In this dissertation, we address the robust stabilization design problem for nonlinear stochastic distributed parameter systems (NSDPSs) with random external disturbances and measurement noises in the spatio-temporal domain. A fuzzy stochastic distributed parameter system is proposed to approximate the NSDPS based on fuzzy interpolation approach. Then a fuzzy stochastic spatial state space model is developed to represent the fuzzy stochastic distributed parameter system via a semi-discretization finite difference scheme. Based on this model, a robust fuzzy estimator-based stabilization controller is proposed to stabilize the NSDPS. Furthermore, the robust stochastic $H_\infty$ stabilization design is proposed to attenuate the effects of random external disturbances and measurement noise in the spatio-temporal domain from the area energy point of view, and the LMI technique is applied to solve the control gains and estimator gains of the controller via a systematic control design procedure. Finally, a simulation example is given to illustrate the design procedure and to confirm the performance of the proposed robust fuzzy estimator-based stabilization design for the NSDPSs.

在本論文中，我們探討非線性隨機分佈參數系統的隨機穩定化問題，和有外部擾動和量測雜訊影響下的非線性隨機分佈參數系統的強健性 $H_\infty$ 穩定化問題。我們更針對外部的擾動和量測雜訊是在空間位置分佈的情況下來探討其穩定化的控制器設計。模糊方法被廣泛的應用於非線性系統的近似。因此，我們利用模糊內插法，提出一個模糊隨機分佈參數系統來近似原本的非線性隨機分佈參數系統。然後使用半離散化的有限差分法，我們發展一個模糊隨機的狀態空間模型，來取代模糊隨機分佈參數系統。模糊隨機的狀態空間模型是被證明可以近似原本的非線性隨機分佈參數系統。因此，基於這個模型，一個強健模糊估測器結合穩定化控制器是被提出來控制非線性隨機分佈參數系統使其穩定。控制器使其系統穩定的條件是被證明只要符合一個矩陣不等式即可被保證。進一步地，強健性 $H_\infty$ 控制設計法則是被提出來消除外部干擾和量測雜訊對系統輸出的影響。因為控制器增益及估測器增益互相偶和的問題，所以設計條件是一個雙線性的矩陣不等式。為了有系統的解決設計的問題，我們簡化BMI的問題成LMI的問題，並使用 LMI 技巧來求解控制器增益和估測器增益。最後，為了呈現設計的性能及方法的實用性，我們給一個神經系統的例子來說明控制器設計的流程，並驗證設計方法的效能。

Bibliography
[1] K. Liu, Stability of Infinite Dimensional Stochastic Differential Equations With Applications. Chapman & Hall/CRC, 2006.
[2] P.-L. Chow, Stochastic Partial Differential Equations. Chapman & Hall/CRC, 2007.
[3] M. Kamrani and S. M. Hosseini, “The role of coefficients of a general SPDE on the
stability and convergence of a finite difference method,” J. Comput. Appl. Math.,
vol. 234, no. 5, pp. 1426–1434, 2010.
[4] M. J. Anabtawi and S. Sathananthan, “Stability and convergence via Lyapunov-like functionals of stochastic parabolic partial differential equations,” Appl. Math.
Comput., vol. 157, pp. 201–218, 2004.
[5] M. J. Anabtawi and S. Sathananthan, “Stability and convergence results for Itˆo-type
parabolic partial differential equations in Hilbert spaces,” Stoch. Anal. Appl., vol. 27,
pp. 671–693, 2009.
[6] P.-L. Chow, “Stability of nonlinear stochastic-evolution equations,” J. Math. Anal.
Appl., vol. 89, pp. 400–419, 1982.
[7] Y. Lou, G. Hu, and P. D. Christofides, “Model predictive control of nonlinear stochas-
tic partial differential equations with application to a sputtering process,” Aiche
Journal, vol. 54, pp. 2065–2081, 2008.
