基於進化計算演繹法，本論文提出一種嶄新的性能指標強健度分析方法，並探討了應用於卡爾曼濾波的實際例子。首先，提出一個進化計算演繹法，求存在有界不確定性、離散且具有時間不變性的可實作最佳卡爾曼濾波器，使得此濾波器的預期最大誤差最小化，而這可以被表示為一個「最小化─最大化」問題。其次，提出改進的「最大化─（最小化─最大化）」差分進化演算法（differential evolution, DE）與「最小化─（最小化─最大化）」差分進化演算法，以求解當濾波器存在著對於某些成分的加權/干擾項時的效能指標強健度邊界與對應的最差狀況之系統參數。前述「最大化─（最小化─最大化）」與「最小化─（最小化─最大化）」問題，可視為本文所提出的改良型「最大化─最小化」與「最小化─最小化」問題的延伸應用。和之前的方法比較，本研究所提出的改良型「最大化─最小化」與「最小化─最小化」演算法可改進精確度、速度與強健度，且本研究提出的演算法不失一般性，可應用在卡爾曼濾波以外的地方。

In this thesis, a novel approach to the robustness analysis of the performance index based on the evolutionary algorithm (EA) with a case study on Kalman filtering is proposed. First, an EA-based filtering algorithm to find a practically implementable “best” Kalman filter is proposed for a linear discrete-time time-invariant plant with bounded uncertainties, such that the maximum filtering error is minimized, denoted as the min-max problem. The improved max-(min-max) differential evolution (DE) and the improved min-(min-max) DE are then proposed to determine the robustness bound of the performance index when some components of the practically implementable “best” Kalman filter are weighted/perturbed in some specified domain. The applications of the proposed max-(min-max) and min-(min-max) DEs to the practical Kalman filtering can be regarded as an extension of the max-min and min-min DEs proposed in this thesis, which also improve the precision, speed, and robustness more than conventional ones. The worst-case parameter set from the stochastic uncertainties is also found in this work. Without loss of generality, the proposed approach works not only for Kalman filtering, but for a wide selection of practical applications.

[1] K. I. Ko and C. L. Lin, “On the complexity of min-max optimization problems and their approximation,” Nonconvex Optimization and Its Applications, vol. 4, pp. 219-240, 1995.
[2] D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning, Addison-wesley, 1989.
[3] N. Hansen, S. D. Müller, and P. Koumoutsakos, “Reducing the time complexity of the derandomized evolution strategy with covariance matrix adaptation (CMA-ES),” Evolutionary Computation, vol. 11, pp. 1-18, 2003.
[4] R. Storn and K. Price, “Differential evolution–a simple and efficient heuristic for global optimization over continuous spaces,” Journal of global optimization, vol. 11, pp. 341-359, 1997.
[5] Y. Ao, H. Chi, E. Ayachi, S. Ihsen, B. Mohamed, M. H. F. Zarandi, M. Khademian, B. Minaei-Bidgoli, I. B. Türkşen, and M. Maraqa, “Differential evolution using opposite point for global numerical optimization,” Journal of Intelligent Learning Systems and Applications, vol. 4, pp. 1-19, 2012.
[6] K. Masuda, K. Kurihara, and E. Aiyoshi, “A novel method for solving min-max problems by using a modified particle swarm optimization,” IEEE International Conference on Systems, Man, and Cybernetics (SMC), pp. 2113-2120, 2011.
[7] G. Chen, J. Wang, and L. S. Shieh, “Interval Kalman filtering,” IEEE Transactions on Aerospace and Electronic Systems, vol. 33, pp. 250-259, 1997.
[8] S. M. Guo, L. S. Shieh, G. Chen, and N. P. Coleman, “Observer-type Kalman innovation filter for uncertain linear systems,” IEEE Transactions on Aerospace and Electronic Systems, vol. 37, pp. 1406-1418, 2001.
[9] S. M. Guo, L. S. Shieh, C. F. Lin, and N. P. Coleman, “Evolutionary‐programming‐based Kalman filter for discrete‐time nonlinear uncertain systems,” Asian Journal of Control, vol. 3, pp. 319-333, 2001.
[10] M. Sion, “On general minimax theorems,” Pacific Journal of mathematics, vol. 8, pp. 171-176, 1958.
[11] A. Petrowski, “A clearing procedure as a niching method for genetic algorithms,” IEEE International Conference on Evolutionary Computation, pp. 798-803, 1996.
[12] K. J. Astrom and B. Wittenmark, Computer-controlled Systems: Theory and Design, 3rd Edition, Prentice Hall, Englewood Cliffs, NJ, 1990.
[13] F. L. Lewis, Applied Optimal Control and Estimation, Prentice Hall PTR, 1992.
[14] X. Zhang, A. Heemink, and J. Van Eijkeren, “Performance robustness analysis of Kalman filter for linear discrete-time systems under plant and noise uncertainty,” International Journal of Systems Science, vol. 26, pp. 257-275, 1995.