在本論文中，新建立了一個適用於多變殊異擾動系統的線性二次納許賽局追蹤器。加入LQR設計方法應用於追蹤器設計的兩個二次估測方程的常態交錯多變數立卡迪函數(GCMARE)需要被解出。首先，新提出了針對GCMARE的漸進擴展式，且提出基於牛頓法的演算法來解GCMARE且保證其收斂性。然後，藉由數位再設計的方法設計出一個低增益且有高效能的控制器。最後，為了增進追蹤器的效能，使用了渾沌進化演算法(CEPA)來調整追蹤器的參數。最後以一個例子來驗證提出的方法是有效的。

In this thesis, a tracker for the linear quadratic Nash game of multiparameter singularly perturbed sampled-data systems is newly established. A generalized cross-coupled multiparameter algebraic Riccati equation (GCMARE) for two quadratic cost functions is needed to be solved by applying the LQR design methodology for the tracker design. Firstly, the asymptotic expansions for the GCMARE are newly established, and the proposed algorithm based on the Newton’s method for solving the GCMARE guarantees the quadratic convergence. Then the low-gain sample-data controller with a high design performance is realized though the digital redesign method. Finally, for further improving the tracking performance, the chaos evolutionary programming algorithm (CEPA) is utilized to tune the parameters of the tracker. An example is presented to demonstrate the effectiveness on the proposed methodology

Reference

[1] P. V Kokotovi´c, H. K. Khalil, and J. O’Reilly, Singular perturbation methods in control: Analysis and design. Philadelphia: SIAM. 1999.
[2] Z. Gajic´, and M-. T. Lim, Optimal control of singularly perturbed linear systems and applications. New York: Marcel Dekker. 2001.
[3] T.-Y. Li and Z. Gajic´, “Lyapunov iterations for solving coupled algebraic lyapunov equations of Nash differential games and algebraic Riccati equations of zero-sum games,” in New Trends in Dynamic Games and Applications. Boston, MA: Birkhauser, pp. 333–351, 1994
[4] J. R. Magnus and H. Neudecker, Matrix Differential Calculus with Applications in Statistics and Econometrics. New York: Wiley, 1999.
[5] A. W. Starr and Y. C. Ho, “Nonzero-sum differential games,” J. Optim. Theory Appl., vol. 3, no. 3, pp. 184–206, 1969.
[6] D. J. N. Limebeer, B. D. O. Anderson, and B. Hendel, “A Nash game approach to mixed control,” IEEE Trans. Autom. Contr., vol. 39, no. 1, pp. 69–82, Jan. 1994.
[7] H. K. Khalil and P. V. Kokotoviæ, “Feedback and well-posedness of singularly perturbed Nash games,” IEEE Trans. Autom. Contr., vol. 24, no. 5, pp. 699–708, Oct. 1979.
[8] H. K. Khalil and P. V. Kokotoviæ, “Control of linear systems with multiparameter singular perturbations,” Automatica, vol. 15, no. 2, pp. 197–207, 1979.
[9] Z. Gajic´, D. Petkovski, and X. Shen, Singularly perturbed and weakly coupled linear system a recursive approach. Lecture Notes in Control and Information Sciences. Berlin:Springer. 1990.
[10] H. Mukaidani, H. Xu, and K. Mizukami, “Recursive approach of control problems. for singularly perturbed systems under perfect and imperfect state measurements.” International Journal of systems Science, vol. 30, 467–477, 1999.
[11] H. Mukaidani, H. Xu, and K. Mizukami, “Numerical algorithm for solving cross-coupled algebraic Riccati equations of singularly perturbed systems,” Advances in Dynamic Games: Applications to Economics, Finance, Optimization and Stochastic Control, A. S. Nowak and K. Szajowski, Eds. Boston: Birkhauser, pp. 545–570, 2005.
[12] H. Mukaidani, T. Shimomura, and H. Xu, “Numerical algorithm for solving cross-coupled algebraic Riccati equations related to Nash games of multimodeling systems,” in Proc. IEEE Conf. Decision and Control, pp. 4167–4172, 2002.
[13] Z. Gajic, D. Petkovski, and X. Shen, “Singularly Perturbed and Weakly Coupled Linear System—A Recursive Approach,” Lecture Notes in Control and Information Sciences Berlin, Germany: Springer-Verlag, vol. 140, 1990.
[14] H. Mukaidani, “A new design approach for solving linear quadratic Nash games of multiparameter singularly perturbed systems,” IEEE Trans. Circuits Syst. I, Regu. papers, vol. 52, no. 5, pp. 960-974, May 2005.
[15] F. L.Lewis and V. L. Syrmos, Optimal Control, 2nd Edition, New York: Wiley, 1986
[16] S. M. Guo, L. S. Shieh, G.. Chen and C. F. Lin, “Effective chaotic orbit tracker: a prediction-based digital redesign approach,” IEEE Transaction on Circuits and Systems-I, Fundamental Theory and Application vol. 47 no.11, 1557-1570, 2000.
[17] G. Jank and G. Kun et al., “Solutions of generalized Riccati differential equations and their approximation,” in Computational Methods and Function, Theory , N. Papamichael et al., Eds. Singapore: World Scientific, pp. 1–18, 1998.
[18] K. Lu, J. H. Sun, R. B. Ou, and L. Y. Huang., Chaos dynamics, Shanghan: shanghai Translation Press Company, 1990
[19] J. H. Halton, “On the efficiency of certain quasi-random sequences of points in evaluating multidimensional integrals”, Numerische Mathematik, vol. 2, pp. 84-90, 1960 with Corrigenda on pp. 196.
[20] T. Yamamoto, “A method for finding sharp error bounds for Newton’s method under the Kantorvich assumptions,” Num. Math., vol. 49, pp. 203–220, 1986.
[21] J.C. Van Der Corput, “Verteilungsfunktionen,” Proc. Kon Akad. Wet., Amsterdam, vol. 38, pp. A12-A21, pp. 1058-1066, 1935.
[22] S. M. Guo, K. T. Liu, J. S. H. Tsai, and L. S. Shieh, “An observer-based tracker for hybrid interval chaotic systems with saturating inputs: the chaos evolutionary programming approach,” Computers and mathematics with Applications (Master Thesis, Nation Cheng Kung University)
[23] J. S. H. Tsai, L. S. Shieh and R. E. Yates, “Fast and stable algorithms for computing the principal nth root of a complex matrix and the matrix sector function,” Comput. Math. Application, vol. 15, no.11, pp. 903-913, 1988.