解析最优控制用于一类无漂移的x-平坦系统
Analytic Optimal Control for a Class of Driftless x-Flat Systems
AI总结：
本文研究无漂移、x-平坦非线性系统的最优轨迹跟踪问题，通过几何性质和权重矩阵关系推导出闭环表达式和显式反馈律，避免求解二点边界值问题，展示了在转向轴运动学模型中的准确轨迹跟踪。
摘要：
本文研究无漂移、x-平坦非线性系统的最优轨迹跟踪问题。跟踪问题以Bolza形式提出，成本函数为跟踪误差及其导数的二次函数。应用庞特里亚金极大值原理得到混合正则-奇异最优控制问题。通过利用几何性质和权重矩阵之间的特定关系，推导出协态的闭式表达式以及两个输入的显式反馈律，从而避免了求解二点边界值问题的数值解。奇异输入导致 bang-singular-bang 最优控制结构，在奇异弧上，跟踪误差动力学简化为二阶线性动力学。该方法在转向轴运动学模型中得到验证，展示了准确的轨迹跟踪性能。
英文摘要：
This paper studies optimal trajectory-tracking for driftless, x-flat nonlinear systems with three states and two inputs. The tracking problem is formulated in Bolza form with a quadratic cost of the tracking error and its derivative. Applying Pontryagin's maximum principle yields a mixed regular-singular optimal control problem. By exploiting geometric properties and a specific relation between the weighting matrices, a closed-form expression for the costate and an explicit feedback law for both inputs is derived. Thereby, the numerical solution of a two-point boundary-value problem is avoided. The singular input leads to a bang-singular-bang optimal control structure, while on the singular arc, the tracking error dynamics reduces to a linear dynamics of order two. The approach is illustrated for the kinematic model of a steerable axle, demonstrating accurate trajectory-tracking.
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