Implicit Communication in Linear Quadratic Gaussian Control Systems

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.