A class of arbitrary time convergence controllers based on a special time-varying scaling function provide designers with a good choice to realize prescribed time stable systems. However, these controllers have the problem of conservative control parameter range and lack a uniform formula for systems of different order. Herein, the arbitrary time convergence controller is improved, the unified formula for -th order system is given, and a more accurate control parameter range is obtained. By constructing an ingenious auxiliary function and a novel Lyapunov function, the arbitrary time convergence of the -th order controller is proved, and the reasonable parameter selection range is obtained using the integrator backstepping and the mathematical induction method. The effectiveness and advantages of the proposed arbitrary time convergence controllers under saturated input constraints are illustrated through numerical simulations by comparative studies in unmanned aerial vehicle (UAV) formation control.

中文翻译：

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三阶系统任意时间收敛控制器的设计

一类基于特殊时变标度函数的任意时间收敛控制器为设计人员实现规定的时间稳定系统提供了很好的选择。然而，这些控制器存在控制参数范围保守的问题，并且对于不同阶的系统缺乏统一的公式。在此，对任意时间收敛控制器进行了改进，给出了-阶系统的统一公式，得到了更精确的控制参数范围。通过构造一个巧妙的辅助函数和一个新颖的 Lyapunov 函数，-证明了三阶控制器，利用积分器反步法和数学归纳法得到了合理的参数选择范围。通过在无人飞行器 (UAV) 编队控制中的比较研究，通过数值模拟说明了所提出的任意时间收敛控制器在饱和输入约束下的有效性和优势。