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Nonlinear Optimal Control Via the Chebyshev Polynomials
以柴比雪夫多項式做最佳軌跡飛行控制
摘要
一種以柴比雪夫多項式近似以求解非線性最佳化控制問題或是兩邊界值問題之方法已經被發展。此一方法主要的特性是假設狀態與控制變數皆可用柴比雪夫級數來展開。因此，存在於動態系統中之微分方程式，行為指標中之積分式與兩點邊界值問題中之邊界值皆可被轉換成一組代數方程式而可大大地簡化所要處理的問題。然而，柴比雪夫法對於解兩點邊界值問題存在一個困難。這個困難主要來自於當一個非線性系統已經被轉換成一組非線性代數方程式時，將會遭遇到如何去決定拉格朗運算子的起始值困難。因此，在本論文中，一個改良後的演算法被提出來克服上述的困難。最後，柴比雪夫近似法與改良後的演算法已經被應用在最佳軌跡飛行控制上。
並列摘要
A polynomial approximation involving the Chebyshev technique for solving the nonlinear optimal control problems or two-point boundary value problems (TPBVP) has been developed. The main characteristic of the technique is based on the assumption that the state and control variables can be expanded in the Chebyshev series. Consequently, the differential and integral equation involved in the system dynamics, performance index, and boundary conditions of the TPBVP can be converted into a set of algebraic equations and greatly simplifying the optimal control problems. Nevertheless, the Chebyshev approach for the TPBVP has presented a major difficulty in determining the starting values of the Lagrangian multiplier when iterative numerical techniques (such as Newton method) are employed. An improved algorithm that overcomes this difficulty is pressented in this paper. Finally, the proposed technique and improved numerical algorithm have been applied to an optimal tracking flight control problem.