In this paper, a unified theory is proposed for the fractional control of the non-minimum phase plus dead time systems via the introduction of the fractional filter-PID control and stability inequality. More precisely, three Theorems from a fractional system-theoretic viewpoint are developed. The first Theorem deals with the controller design through the notion of generalised internal model control. That embeds the fractional filter-PID controller coupled with the unified form of the Smith predictor and the inverse compensator. The second Theorem designs a bound on the fractional filter parameter that leads to a stable feedback system. The third Theorem deals with the stability of a fractional quasi-characteristic polynomial of the commensurate order arising from the control of non-minimum phase systems with dead time. Two illustrative second-order plus dead time non-minimum phase systems are presented to test the efficacy of the resulting fractional filter-PID controller that obeys the stability bound. Controller and sensitivity performance indices for the nominal as well as mismatch cases reveal the effectiveness and robustness of the proposed method for enhanced closed-loop performance. Thus, the generalised inverse compensator, the fractional filter-PID controller and the stability inequality in corroboration with numerical simulations demonstrate the better control of non-minimum phase systems with dead time.

中文翻译：

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具有死区时间的非最小相位系统的控制：分数系统观点

在本文中，通过引入分数滤波器-PID控制和稳定性不等式，提出了非最小相位加死区系统的分数控制的统一理论。更准确地说，从分数系统理论的角度发展了三个定理。第一个定理通过广义内部模型控制的概念来处理控制器设计。它嵌入了分数滤波器-PID 控制器以及史密斯预测器和逆补偿器的统一形式。第二个定理设计了导致稳定反馈系统的分数滤波器参数的界限。第三定理涉及由具有死区时间的非最小相位系统的控制产生的具有相应阶数的分数阶准特征多项式的稳定性。提出了两个说明性的二阶加死区时间非最小相位系统来测试所得分数滤波器-PID 控制器的功效，该控制器服从稳定性界限。标称和失配情况下的控制器和灵敏度性能指标揭示了所提出的用于增强闭环性能的方法的有效性和鲁棒性。因此，广义逆补偿器、分数滤波器-PID 控制器和稳定性不等式与数值模拟证实了对具有死区时间的非最小相位系统的更好控制。标称和失配情况下的控制器和灵敏度性能指标揭示了所提出的用于增强闭环性能的方法的有效性和鲁棒性。因此，广义逆补偿器、分数滤波器-PID 控制器和稳定性不等式与数值模拟证实了对具有死区时间的非最小相位系统的更好控制。标称和失配情况下的控制器和灵敏度性能指标揭示了所提出的用于增强闭环性能的方法的有效性和鲁棒性。因此，广义逆补偿器、分数滤波器-PID 控制器和稳定性不等式与数值模拟证实了对具有死区时间的非最小相位系统的更好控制。