We provide a comprehensive impossibility result toward achieving string stability, i.e., keeping local relative errors in check with local controllers independently of the size of a chain of subsystems. We significantly extend the existing results, from the linear time-invariant (LTI) setting to any homogeneous controllers that can be nonlinear, time varying, and locally communicating. We prove this impossibility for a set of definitions with various norm choices, including the $L_2$-type standard and a bounded-input implies bounded-output (BIBO) type criterion. All the results hold for a general discrete-time controller, which should cover most applications.

中文翻译：

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对弦稳定性的全面不可能结果

我们为实现串稳定性提供了一个全面的不可能结果，即保持局部相对误差与局部控制器的检查无关，而与子系统链的大小无关。我们显着扩展了现有结果，从线性时不变 (LTI) 设置到可以是非线性、时变和本地通信的任何齐次控制器。我们证明了一组具有各种规范选择的定义的不可能性，包括 $L_2$ 类型标准和有界输入隐含有界输出 (BIBO) 类型标准。所有结果都适用于一般离散时间控制器，它应该涵盖大多数应用。