以李群打靶法求解達芬非線性振子的最佳控制問題

Abstract

結構主動控制就是由控制元件對結構系統施加額外作用力，以改善土木結構之動力特性，或提高勁度，或提高阻尼，來達到減震消能之效果。結構主動控制系統包含三個核心部分：感應計、控制律及控制器。感應計佈設在土木結構及控制器上，以量測其動力反應；控制律就是控制力施加之法則，根據結構及控制器之反應，決定控制力施加之時機、大小及方向；控制器就是控制力產生之機構，透過一系列的動力系統，將控制力施加在土木結構上。而用於非線性結構物上最佳控制理論研究上，通常將此最佳控制問題轉換為兩點邊界值問題，而在本論文用於求解兩點邊界值問題的數值解法為李群打靶法，再配合四階精度龍格-庫塔法求出數值結果。而李群打靶法對於求解未知的邊界值問題是一個強而有力的數值解法，此法根據保群算法中群的封閉性質、李群的性質、保長性質以及一些簡單數學的推導，例如：中值原理的觀念推導而來。再將此數值解法運用在線性最佳化控制、單自由度達芬非線性振子以及雙自由度達芬非線性振子上。本文當中使用到程式語言FORTRAN進行數值分析，並透過繪圖軟體GRAPHER將數值模擬圖形呈現出來，並且期望未來能應用在土木工程發展上。

In order to improve the dynamic characteristics, the stiffness and the damping of civil engineering structures to achieve a certain energy dissipation effect, and the active structural control is exerted additional force by the control elements to the traditional structural system. The active structural control system consists of three core components: the sensor, the control law and actuator. Sensors are used to measure the dynamic response which layout in structures and controllers of the civil engineering. According to the response of the structure and controllers, decide the timing, size and the direction of control being imposed on the structure. The actuator is developed the institutions by the control force and which is applied to structures of civil engineering through a series of dynamical systems. In the study of optimal control theory for nonlinear structures, one often encounters two-point boundary-value problems (TPBVPs). In this study, the numerical solution for two-point boundary value problem is the Lie group shooting method (LGSM), and then with the fourth-order Runge-Kutta method (RK4). The LGSM is a powerful technique to search the unknown initial conditions. These methods are gradually derived based on the closure property of the group, the Lie group property and the length preserving property in GPS, some simple mathematical derivation, the mid-point rule. And it will be used to this numerical solution of the linear optimal control, the single degree nonlinear Duffing osillator, as well as two degrees nonlinear Duffing osillator. In this thesis, we use programming language FORTRAN for the numerical analysis and plot the numerical results by the GRAPHER. Finally, we want to apply this method on the development of the civil engineering in the future.