Legendre Pseudospectral Approximations of Optimal Control Problems
Legendre Pseudospectral Approximations of Optimal Control Problems
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DOI：
10.1007/978-3-540-45056-6_21
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发表时间：
2003
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影响因子：
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通讯作者：
I. Michael Ross;F. Fahroo
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文献类型：
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作者：
I. Michael Ross;F. Fahroo
本文研究了具有混合状态控制约束的非线性最优控制问题. Bolza问题的Legendre伪谱方法的离散化。证明了离散化和对偶化的操作是不可交换的。引入一组闭包条件来交换这些操作。一个直接的后果，这是一个共向量映射定理（CMT），提供了一个保序变换的拉格朗日乘子与离散化的问题，离散的最优控制问题相关联的共向量。CMT的一个自然结果是，对于纯状态约束问题，对偶变量可以很容易地与哈密顿量的拉格朗日量的D-形式有关。我们通过数值求解一个状态约束最优控制问题，证明了我们的结果的实际优势，而无需推导必要条件。我们的CMT的应用程序获得的costates显示出良好的协议与精确的解析解。
We consider nonlinear optimal control problems with mixed statecontrol constraints. A discretization of the Bolza problem by a Legendre pseudospectral method is considered. It is shown that the operations of discretization and dualization are not commutative. A set of Closure Conditions are introduced to commute these operations. An immediate consequence of this is a Covector Mapping Theorem (CMT) that provides an order-preserving transformation of the Lagrange multipliers associated with the discretized problem to the discrete covectors associated with the optimal control problem. A natural consequence of the CMT is that for pure state-constrained problems, the dual variables can be easily related to theD-form of the Lagrangian of the Hamiltonian. We demonstrate the practical advantage of our results by numerically solving a state-constrained optimal control problem without deriving the necessary conditions. The costates obtained by an application of our CMT show excellent agreement with the exact analytical solution.