Solving the Linearly Quadratic Optimal Control Problems of Elastic Structure by the Symplectic Group

結構主動控制研究與應用至今已有三十餘年的歷史，在土木工程應用上，能夠有效減輕結構在車輛、風力、地震力等動力作用下的反應與累積損害，提高結構的抗震能力與防災性能，為結構抗減震與防滅災之有效的方法及技術。土木工程結構振動控制大體上分為三個領域：基礎隔震、被動耗能減震與主動、半主動和智能控制。線性二次經典最佳控制中，求解黎卡提矩陣微分方程是無可避免的，因此本文之研究主旨便是透過具有辛群性質之數值方法，其可避開求解黎卡提微分方程，探討結構主動控制算法之線性二次經典最佳控制。文中所應用之數值方法為保群算法與辛群打靶法，此兩方法以李群的為基礎。使用這些數值方法可以在主動控制算法上達到迅速、經濟及精確的目的。將會設計幾個數值算例，並使用程式語言FORTRAN進行數值分析，展示此方法的結果，最後做出整體性的歸納以及未來發展的方向。The research and application of active structure control have been studied for almost thirty years. For the civil engineering, the active control can reduce the damage from the dynamical response of vehicles, wind, earthquake, and increase the resistance of the earthquake and disaster. It is an effective way for the structure to resist from the earthquake and decrease the damage from the disaster. Moreover, the active control of civil engineering can be divided into three parts: base isolation systems, passive energy dissipation systems, active, semi-active and intelligent control systems. In the classical linear quadratic optimal control (LQR), it is unavoidable to solve the Riccati differential equations. Therefore, the main purpose of this paper is to use the numerical method with the properties of symplectic group and avoid solving the Riccati differential equations and investigate the classical linear quadratic optimal control problems. We employ the group preserving schemes (GPS) and symplectic group shooting method in this paper, which are on the basis of the Lie group. Using these two methods, we can solve the active control problems quickly, economically and accurately. Besides, we design several examples and utilize the programming language, FORTRAN, to analyze them. Then, we will show the numerical results. Finally, the conclusions and the future work are addressed