论文标题
非线性最佳调节剂的递归分析解决方案
Recursive Analytic Solution of Nonlinear Optimal Regulators
论文作者
论文摘要
该论文为一类多输入仿射非线性系统的一类最佳调节剂最大程度地减少了无限范围的非线性成本功能。成本功能足够一般，如果需要，可以在控制输入上强制饱和限制。利用张量代数的有效算法用于计算泰勒级数膨胀的张量系数（即最佳成本成本）。通过求解一组非线性基质方程来递归概括众所周知的线性二次溶液，可以找到张量系数。最终的解决方案将最佳控制器作为状态矢量的非线性函数生成，直到规定的截断顺序。此外，还提供了计算解决方案的完整收敛以及对其适用性域的估计，以进一步指导用户。该算法的计算复杂性显示仅相对于串联顺序多一级生长。最后，提出了几个非线性示例，包括一些输入饱和的示例，以证明该算法的功效以生成最佳控制器的高阶泰勒串联解决方案。
The paper develops an optimal regulator for a general class of multi-input affine nonlinear systems minimizing a nonlinear cost functional with infinite horizon. The cost functional is general enough to enforce saturation limits on the control input if desired. An efficient algorithm utilizing tensor algebra is employed to compute the tensor coefficients of the Taylor series expansion of the value function (i.e., optimal cost-to-go). The tensor coefficients are found by solving a set of nonlinear matrix equations recursively generalizing the well-known linear quadratic solution. The resulting solution generates the optimal controller as a nonlinear function of the state vector up to a prescribed truncation order. Moreover, a complete convergence of the computed solution together with an estimation of its applicability domain are provided to further guide the user. The algorithm's computational complexity is shown to grow only polynomially with respect to the series order. Finally, several nonlinear examples including some with input saturation are presented to demonstrate the efficacy of the algorithm to generate high order Taylor series solution of the optimal controller.
