This paper deals with time-optimal control of nonlinear continuous-time systems based on direct collocation. The underlying discretization grid is variable in time, as the time intervals are subject to optimization. This technique differs from approaches that are usually based on a time transformation. Hermite-Simpson collocation is selected as common representative in the field of optimal control and trajectory optimization. Hereby, quadratic splines approximate the system dynamics. Several splines of different order are suitable for the control parameterization. A comparative analysis reveals that increasing the degrees of freedom in control, e.g. quadratic splines, is not suitable for time-optimal control problems due to constraint violation and inherent oscillations. However, choosing constant or linear control splines points out to be very effective. A major advantage is that the implicit solution of the system dynamics is suited for stiff systems and often requires smaller grid sizes in practice.

中文翻译：

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直接搭配和变量离散化的时间最优控制

本文研究基于直接搭配的非线性连续时间系统的时间最优控制。底层离散化网格随时间变化，因为时间间隔需要优化。这种技术不同于通常基于时间变换的方法。选择Hermite-Simpson搭配作为最优控制和轨迹优化领域的共同代表。因此，二次样条近似于系统动力学。多个不同阶次的样条适用于控制参数化。比较分析表明，由于违反约束和固有振荡，增加控制的自由度，例如二次样条，不适用于时间最优控制问题。然而，选择恒定或线性控制样条表明非常有效。一个主要优点是系统动力学的隐式解决方案适用于刚性系统，并且在实践中通常需要较小的网格尺寸。