduced only after more short days of drug treatment

**KEYWORDS:**
Almost Disturbance Decoupling; Feedback Linearization Approach; Acquired Immunity
Deficiency Syndrome; Human Immunodeficiency Virus

**REFERENCES:**

[1] A. Alleyne, A systematic approach to the control of electrohydraulic servosystems, American Control Conference, Philadelphia. Pennsylvania, 1998, pp. 833-837.

[2] J.A. Ball, J.W. Helton M.L. Walker, H∞ control for nonlinear systems with output feedback, IEEE Trans. Automat. Contr., Vol.38, April, 1993, pp. 546-559.

[3] N.S. Bedrossian, Approximate feedback linearization: the car-pole example, IEEE International Conference Robotics and Automation, France, 1992, pp. 1987-1992.

[4] Z. Bentwich, CD4 measurements in patients with HIV: Are they feasible for poor settings?, PLoS Medicine, Vol.2, No.7, 2005.

[5] F.L. Biafore, C.E. D’Attellis, Exact linearisation and control of a HIV-1 predator-prey model, Proceedings of the 2005 IEEE Engineering Medicine and Biology 27th Annual Conference, Shanghai, China, 2005. pp. 2367-2370.

[6] M.E. Brandt, G. Chen, Feedback control of a biomedical model of HIV-1, IEEE Trans. Biomedical Eng., Vol.48, pp. 754-759, 2001.

[7] F.C. Chen, Back-propagation neural networks for nonlinear self-tuning adaptive control, IEEE Control Systems Magazine, Vol.10, No.3, 1990, pp. 44-48.

[8] F.C. Chen, H.K. Khalil, Adaptive control of a class of nonlinear discrete-time systems using neural networks, IEEE Trans. Automat. Contr., Vol.40, No.5, 1995, pp. 791-801.

[9] B.S. Chen, C.H. Lee, Y.C. Chang, H∞ tracking design of uncertain nonlinear SISO systems: Adaptive fuzzy approach, IEEE Trans. Fuzzy System, Vol. 4, No.1, 1996, pp. 32-43.

[10] M.J. Corless, G. Leitmann, Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems, IEEE Trans. Automat. Contr., Vol.26, No.5, 1981, pp. 1139-1144.

[11] A. Floares, Feedback linearization using neural networks applied to advanced pharmacodynamic and pharmacogenomic systems, IEEE International Joint Conference on Neural Networks, Vol.1, 2005, pp. 173-178.

[12] A.G. Floares, Adaptive neural networks control of drug dosage regimens in cancer chemotherapy, Proceedings of the International Joint Conference on Neural Networks, Canada, 2006, pp. 3820-3827.

[13] A. Floares, C. Floares, M. Cucu et,al, Adaptive neural networks control of drug dosage regimens in cancer chemotherapy, Proceedings of the IJCNN, Porland OR, 2003, pp. 154-159.

[14] F. Garces, V.M. Becerra et,al, Strategies for feedback linearization: a dynamic neural network approach, London: Springer, 2003.

[15] S.S. Ge, Z. Tian and T.H. Lee, Nonlinear control of a dynamic model of HIV-1, IEEE Trans. Biomedical Eng., Vol.52, 2005, pp. 353-361.

[16] S. Gopalswamy and J.K. Hedrick, Tracking nonlinear nonminimum phase systems using sliding control, Int. J. Contr., Vol.57, 1993, pp. 1141-1158.

[17] P. Guo, Nonlinear predictive functional control based on hopfield network and its application in CSTR, in Proc. of the International Conference on Machine Learning and Cybernetics, Dalian, 2006, pp. 3036-3039.

[18] A.T. Haase, K. Henry, M. Zupancic, et,al, Quantitative image analysis of HIV-1 infection in lymphoid tissue, Science, Vol.274, 1996, pp. 985-989.

[19] M.A. Henson, D.E. Seborg, Critique of exact linearization strategies for process control, Journal Process Control, Vol.1, 1991, pp. 122- 139.

[20] D.D. Ho, A.U. Neumann, et,al, Rapid turnover of plasma virions and CD4 lymphocytes in HIV1 infection, Nature, Vol.373, 1995, pp. 123-126.

[21] J. Huang, W.J. Rugh, On a nonlinear multivariable servomechanism problem, Automatica, vol. 26, pp. 963-992, June, 1990.

[22] A. Isidori, Nonlinear control system, New York: Springer Verlag, 1995.

[23] A. Isidori, H ∞ control via measurement feedback for affine nonlinear systems, International Journal of Robust Nonlinear Control, Vol.4, No.4, 1994, pp. 553-574.

[24] A. Isidori, C.I. Byrnes, Output regulation of nonlinear systems, IEEE Trans. Automat. Contr., Vol.35, 1990, pp. 131-140.

