Implicit Communication in Linear Quadratic Gaussian Control Systems
Gongpu Chen, Deniz Gunduz
TL;DR
This work introduces implicit communication in discrete-time LQG control systems, where the controller’s inputs serve as an implicit signaling channel to a receiver observing the state, subject to a control-cost constraint $J_n\le J^*_n+V$ and yielding an implicit-channel capacity $C(V)$. In the noiseless-observation setting, the authors derive a closed-form, memoryless Gaussian-MIMO capacity and show a separation principle: the optimal input is $u_t=-Kx_t+s_t$ with $s_t$ Gaussian, allowing control and channel coding to be designed independently; the capacity expression mirrors classic Gaussian MIMO capacity with a water-filled power allocation. In the noisy-observation setting, they provide a computable lower bound via stationary linear Gaussian policies and demonstrate a channel-translation that reduces the problem to a Gaussian channel with memory but without state information or feedback. The results illuminate when implicit communication can be practically realized, establish a principled trade-off between control and communication, and pave the way for extensions to networks and multi-agent systems. Overall, the paper connects LQG control with information-theoretic channel capacity, offering actionable insights for encoding information in control signals and leveraging established Gaussian-channel coding techniques in embedded control contexts.
Abstract
This paper studies implicit communication in linear quadratic Gaussian control systems. We show that the control system itself can serve as an implicit communication channel, enabling the controller to transmit messages through its inputs to a receiver that observes the system state. This communication is considered implicit because (i) no explicit communication channels are needed; and (ii) information is transmitted while simultaneously fulfilling the controller's primary objective--maintaining the control cost within a specified level. As a result, there exists an inherent trade-off between control and communication performance. This trade-off is formalized through the notion of implicit channel capacity, which characterizes the supremum reliable communication rate subject to a constraint on control performance. We investigate the implicit channel capacity in two settings. When both the controller and the receiver have noiseless observations of the system state, the channel capacity admits a closed-form expression. When both the controller and the receiver have noisy observations, we establish a lower bound on the implicit channel capacity. Surprisingly, in the noiseless observation case, the capacity-achieving input policy adheres to a separation principle, allowing the control and channel coding tasks to be addressed independently, without loss of optimality. While this separation principle no longer holds in the noisy observation setting, we show that linear Gaussian input policies still decouple the channel coding problem from control, and can thus greatly simplify the practical implementation of implicit communication.