Lower Bound of Networked Linear Quadratic Gaussian Plant with Two Linear Sensors and One Controller

This paper investigates the causal rate-distortion function for networked Linear Quadratic Gaussian (LQG) control systems with two encoders and a single decoder, a longstanding open problem in information/control theory. While previous work has explored the causal rate-distortion function for single-encoder and feedback-enabled networked settings, the case of networks without feedback remains unaddressed. We establish a novel directed information lower bound, the first derived for the networked LQG setting. We further demonstrate the optimality of linear, independent encoders and linear decoders for optimizing this lower bound. By reducing the original infinite-dimensional optimization problem to a finite-dimensional one, our approach simplifies the analysis. Additionally, our directed information lower bound provides an alternate proof for the sufficiency of linear encoders in point-to-point settings, both with and without side information, extending prior results in the literature.