68[8] A. M. Davie and J. G. Gaines, “Convergence of numerical schemes for the solution of
parabolic stochastic partial differential equations,” Math. Comput., vol. 70, no. 233,
pp. 121–134, 2001.
[9] I. Gy¨ongy, “Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise I,” Potential Analysis, vol. 9,
pp. 1–25, 1998.
[10] I. Gy¨ongy, “Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise II,” Potential Analysis, vol. 11,
pp. 1–37, 1999.
[11] H. Yoo, “Semi-discretization of stochastic partial differential equations on R 1 by a
finite-difference method,” Math. Comput., vol. 69, no. 230, pp. 653–666, 1999.
[12] P.-L. Lions and P. E. Souganidis, “Fully nonlinear stochastic partial differential
equations,” Comptes Rendus de l’Acad´emie des Sciences-Series I - Mathematics,
vol. 326, pp. 1085–1092, 1998.
[13] P.-L. Lions and P. E. Souganidis, “Uniqueness of weak solutions of fully nonlinear
stochastic partial differential equations,” Comptes Rendus de l’Acad´emie des Sciences - Series I - Mathematics, vol. 331, pp. 783–790, 2000.
[14] M. J. Balas, “Feedback control of flexible systems,” IEEE Trans. Automat. Contr.,
vol. 23, pp. 673–679, 1978.
[15] C.-L. Lin and B.-S. Chen, “Robust observer-based control of large flexible struc-
tures,” J. Dyn. Syst. Meas. Control-Trans. ASME, vol. 116, pp. 713–722, Dec. 1994.
[16] P. D. Christofides, Nonlinear and Robust Control of PDE Systems: Methods and Applications to Transport-Reaction Processes. Birkhauser, 2001.
[17] J. Keener and J. Sneyd, Mathematical Physiology. Springer-Verlag, 1998.
[18] H. T. Banks, Modeling and Control in the Biomedical Sciences. Springer-Verlag, 1975.
[19] C. V. Pao, Nonlinear Parabolic and Elliptic Equations. New York :Plenum Press, 1992.
[20] P. L. Chow, “Stochastic partial differential equations in turbulence-related prob-
lems,” in Probabilistic Analysis and Related Topics, Vol. I, pp. 1–43, Academic Press,
New York, 1978.
[21] D. Dawson, “Stochastic evolution equation,” Math. Bioseciences, vol. 15, pp. 287–316, 1972.
[22] W. Fleming, Distributed Parameter Stochastic Systems in Population Biology,
vol. 107 of Lecture Notes in Economics and Mathematical Systems. Springer, Berlin,
1975.
[23] C. Koch, Biophysics of Computation: Information Processing in Single Neurons.
Oxford University Press, Inc., 1999.
[24] P. D. Christofides, “Robust control of parabolic PDE systems,” Chemical Engineer-
ing Science, vol. 53, pp. 2949–2965, 1998.
[25] R. F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems
Theory. Springer-Verlag, 1995.
[26] J. Robinson, Infinite-Dimensional Dynamical Systems. Cambridge University Press,
2001.
[27] C. I. Byrnes, I. G. Lauk´o, D. S. Gilliam, and V. I. Shubov, “Output regulation
for linear distributed parameter systems,” IEEE Trans. Automat. Contr., vol. 45, pp. 2236–2252, 2000.
[28] M. J. Balas, “Nonlinear finite-dimensional control of a class of nonlinear distributed
parameter systems using residual mode filters: A proof of local exponential stability,”
J. Math. Anal. Appl., vol. 162, pp. 63–70, 1991.
[29] J. Baker and P. D. Christofides, “Finite-dimensional approximation and control of non-linear parabolic PDE systems,” Int. J. Control, vol. 73, no. 5, pp. 439–456, 2000.
[30] H.-N. Wu and H.-X. Li, “H∞ fuzzy observer-based control for a class of nonlinear distributed parameter systems with control constraints,” IEEE Trans. Fuzzy Syst., vol. 16, no. 2, pp. 502–516, 2008.
