面向参数依赖闭环律的邻近极值最优控制理论
Neighboring Extremal Optimal Control Theory for Parameter-Dependent Closed-loop Laws
AI总结：
本文提出参数依赖闭环最优控制律的NEOC理论，用线性PDE与Galerkin算法求参数小变化下的控制律修正，并以同伦法处理大扰动。
摘要：
本研究提出了一种为闭环最优控制问题获取邻近极值最优控制（NEOC）解的方法，该方法适用于广泛的非线性系统，且性能指标不必为二次型。该方法研究由于系统方程或性能指标中已知的小参数变化而导致的已知闭环最优控制律函数形式所发生的变化。NEOC解在形式上可以通过求解一个线性偏微分方程得到，这类似于在非线性Hamilton-Jacobi方程迭代求解过程中遇到的方程。受求解后一类方程的数值过程启发，我们还提出了一种基于Galerkin算法的数值算法，利用基函数来求解原始最优控制问题所对应的底层Hamilton-Jacobi方程。所提出的方法通过将NEOC问题简化为一组简单线性方程的求解，从而无需对调整后的最优控制问题进行完整重新求解。此外，最优性能指标的变化可以作为系统状态和参数小变化两者的函数获得，从而能够在系统或性能指标参数发生小调整时确定最优控制律的调整量。此外，为了处理较大的已知参数扰动，我们提出了一种同伦方法，将NEOC的单次计算分解为有限多个步骤。最后，理论分析和数值仿真支持了相关论断和理论的有效性。
英文摘要：
This study introduces an approach to obtain a neighboring extremal optimal control (NEOC) solution for a closed-loop optimal control problem, applicable to a wide array of nonlinear systems and not necessarily quadratic performance indices. The approach involves investigating the variation incurred in the functional form of a known closed-loop optimal control law due to small, known parameter variations in the system equations or the performance index. The NEOC solution can formally be obtained by solving a linear partial differential equation, akin to those encountered in the iterative solution of a nonlinear Hamilton-Jacobi equation. Motivated by numerical procedures for solving these latter equations, we also propose a numerical algorithm based on the Galerkin algorithm, leveraging the use of basis functions to solve the underlying Hamilton-Jacobi equation of the original optimal control problem. The proposed approach simplifies the NEOC problem by reducing it to the solution of a simple set of linear equations, thereby eliminating the need for a full re-solution of the adjusted optimal control problem. Furthermore, the variation to the optimal performance index can be obtained as a function of both the system state and small changes in parameters, allowing the determination of the adjustment to an optimal control law given a small adjustment of parameters in the system or the performance index. Moreover, in order to handle large known parameter perturbations, we propose a homotopic approach that breaks down the single calculation of NEOC into a finite set of multiple steps. Finally, the validity of the claims and theory is supported by theoretical analysis and numerical simulations.
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