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.

翻译：本文研究了线性二次高斯控制系统中的隐式通信问题。我们证明，控制系统本身可作为隐式通信信道，使控制器能够通过其输入向观测系统状态的接收端传输信息。这种通信之所以称为隐式，是因为（i）无需显式通信信道；（ii）信息传输过程同时满足控制器的主要目标——将控制成本维持在指定水平。因此，控制性能与通信性能之间存在固有的权衡关系。该权衡关系通过隐式信道容量的概念形式化定义，该容量表征了在控制性能约束下可达到的最高可靠通信速率。我们在两种场景下探究了隐式信道容量：当控制器与接收端均能无噪观测系统状态时，信道容量存在闭式解；当双方均存在观测噪声时，我们建立了隐式信道容量的下界。值得注意的是，在无噪观测场景中，达到容量的输入策略遵循分离原理，使得控制任务与信道编码任务可独立处理而不损失最优性。虽然在有噪观测场景中该分离原理不再成立，但我们证明线性高斯输入策略仍能将信道编码问题与控制问题解耦，从而显著简化隐式通信的实际实现。