[31] K. Yuan, H.-X. Li, and J. Cao, “Robust stabilization of the distributed parameter system with time delay via fuzzy control,” IEEE Trans. Fuzzy Syst., vol. 16, no. 3, pp. 567–584, 2008.
[32] B.-S. Chen and Y.-T. Chang, “Fuzzy state space modeling and robust stabilization design for nonlinear partial differential systems,” IEEE Trans. Fuzzy Syst., vol. 17, no. 5, pp. 1025–1043, 2009.
[33] C.-S. Tseng, B.-S. Chen, and H.-J. Uang, “Fuzzy tracking control design for nonlinear dynamic systems via T-S fuzzy model,” IEEE Trans. Fuzzy Syst., vol. 9, pp. 381–392, 2001.
[34] B.-S. Chen, C.-S. Tseng, and H.-J. Uang, “Robustness design of nonlinear uncertain system via fuzzy linear control,” IEEE Trans. Fuzzy Syst., vol. 7, no. 5, pp. 571–585, 1999.
71[35] L. X. Wang, A Course in Fuzzy Systems and Control. Prentice Hall, 1997.
[36] K. Tanaka and H. O. Wang, Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach. New York: Wiley, 2001.
[37] H. O. Wang, K. Tanaka, and M. F. Griffn, “An approach to fuzzy control of nonlinear systems: Stability and design issues,” IEEE Trans. Fuzzy Syst., vol. 4, no. 1, pp. 14–23, 1996.
[38] T.-H. S. Li and S.-H. Tsai, “T-S fuzzy bilinear model and fuzzy controller design for a class of nonlinear systems,” IEEE Trans. Fuzzy Syst., vol. 15, pp. 494–506, June 2007.
[39] T.-H. S. Li and K.-J. Lin, “Composite fuzzy control of nonlinear singularly perturbed systems,” IEEE Trans. Fuzzy Syst., vol. 15, no. 2, pp. 176–187, 2007.
[40] Y.-Y. Chen, Y.-T. Chang, and B.-S. Chen, “Fuzzy solutions to partial differential equations: Adaptive approach,” IEEE Trans. Fuzzy Syst., vol. 17, no. 1, pp. 116–127, 2009.
[41] Y.-T. Chang and B.-S. Chen, “A fuzzy approach for robust reference tracking control design of nonlinear distributed parameter time-delayed systems and its application,” IEEE Trans. Fuzzy Syst., in press.
[42] G. D. Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions. Cambridge University Press, 1992.
[43] R. F. Curtain, “Stability of stochastic partial differential equations,” J. Math. Anal. Appl., vol. 79, pp. 352–369, 1981.
[44] G. Ladde and V. Laksmikantham, Random Differential Inequalities. Academic Press, New York, 1980.
[45] A.Jentzen and P. E. Kloeden, “The numerical application of stochastic partial equations,” Milan J. Math., vol. 77, pp. 205–244, 2009.
[46] J. G. Verwer and J. M. Sanz-Serna, “Convergence of method of linear approximations to partial differential equations,” Computing, vol. 33, pp. 297–313, 1984.
[47] Z. Kamont and K. Kropielnicka, “Numerical method of lines for parabolic functional differential equations,” Applicable Analysis, vol. 88, pp. 1637–1650, 2009.
[48] E. Kim and S. Kim, “Stability analysis and synthesis for an affine fuzzy control system via LMI and ILMI: A continuous case,” IEEE Trans. Fuzzy Syst., vol. 10, no. 3, pp. 391–400, 2002.
[49] D. Huang and S. K. Nguang, “Robust H∞ static output feedback control of fuzzy systems: An ILMI approach,” IEEE Trans. Syst., Man, Cybern. B, vol. 36, no. 1, pp. 216–222, 2006.
