In this paper, the robust stability of interval fractional order plants with one time delay controlled by fractional order controllers is investigated in a general form. For robust stability analysis of the closed loop system by the zero exclusion principle, the distance between the origin and the value set of the characteristic function needs to be checked. It is known that the outer vertices of this value set may change at some switching frequencies and the repetitive calculation of these vertices at switching frequencies leads to additional calculations. In this study initially, new necessary and sufficient conditions are proposed to check the robust stability of a delayed fractional order closed loop system. Then, a novel robust stability testing function is presented based on some vertices, which are fixed for all positive frequencies. Therefore, no additional calculation is needed to obtain the outer vertices of the characteristic function value set for any pair of the switching frequencies. Also, a finite frequency range is presented to reduce the computational cost noticeably. Eventually, three numerical examples are given to verify the efficiency of the results of this paper.

中文翻译：

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一类间隔时滞分数阶设备的一般形式的分数阶控制器的鲁棒稳定性分析

在本文中，以一般形式研究了由分数阶控制器控制的具有一个时滞的区间分数阶植物的鲁棒稳定性。利用零排斥原理对闭环系统进行鲁棒稳定性分析，需要检查原点与特征函数值集之间的距离。众所周知，该值集的外部顶点可能会在某些切换频率处发生变化，并且这些顶点在切换频率处的重复计算会导致额外的计算。在这项研究中，最初提出了新的充分必要条件来检查延迟分数阶闭环系统的鲁棒稳定性。然后，基于一些顶点提出了一种新颖的鲁棒稳定性测试函数，这些顶点对于所有正频率都是固定的。因此，不需要额外的计算来获得为任何一对开关频率设置的特征函数值的外顶点。此外，还提出了一个有限的频率范围，以显着降低计算成本。最后给出三个数值算例来验证本文结果的有效性。