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.

中文翻译：

---

非线性最优调节器的递归解析解

该论文为一般类别的多输入仿射非线性系统开发了一个最优调节器，该系统最小化具有无限视野的非线性成本函数。如果需要，成本泛函足以对控制输入实施饱和限制。一种利用张量代数的有效算法被用来计算价值函数的泰勒级数展开式的张量系数（即，最优成本）。张量系数是通过求解一组非线性矩阵方程来找到的，这些方程递归地推广了众所周知的线性二次解。由此产生的解决方案生成最优控制器作为状态向量的非线性函数，直到规定的截断阶数。而且，提供了计算解决方案的完全收敛及其适用域的估计，以进一步指导用户。该算法的计算复杂度显示仅相对于序列顺序以多项式方式增长。最后，给出了几个非线性例子，包括一些输入饱和的例子，以证明该算法生成最优控制器的高阶泰勒级数解的有效性。