[50] M. Kocvara and M. Stingl, “PENNON: a code for convex nonlinear and semidefinite programming,” Optim. Method Softw., vol. 18, pp. 317–333, 2003.
[51] C. Lin, Q.-G. Wang, T. H. Lee, and Y. He, “Design of observer-based H∞ control for fuzzy time-delay systems,” IEEE Trans. Fuzzy Syst., vol. 16, no. 2, pp. 534–543, 2008.
[52] L. Zambotti and P. di Milano, “Itˆo-Tanaka’s formula for stochastic partial differential equations driven by additive space-time white noise,” in Stochastic Partial Differential Equations and Applications - VII, pp. 350–360, 2006.
[53] G. Dong, Nonlinear Partial Differential Equations of Second Order. American Mathematical Society, 1991.
[54] J. C. Strikwerda, Finite Difference Schemes and Partial Differential Equations Second Edition. SIAM, 2004.
[55] G. Evans, J. Blackledge, and P. Yardley, Numerical Methods for Partial Differential Equations. Springer-Verlag, 2000.
[56] P. Lancaster and M. Tismenetsky, The Theory of Matrices: with Application Second Edition. Academic Press, 1985.
[57] S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. SIAM, 1994.
[58] B. S. Chen and W. Zhang, “Stochastic H2/H∞ control with state-dependent noise,” IEEE Trans. Autom. Control, vol. 49, pp. 45–57, 2004.
[59] W. Zhang and B. S. Chen, “H∞ control for nonlinear stochastic systems,” SIAM J. Control and Optimization, vol. 44, no. 6, pp. 1973–1991, 2006.
[60] J. E. Marsden and M. J. Hoffman, Elementary Classical Analysis Second Edition. W. H. Freeman and Company, New York, 1993.
[61] K. Tanaka, T. Hori, and H. O. Wang, “A multiple Lyapunov function approach to stabilization of fuzzy control systems,” IEEE Trans. Fuzzy Syst., vol. 11, no. 4, pp. 582–589, 2003.
[62] S. G. Cao, N. W. Rees, and G. Feng, “Analysis and design of a class of continuous time fuzzy control systems,” Int. J. Control, vol. 64, pp. 1069–1087, 1996.
[63] K. Tanaka, T. Ikeda, and H. O. Wang, “Fuzzy regulators and fuzzy observers: Relaxed stability conditions and LMI-based designs,” IEEE Trans. Fuzzy Syst., vol. 6, no. 2, pp. 250–265, 1998.
[64] W.-J. Wang and C.-H. Sun, “Relaxed stability and stabilization conditions for a T-S fuzzy discrete system,” Fuzzy Sets Syst., vol. 156, no. 2, pp. 208–225, 2005.
[65] W.-J. Wang and C.-H. Sun, “A relaxed stability criterion for T-S fuzzy discrete systems,” IEEE Trans. Syst., Man, Cybern. B, vol. 34, no. 5, pp. 2155–2158, 2004.
[66] J. Qiu, G. Feng, and J. Yang, “A new design of delay-dependent robust H∞ filtering for discrete-time T-S fuzzy systems with time-varying delay,” IEEE Trans. Fuzzy Syst., vol. 17, no. 5, pp. 1044–1058, 2009.
[67] M. Chen, G. Feng, H. Ma, and G. Chen, “Delay-dependent H∞ filter design for discrete-time fuzzy systems with time-varying delays,” IEEE Trans. Fuzzy Syst., vol. 17, no. 3, pp. 604 – 616, 2009.
[68] B.-S. Chen, C.-S. Tseng, and H.-C. Wang, “Mixed H2/H∞ fuzzy output feedback control for nonlinear uncertain systems: LMI approach,” IEEE Trans. Fuzzy Syst., vol. 8, no. 3, pp. 249–265, 2000.
[69] M. de Oliveira, J. Bernussou, and J. Geromel, “A new discrete-time robust stability condition,” Syst. Control Lett., vol. 37, pp. 261–265, 1999.
