**Commenced**in January 2007

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**Edition:**International

**Paper Count:**30576

##### An LMI Approach of Robust H∞ Fuzzy State-Feedback Controller Design for HIV/AIDS Infection System with Dual Drug Dosages

**Authors:**
Wudhichai Assawinchaichote

**Abstract:**

This paper examines the problem of designing robust H controllers for for HIV/AIDS infection system with dual drug dosages described by a Takagi-Sugeno (S) fuzzy model. Based on a linear matrix inequality (LMI) approach, we develop an H controller which guarantees the L2-gain of the mapping from the exogenous input noise to the regulated output to be less than some prescribed value for the system. A sufficient condition of the controller for this system is given in term of Linear Matrix Inequalities (LMIs). The effectiveness of the proposed controller design methodology is finally demonstrated through simulation results. It has been shown that the anti-HIV vaccines are critically important in reducing the infected cells.

**Keywords:**
linear matrix inequalities (LMIs),
H∞ Fuzzy control,
Takagi-Sugeno (TS) fuzzy model,
HIV/AIDS infection system

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1061216

**References:**

[1] Z.X. Han, and G. Feng, "State-feedback H Controller Design of Fuzzy Dynamic System Using LMI Techniques,- Fuzzy-IEEE-98, pp. 538-544, 1998.

[2] B.S. Chen, C.S. Tseng, and Y.Y. He, "Mixed H2/H Fuzzy Output Feedback Control Design for Nonlinear Dynamic Systems: An LMI Approach,- IEEE Trans. Fuzzy Syst., vol. 8, pp. 249-265, 2000.

[3] K. Tanaka, T. Ikeda, and H.O. Wang, "Robust Stabilization of a Class of Uncertain Nonlinear Systems via Fuzzy Control: Quadratic Stability, H Control Theory, and Linear Martix Inequality,- IEEE Trans. Fuzzy. Syst., vol. 4, pp. 1-13, 1996.

[4] Z.X. Han, G. Feng, B.L. Walcott, and Y.M. Zhang, "H Controller Design of Fuzzy Dynamic Systems with Pole Placement Constraints,- Proc. American Control Conf., pp. 1939-1943, 2000.

[5] T. Takagi, and M. Sugeno, "Fuzzy Identification of Systems and Its Applications to Model and Control,- IEEE Trans. Syst., Man, Cybern., vol. SMC-15, no. 1, pp. 116-132, 1985.

[6] L.X. Wang, "Design and Analysis of Fuzzy Identifiers of Nonlinear Dynamic Systems,- IEEE Trans. Automat. Control, vol. 40, pp. 11-23,1995.

[7] M. Sugeno, and G.T. Kang, "Structure Identification of Fuzzy Model,- Fuzzy Sets Syst., vol. 28, pp. 15-33, 1988.

[8] H.O. Wang, K. Tanaka, and M.F. Griffin, "An Approach to Fuzzy Control of Nonlinear Systems: Stability and Design Issues,- IEEE Trans. Fuzzy Syst., vol. 4, no. 1, pp. 14-23, 1996.

[9] J. Joh, Y.H. Chen, and R. Langari, "On the Stability Issues of Linear Takagi-Sugeno Fuzzy Models,- IEEE Trans. Fuzzy Syst., vol. 6, pp. 402-410, 1998.

[10] S.G. Cao, N.W. Ree, and G. Feng, "Quadratic Stabilities Analysis and Design of Continuous-Time Fuzzy Control Systems,- Int. J. Syst. Sci., vol. 27, pp. 193-203, 1996.

[11] K. Tanaka, T. Ikeda, and H.O. Wang, "Robust Stabilization of A Class of Uncertain Nonlinear Systems Via Fuzzy Control: Quadratic Stabilizability, H Control Theory, and Linear Matrix Inequalities,- IEEE Trans. Fuzzy Syst., vol. 4, 1996.

[12] K. Tanaka, T. Taniguchi, and H.O. Wang, "Fuzzy Control based on Quadratic Performance Function - A Linear Matrix Inequality Approach,- Proc. IEEE Conf. Decision and Control, pp. 2914-2919, 1998.

[13] K. Tanaka, and H.O. Wang, "Fuzzy Regulators and Fuzzy Observers: A Linear Matrix Inequality Approach,- Proc. IEEE Conf. Decision and Control, pp. 1315-1320, 1997.

[14] S.K. Nguang, and P. Shi, "H Fuzzy Output Feedback Control Design for Nonlinear Systems: An LMI Approach,- Proc. IEEE Conf. Decision and Control, pp. 4352-4357, 2001.

[15] W. Assawinchaichote, and S.K. Nguang, "Fuzzy Control Design for Singularly Perturbed Nonlinear Systems: An LMI Approach,- ICAIET, Malaysia, 2002.

