以李群打靶法求解杜芬非線性振子的最佳化控制問題

Abstract

在最佳化控制理論中，通常使用哈密頓函數，利用其方便找尋控制力函數的特點來設計控制力。然而，當狀態函數形式較為複雜時，哈密頓函數將構成一非線性微分代數方程組的兩點邊界值問題而難以找出封閉解，因此需使用其他數值方法輔助求解。 本篇論文將杜芬非線性振子代入兩點邊界值問題模擬非線性微分代數方程組，藉此探討上述議題，並建立一套數值方法利用李群 及 打靶法配合李群微分代數方程法對杜芬非線性振子的最佳化控制問題求出數值近似解。在論文中將演示如何使用上述方法求解六個單自由度以及一個雙自由度的杜芬非線性振子最佳化控制問題，並分析其數值結果。

In the optimal control theory, the Hamiltonian formulation is a famous one which is convenient to find an optimally designed control force. However, when the performance index is a complicated function of control force, the Hamiltonian method is not easy to find the optimal closed-form solution, because one may encounter a two-point boundary value problem of nonlinear differential algebraic equations (DAEs). In this thesis, we address this issue via an novel approach, of which the optimal vibration control problem of Duffing oscillator is recast into a two-point nonlinear DAEs. We develop the corresponding and shooting methods, as well as a Lie-group differential algebraic equations (LGDAE) method to numerically solve the optimal control problems of nonlinear Duffing oscillators. Seven examples of a single Duffing oscillator and one coupled Duffing oscillators are used to test the performance of the present method.