[25] A. Isidori, W. Kang, H ∞ control via measurement feedback for general nonlinear systems, IEEE Trans. Automat. Contr., Vol.40, 1995, pp. 466-472.

[26] S.J. Joo and J.H. Seo, Design and analysis of the nonlinear feedback linearizing control for an electromagnetic suspension system, IEEE Trans. Automat. Contr., Vol.5, No.1, 1997, pp. 135-144.

[27] H.R. Joshi, Optimal control of an HIV immunology model, Optimal Control Applications and Methods, Vol.23, 2002, pp. 199-213.

[28] J. Karrakchou, M. Rachik, S. Gourari, Optimal control and infectiology: Application to an HIV/AIDS model, Applied Mathematics and Computation, Vol.177, 2006.

[29] H.K. Khalil, Nonlinear systems, New Jersey: Prentice-Hall, 1996.

[30] K. Khorasani, P.V. Kokotovic, A corrective feedback design for nonlinear systems with fast actuators, IEEE Trans. Automat. Contr., Vol.31, 1986, pp. 67-69.

[31] D. Kirschner, Using mathematics to understand HIV immune dynamics, Notices Amer. Math. Soc., 1996, pp. 191-202.

[32] S.Y. Lee, J.I. Lee, I.J. Ha, A new approach to nonlinear autopilot design for bank-to-turn missiles, in Proc. of the 36th Conference on Decision and Control, San Diego. California, 1997, pp. 4192-4197.

[33] C.M. Lin, Y.F. Peng, Missile guidance law design using adaptive cerebellar model articulation controller, IEEE Trans. Neural Networks, Vol.16, No.3, 2005, pp. 636-644.

[34] F.J. Lin, H.J. Shieh, L.T. Teng and P.H. Shieh, Hybrid controller with recurrent neural network for magnetic levitation system, IEEE Trans. Magnetics., Vol.41, No.7, 2005, pp. 2260-2269.

[35] R. Marino, P.V. Kokotovic, A geometric approach to nonlinear singularly perturbed systems, Automatica, Vol.24, 1988, pp. 31-41.

[36] R. Marino, P. Tomei, Nonlinear output feedback tracking with almost disturbance decoupling, IEEE Trans. Automat. Contr., Vol.44, No.1, 1999, pp. 18-28.

[37] R.K. Miller, A.N. Michel, Ordinary differential equations, New York: Academic Press, 1982.

[38] H. Nijmeijer, A.J. Van Der Schaft, Nonlinear dynamical control systems, New York: Springer Verlag, 1990.

[39] A.S. Perelson, P. Essunger, Y. Cao, et,al, Decay characteristics of HIV-1-infected compartments during combination therapy, Nature, Vol.387, 1997, pp. 188-191.

[40] A.S. Perelson, W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Review, Vol.41, No. 1, 1999, pp. 3-44.

[41] H. Peroz, B. Ogunnaike, S. Devasia, Output tracking between operating points for nonlinear processes: Van de Vusse example, IEEE Trans. Control Systems Technology, Vol.10, No.4, 2002, pp. 611-617.

[42] C. Qian, W. Lin, Almost disturbance decoupling for a class of high-order nonlinear systems, IEEE Trans. Automat. Contr., Vol.45, No.6, 2000, pp. 1208-1214.

[43] N. Sachsenberg, A.S. Perelson, S. Yerly, et,al, Turnover of CD4+ and CD8+ T lymphocytes in HIV-1 infection as measured by ki-67 antigen, J. Exp. Med., Vol.187, 1998, pp. 1295-1303.

[44] J.J. Sheen, R.H. Bishop, Adaptive nonlinear control of spacecraft, American Control Conference, Baltlmore. Maryland, 1998, pp. 2867-2871.

[45] J.J.E. Slotine , W. Li, Applied nonlinear control, New York: Prentice-Hall, 1991.

[46] D. Swaroop, J.K. Hedrick, P.P. Yip, et,al, Dynamic surface control for a class of nonlinear systems, IEEE Trans. Automat. Contr., Vol.45, No.10, 2000, pp. 1893-1899.

[47] A.J. Van der Schaft, L2-gain analysis of nonlinear systems and nonlinear state feedback H ∞control, IEEE Trans. Automat. Contr., Vol.37, 1992, pp. 770-784.

[48] S. Weiland, J. C. Willems. Almost disturbance decoupling with internal stability, IEEE Trans. Automat. Contr., Vol.34, No.3, 1989, pp. 277- 286.

[49] J.C. Willems, Almost invariant subspace: An approach to high gain feedback design –Part I: Almost controlled invariant subspaces, IEEE Trans. Automat. Contr., Vol.AC-26, No.1, 1981, pp. 235-252.

[50] P.P. Yip and J.K. Hedrick, Adaptive dynamic surface control: a simplified algorithm for adaptive backstepping control of nonlinear systems, International Journal of Control, Vol.71, No.5, 1998, pp. 959-979.