Lower Bound of Networked Control with Multiple Sensors and One Controller And The Application to Tracking Gaussian-Markov

Jul 16, 2026

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Investigates the causal rate-distortion function in networked control systems with multiple encoders and a single decoder, an open problem at the intersection of information theory and control. Introduces a directed information lower bound tailored to the networked control setting, and proves that linear, independent encoders with linear decoders can optimize this bound for LQG plants under quadratic cost when the full plant state is observed by co-located sensors. Reduces the original infinite-dimensional optimization to finite dimensions, provides alternative sufficiency proofs for linear encoders, and derives semidefinite programming formulations for Gaussian-Markov sources with linear side information and singular noise.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages36

Reading Time1 h 31 min

Introduction

Networked LQG model and key metrics

Background and related work

arXiv :2607 .04 172v 1 [ cs .IT] 5 Jul 2026 Lower Bound of Networked Control with Multiple Sensors and One Controller And The Application to Tracking Gaussian-Markov Source Sijie Li, Takashi Tanaka, and Hyeji Kim Abstract This paper investigates the causal rate-distortion function for networked control systems with multiple encoders and a single decoder, a longstanding open problem in information and control theory. While previous work has explored the causal rate-distortion function for single-encoder and feedbackenabled networked settings, the case of networks without feedback remains unaddressed. We establish a novel directed information lower bound, the first derived for the networked control setting. We further demonstrate the optimality of linear, independent encoders and linear decoders for optimizing this lower bound for Linear Quadratic Gaussian (LQG) plant and quadratic cost, with the condition that the full plant state is observed when sensors are sitting together. 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 the single encoder and single decoder setting with side information, extending prior results in the literature. We present Semidefinite Programming formulations for the causal rate distortion function of Gaussian-Markov sources with linear side information and the singular noise matrix. Index Terms Rate-Distortion tradeoff, Gaussian-Markov source, Linear quadratic Gaussian. Funding ackowledgement Sijie Li and Hyeji Kim are with the University of Texas at Austin, Austin, TX 78712 USA (email: [sijieli@utexas.edu](sijieli@utexas.edu); [hyeji@utexas.edu](hyeji@utexas.edu)). Takashi Tanaka is with the Purdue University, West Lafayette, IN 47907 USA (e-mail: [tanaka16@purdue.edu](tanaka16@purdue.edu)). July 7, 2026 DRAFT I. INTRODUCTION Control with communication constraints has been widely studied in the past decades. The tradeoff between data rate and control performance is an interesting problem, as it lies at the intersection of information theory and control theory. The solution typically leads to the design of the encoder and the controller, as the goal is to transmit sufficient information so that the controller can output control signals that satisfy the distortion constraints. From this perspective, the problem can be viewed as a causal compression for the function computation problem, which leads to solving the corresponding causal rate-distortion function. For the single encoder and single decoder case, the problem has some nice solutions [1]–[5] . However, this type of problem is generally challenging in a setting with multiple encoders and a single decoder as Figure 1 . Even for the i.i.d. binary sources, the rate region of computing the modulo sum [6] is generally unknown. The optimal rate region is only known for some of the source distributions [7] . Fig. 1: Linear Quadratic Gaussian model with k-linear sensors. This paper considers a networked control system, as depicted in Figure 1 . Specifically, the system has an LQG plant, k sensors/encoders, and one decoder. The k encoders receive signals of the plant’s output and transmit a message independently through a lossless channel with the prefix code. The decoder decodes these messages and outputs a control signal to the plant, which evolves with the plant’s output and the control signal. Two critical metrics are rates and the control cost. Rates measure the expected average bit length of the encoders’ codewords, and the control cost measures the price of control signals. The detailed definitions can be found in Section II. Characterizing the causal rate-distortion function is a fundamental and intriguing problem. The lower bound and the achievable schemes have been widely studied for cases with one encoder Jul

FAQs

What problem does the paper address in networked control systems?

It studies the causal rate-distortion function for networked control with multiple encoders and one decoder, focusing on how to balance communication rates against control performance.

What is the main contribution regarding the lower bound?

The paper establishes a novel directed-information lower bound specifically for the networked control setting, which is the first derived for this configuration.

Under what conditions are linear encoders and decoders optimal for the LQG setting?

The paper shows optimality of linear, independent encoders and linear decoders for optimizing the lower bound for an LQG plant with quadratic cost when the full plant state is observed by sensors sitting together.

Jul 16, 2026

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