# LQG Control with Minimum Directed Information: Semidefinite Programming Approach

arXiv: https://arxiv.org/abs/1510.04214 · HTML: https://mlsanity.com/paper/1510.04214/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Oct 2015.

## Built on by (most cited first, 8 of 8)

- [Computing Complexity-aware Plans Using Kolmogorov Complexity](https://mlsanity.com/paper/2109.10303/) (2021)
- [Time-invariant Prefix Coding for LQG Control](https://mlsanity.com/paper/2204.00588/) (2022)
- [Mutual Information Optimal Control of Discrete-Time Linear Systems](https://mlsanity.com/paper/2507.04712/) (2025)
- [Entropy-Regularized Partially Observed Markov Decision Processes](https://mlsanity.com/paper/2112.12255/) (2021)
- [Simultaneous Perception-Action Design via Invariant Finite Belief Sets](https://mlsanity.com/paper/2109.05073/) (2021)
- [Online variable-length source coding for minimum bitrate LQG control](https://mlsanity.com/paper/2304.00593/) (2023)
- [Reinforcement Learning for Jointly Optimal Coding and Control Policies for a Controlled Markovian System over a Communication Channel](https://mlsanity.com/paper/2411.13884/) (2024)
- [Deep Reinforcement Learning-Empowered Wireless Sensor Networking for 6G Closed-Loop Controls](https://mlsanity.com/paper/2607.08272/) (2026)

## Abstract (arXiv)

We consider a discrete-time Linear-Quadratic-Gaussian (LQG) control problem in which Massey's directed information from the observed output of the plant to the control input is minimized while required control performance is attainable. This problem arises in several different contexts, including joint encoder and controller design for data-rate minimization in networked control systems. We show that the optimal control law is a Linear-Gaussian randomized policy. We also identify the state space realization of the optimal policy, which can be synthesized by an efficient algorithm based on semidefinite programming. Our structural result indicates that the filter-controller separation principle from the LQG control theory, and the sensor-filter separation principle from the zero-delay rate-distortion theory for Gauss-Markov sources hold simultaneously in the considered problem. A connection to the data-rate theorem for mean-square stability by Nair and Evans is also established.