Stabilizing a linear system using phone calls when time is information

Abstract

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.

Figures (9)

• Figure 1: Model of a networked control system where the feedback loop is closed over a timing channel \ref{['channelm']}.
• Figure 2: a) The timing channel. b) $i$th transmission. Subscripts $s$ and $r$, for symbol $\spadesuit$, are used to denote sent and received symbols, respectively. After reception of the $(i-1)$th symbol, the encoder selects a waiting time $W_i$, which carries information. The channel causes a random delay $S_i$, and the decoder observes the inter-reception time $D_i=W_i+S_i$ .
• Figure 3: The estimation problem.
• Figure 4: Evolution of the channel used in the simulation in an error-free case. Each time $\spadesuit$ is received, a new codeword is decoded using all the symbols received up to that time. The decoded codeword represents the initial state $X[0]$ with a precision that increases by $\mathbb{E}(D)C$ bits at each symbol reception. In the figure, for illustration purposes we have assumed $\mathbb{E}(D)C=3$ bits.
• Figure 5: Here we show the fraction of times stabilization was achieved versus the capacity of the channel across a run of 500 simulations for each value of the capacity. Successful stabilization is defined in these simulations as $|X[250]|\le 0.05$. In the case of a decoding error, no control input is applied and we let the system evolve in open loop. The simulation parameters were chosen as follows: $a=1.2$, $\mathbb{E}(D)=2$, $P_{e} = e^{-\eta k_m}$, where $\eta=0.09$, and the control gain is $K=0.4$.

Theorems & Definitions (31)

• Definition 1
• Definition 2
• Definition 3
• Theorem 1: Anantharam and Verdú
• Remark 1
• Lemma 1
• proof
• Theorem 2
• Theorem 3
• Remark 2