[16] W. Assawinchaichote, S.K. Nguang, P. Shi and E.K. Boukas, "H Fuzzy State-Feedback Control Design for Nonlinear Systems with Stability Constraints: An LMI Approach,- Int. J. Mathematics and Computers in Simulation, vol. 78, pp. 514-531, 2008.

[17] W. Assawinchaichote, "A New Approach to Non-Fragile H Fuzzy Filter of Uncertain Markovian Jump Nonlinear Systems,- Int. J. Mathematics and Computers in Simulation, vol. 2, pp. 21-33, 2010.

[18] W. Assawinchaichote, "A New Approach to Non-Fragile H Fuzzy Controller for Uncertain Fuzzy Dynamical Systems with Multiple Time- Scales,- Int. J. of Signal System Control and Engineering Application, vol. 3, pp. 49-64, 2010.

[19] Sebastian. B., John M. Coffin, and Martin A. Nowak, "Human Immunodeficiency Virus Drug Therapy and Virus Load,- American Society for Microbiology, pp. 3275-3278, 1997.

[20] J. Guedj, R. Thiebaut, and D. Commenges, "Practical identifiability of HIV dynamics models,- Bulletin of Mathematical Biology, vol. 69, pp. 2493-2513, 2007.

[21] R.A. Filter, X. Xia, and C.M. Gray, "Dynamic HIV/AIDS parameter estimation with application to a vaccine readiness study in southern Africa,- IEEE Transactions on Biomedical Engineering, vol. 52, pp. 784-791,2005.

[22] R.M. Jafelice, B.F.Z. Bechara, L.C. Barros R.C. Bassanezi, and F. Gomide. "Cellular automata with fuzzy parameters in microscopic study of positive HIV individuals,- Mathematical and Computer Modeling, vol. 50, pp. 32-44, 2009.

[23] M. Barao a, and J.M. Lemos, "Nonlinear control of HIV-1 infection with a singular perturbation model,- Biomedical Signal Processing and Control, vol. 2, pp. 248-257, 2007.

[24] Christine Brennan, and Demetrius J. Porche., ‘HIV immunopathogenesis,’ Journal of the Association of Nurses in AIDS care, vol. 8,pp. 7–22, 1997.

[25] S. Junhom, and W. Assawinchaichote, ‘A design of H fuzzy controller for HIV/AIDS infection system with dual drug dosages,’ 5th International Symposium on Medical Information and Communication Technology, pp. 152–155, 2011.

[26] S. Junhom, and W. Assawinchaichote, ‘A new approach of H fuzzy controller design for nonlinear positive HIV/AIDS infection dynamic model,’ 3rd International Conference on Machine Learning and Computing, pp. 420–424, 2011.

[27] J.A. Ball, and J.W. Helton, ‘H Control for Nonlinear Plants: Connection with Differential Games,’ IEEE Conf. Decision and Control, pp. 956–962, 1989.

[28] A.J. van der Schaft, ‘L2-Gain Analysis of Nonlinear Systems and Nonlinear State Feedback H Control,’ IEEE Trans. Automat. Control, vol. 37, pp. 770–784, 1992.

[29] A. Isidori, and A. Astofi, ‘Disturbance Attenuation and H-Control via Measurement Feedback in Nonlinear Systems,’ IEEE Trans. Automat. Control, vol. 37, pp. 1283–1293, 1992.

[30] A. Isidori, ‘Feedback Control of Nonlinear Systems,’ Proc. First European Contr. Conf., pp. 1001–1012, 1991.

[31] D.J. Hill, and P.J. Moylan, ‘Dissipative Dynamical Systems: Basic Input- Output and State Properties,’ J. Franklin Inst., vol. 309, pp. 327–357, 1980.

[32] A.J. van der Schaft, ‘A State-Space Approach to Nonlinear H Control,’ Syst. Contr. Letters., vol. 16, pp. 1-8, 1991.

[33] J.C. Willems, ‘Dissipative Dynamic Systems Part I: General Theory,’ Arch. Raitonal Mech. Anal., vol. 45, pp. 321–351, 1972.

[34] B.D.O. Anderson, and J.B. Moore, Optimal Control: Linear Quadratic Methods, Prentice-Hall, New Jersey, 1990.

[35] P. Gahinet, A. Nemirovski, A.J. Laub, and M. Chilali, LMI Control Toolbox – For Use with MATLAB, The MathWorks,Inc., MA, 1995.

[36] M. Chilali, and P. Gahinet, ‘H Design with Pole Placement Contraints: An LMI Approach,’ IEEE Trans. Automat. Control, vol. 41, pp. 358– 367, 1996.

[37] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory, vol. 15, Philadelphia: SIAM, 1994.

[38] B. Anderson, and S. Vongpantlerd, Network Analysis and Synthesis: A Modern System Theory Approach, Prentice-Hall, Englewood Cliffs, NJ, 1973.