We consider the problem of stabilizing an undisturbed, scalar, linear system over a "timing" channel, namely a channel where information is communicated through the timestamps of the transmitted symbols. Each symbol transmitted from a sensor to a controller in a closed-loop system is received subject to some to random delay. The sensor can encode messages in the waiting times between successive transmissions and the controller must decode them from the inter-reception times of successive symbols. This set-up is analogous to a telephone system where a transmitter signals a phone call to a receiver through a "ring" and, after the random delay required to establish the connection; the receiver is aware of the "ring" being received. Since there is no data payload exchange between the sensor and the controller, this set-up provides an abstraction for performing event-triggering control with zero-payload rate. We show the following requirement for stabilization: for the state of the system to converge to zero in probability, the timing capacity of the channel should be, essentially, at least as large as the entropy rate of the system. Conversely, in the case the symbol delays are exponentially distributed, we show an "almost" tight sufficient condition using a coding strategy that refines the estimate of the decoded message every time a new symbol is received. Our results generalize previous zero-payload event-triggering control strategies, revealing a fundamental limit in using timing information for stabilization, independent of any transmission strategy.

翻译：我们考虑在“定时”信道上稳定一个无扰动的标量线性系统的问题，该信道通过传输符号的时间戳来传递信息。在闭环系统中，从传感器传输到控制器的每个符号都会经历随机延迟。传感器可以通过连续传输之间的等待时间对消息进行编码，而控制器则需从连续接收符号的时间间隔中解码这些消息。这一设置类似于电话系统：发射器通过“振铃”向接收器发出呼叫信号，在连接建立所需的随机延迟后，接收器知晓“振铃”已被接收。由于传感器与控制器之间不存在数据载荷交换，该设置为零数据速率的事件触发控制提供了一种抽象模型。我们证明了如下稳定性要求：要使系统状态依概率收敛到零，信道的定时容量本质上至少需达到系统的熵率。反之，在符号延迟服从指数分布的情况下，我们采用一种每次接收新符号时精化解码消息估计的编码策略，给出了“近似”紧的充分条件。我们的结果推广了先前的零数据载荷事件触发控制策略，揭示了利用定时信息实现稳定性的基本极限，该极限与具体传输策略无关。