In this article we consider the -stabilization of nth-order linear time-invariant dynamical systems using multiplicity-induced-dominancy-based controller design in the presence of delays in the input or the output channels. A sufficient condition is given for the dominancy of a real root with multiplicity at least and at least n using an integral factorization of the corresponding characteristic function. A necessary condition for -stabilizability is analyzed utilizing the property that the derivative of a -stable quasipolynomial is also -stable under certain conditions. Sufficient and necessary conditions are given for systems with real-rooted open-loop characteristic function: the delay intervals are determined where the conditions for dominancy and -stabilizability are satisfied. The efficiency of the proposed controller design is shown in the case of a multilink inverted pendulum.

中文翻译：

---

真根植物时滞系统稳定的条件

在本文中，我们考虑在输入或输出通道存在延迟的情况下，使用基于多重性诱导支配的控制器设计来稳定n阶线性时不变动力系统。使用对应特征函数的积分因式分解，给出了具有重数至少且至少为n的实根的优势的充分条件。利用-稳定拟多项式的导数也是- 在某些条件下稳定。给出了具有实根开环特征函数的系统的充分必要条件：在满足支配性和-稳定性条件的情况下确定延迟间隔。所提出的控制器设计的效率显示在多链路倒立摆的情况下。