[70] A. L. Hodgkin and A. F. Huxley, “A quantitative description of membrane current and its application to conduction and excitation in nerve,” Journal of Physiology, vol. 177, pp. 500–544, 1952.
[71] J. Wang, L. Chen, and X. Fei, “Analysis and control of the bifurcation of Hodgkin-Huxley model,” Chaos, Solutions and Fractals, vol. 31, pp. 247–256, 2007.
[72] J. Wang, L. Chen, and X. Fei, “Bifurcation control of the Hodgkin-Huxley equations,” Chaos, Solitions and Fractals, vol. 33, pp. 217–224, 2007.
75[73] H. Fukai, S. Doi, T. Nomura, and S. Sato, “Hopf bifurcations in multiple-parameter space of the Hodgkin-Huxley equations I: Global organization of bistable periodic solutions,” Biol. Cybern., vol. 82, pp. 215–222, 2000.
[74] D. M. Durand and M. Bikson, “Suppression and control of epileptiform activity by electrical simulation: A review,” Proceeding of the IEEE, vol. 89, pp. 1065–1082, 2001.
[75] F. Fr¨ohlich and S. Jezernik, “Feedback control of Hodgkin-Huxley nerve cell dynamics,” Control Eng. Practice, vol. 13, pp. 1195–1206, 2005.
[76] L. Ding and C. Hou, “Stabilizing control of Hopf bifurcation in the Hodgkin-Huxley
model via washout filter with linear control term,” Nonlinear Dyn., vol. 60, pp. 131–139, 2010.
[77] P. H¨anggi, G. Schmid, and I. Goychuk, “Excitable membranes: Channel noise and synchronization, and stochastic resonance,” Adv. in Solid State Phys., vol. 42, pp. 359–370, 2002.
[78] G. Schmid, I. Goychuk, and P. H¨anggi, “Channel noise and synchronization in excitable membranes,” Physica A, vol. 325, pp. 165–175, 2003.
[79] R. F. Fox and Y. nan Lu, “Emergent collective behavior in large numbers of globally coupled independently stochastic ion channels,” Physical Review E, vol. 49, no. 4, pp. 3421–3431, 1994.
[80] J. Moehlis, “Canards for a reduction of the Hodgkin-Huxley equations,” Journal of Mathematical Biology, vol. 52, no. 2, pp. 141–153, 2006.
76[81] D. V. Vavoulis, V. A. Straub, I. Kemenes, J. Feng, and P. R. Benjamin, “Dynamic control of a central pattern generator circuit: a computational model of the snail
feeding network,” Eur. J. Neurosci., vol. 25, pp. 2805–2818, 2007.
[82] T. Takahata, S. Tanabe, and K. Pakdaman, “White-noise simulation of the Hodgkin-Huxley model,” Biol. Cybern., vol. 86, pp. 403–417, 2002.
[83] Y. Xie, L. Chen, Y. M. Kang, and K. Aihara, “Controlling the onset of Hopf bifurcation in the Hodgkin-Huxley model,” Physical Review E, vol. 77, p. 061921, 2008.
[84] E. Rossoni, Y. Chen, M. Ding, and J. Feng, “Stability of synchronous oscillations in a system of Hodgkin-Huxley neurons with delayed diffusive and pulsed coupling,”
Physical review E, vol. 71, p. 061904, 2005.
[85] J. Rinzel, “On repetitive activity in nerve,” Fed. Proc., vol. 37, pp. 2793–2802.
[86] W. C. Troy, “The bifurcation of periodic solutions in the Hodgkin-Huxley equations,” Q. Appl. Math., vol. 36, pp. 73–83, 1978.
[87] J. Rinzel and R. N. Miller, “Numerical calculation of stable and unstable periodic solutions to the Hodgkin-Huxley equations,” Math. Biosci., vol. 49, pp. 27–59, 1980.