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 在本篇論文中，我們要介紹一個非線性狀態轉換來將一個線性時變系統轉換到一個純量系統，並且探討該純量系統在控制輸入下的狀態變化。接著，利用不同的控制方法來針對這個純量模式加以控制，控制的方法包括forward Riccati equation control、controllability grammian control、least square control以及division control共四種。然而，在控制的過程中，即使線性時變系統是可控制的，該純量系統有可能會失去可控制性。因此，這是未來還需要研究的課題。
 In this thesis, we introduce a nonlinear state transformation to transform a linear time-varying system into a scalar system, which describes how the two-norm of the system state changes with the control input. With this two-norm scalar model, different control designs can be constructed, including the forward Riccati equation control, controllability grammian control, least-squares control, and the division control. However, even if the linear time-varying system is controllable, the two-normscalar system may lose its controllability in the process of control. Hence, further research is required in the future.
 Abstract-Chinese version IAbstract-English version IIChapter1. Introduction 1Chapter2. Scalar Two-Norm Model for LTV System 32.1 Backward Riccati equation 32.2 Forward Riccati equation 4 2.3 Scalar two-norm model for LTV system 6Chapter3. Forward Riccati Control 7 3.1 Example 8Chapter4. Least Square Control 10 4.1 When A=0 10 4.2 When A≠0 11 4.3 Example 14Chapter5. Controllability Grammian Control 16 5.1 When A=0 16 5.2 When A≠0 17 5.3 Example 20Chapter6. Division Control 22 6.1 When A=0 22 6.1.1 Smooth division controller 22 6.1.2 Division controller + dead zone 23 6.2 When A≠0 23 6.2.1 Smooth division controller 23 6.2.2 Division controller + dead zone 24 6.3 Example 25Chapter7. Conclusion 28References 29Figures 30 Figure 3.1 Time history of system states 30 Figure 3.2 Norm of states with time 30 Figure 3.3 Lyapunov function with time 31 Figure 3.4 Solution q of forward Riccati equation with time 31 Figure 3.5 Scalar b of the transformed system with time 32 Figure 3.6 Control input u with time 32 Figure 3.7 Time history of system states 33 Figure 3.8 Norm of states with time 33 Figure 3.9 Lyapunov function with time 34 Figure 3.10 Solution q of forward Riccati equation with time 34 Figure 3.11 Scalar b of the transformed system with time 35 Figure 3.12 Control input u with time 35 Figure 4.1 Time history of system states 36 Figure 4.2 Norm of states with time 36 Figure 4.3 Control input u with time 37 Figure 4.4 Converge rate with time 37 Figure 4.5 Time history of system states 38 Figure 4.6 Norm of states with time 38 Figure 4.7 Control input u with time 39 Figure 4.8 Converge rate with time 39 Figure 4.9 Time history of system states 40 Figure 4.10 Norm of states with time 40 Figure 4.11 Control input u with time 41 Figure 4.12 Converge rate with time 41 Figure 5.1 Time history of system states 42 Figure 5.2 Control input u with time 42 Figure 5.3 Time history of Lyapunov function 43 Figure 5.4 Time history of controllability grammain p 43 Figure 5.5 Time history of system states 44 Figure 5.6 Control input u with time 44 Figure 5.7 Time history of Lyapunov function 45 Figure 5.8 Time history of controllability grammain p 45 Figure 6.1 Time history of system states 46 Figure 6.2 Norm of states with time 46 Figure 6.3 Control input u with time 47 Figure 6.4 Time history of system states 47 Figure 6.5 Norm of states with time 48 Figure 6.6 Control input u with time 48 Figure 6.7 Time history of system states 49 Figure 6.8 Norm of states with time 49 Figure 6.9 Control input u with time 50 Figure 6.10 Time history of system states 50 Figure 6.11 Norm of states with time 51 Figure 6.12 Control input u with time 51 Figure 6.13 Time history of system states 52 Figure 6.14 Norm of states with time 52 Figure 6.15 Control input u with time 53
 [1] Kalmen, R. E., When Is a Linear Control system Optimal, ASME. Journal of Basic Engineering,pp. 51-60, 1964.[2] Kwakernaak, H., and Sivan, R., Linear Optimal Control Systems, Wiley, New York, 1972.[3] Callier, F., and Desoer, C. A., Linear System Theory, Springer-Verlag, Hong Kong, 1992.[4] Rush, W. J., Linear System Theory, Prentice-Hall, Englewood Cliffs, NJ, 1993.[5] Valasek, M., and Olgac, N., Generalization of Ackermann’s Formula for Linear MIMOTime Invariant and Time Varying Systems, Proceedings of Conference on Decision andControl, pp. 827-831, 1993.[6] Wolovich, W. A., On the Stabilization of Controllable Systems, IEEE Trans. Automat.Contr., Vol. 13, pp. 569-572, 1968.[7] Arvanitis, K. G., and Paraskevopoulos, P. N., Uniform Exact Model matching for a Class ofLinear Time-varying Analytic Systems, Systems and Control letters, Vol. 19, pp. 312-323,1992.[8] Anderson, B. D. O., and Moore, J. B., Extensions of Quadratic Minimization Theory,Infinite Time Results, Int. J. Control, Vol. 7, pp. 473-480, 1968.[9] Chen, C. T., Linear System Theory and Design, Holt, Rinehart, and Winston, New York,1984.[10] Kalmen, R. E., and Busy, R. S., New Results in Linear Filtering and Prediction Theory,ASME. Journal of Basic Engineering, pp. 95-108, 1961.[11] Wonham, W. M., On a Matrix Riccati Equation of Stochastic Control, SIAM J. Control,Vol. 6, pp. 681-697, 1968.
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