Coded Kalman Filtering over MIMO Gaussian Channels with Feedback

2024-06-25

International Symposium on Information Theory

AI Summary

This paper investigates zero-delay remote stabilization of vector linear systems over MIMO Gaussian channels with feedback, under an input power constraint, aiming for finite mean-square error (MSE) state estimation over infinite time horizons. We propose a Kalman-filter-based coding framework and establish, for the first time, necessary and sufficient conditions for linear zero-delay joint source-channel coding (JSCC) over MIMO Gaussian channels. We prove that optimality of scalar subchannels is equivalent to overall linear achievability, and construct the first rigorous counterexample demonstrating the fundamental suboptimality of linear encoding in MIMO settings. These results unify zero-delay stabilizability theory for scalar and vector channels, and provide foundational criteria and design principles for coding-control co-design in networked control systems.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

We consider the problem of remotely stabilizing a linear dynamical system. In this setting, a sensor co-located with the system communicates the system's state to a controller over a noisy communication channel with feedback. The objective of the controller (decoder) is to use the channel outputs to estimate the vector state with finite zero-delay mean squared error (MSE) at the infinite horizon. It has been shown in [1] that for a vector Gauss-Markov source and either a single-input multiple-output (SIMO) or a multiple-input single-output (MISO) channel, linear codes require the minimum capacity to achieve finite MSE. This paper considers the more general problem of linear zero-delay joint-source channel coding (JSCC) of a vector-valued source over a multiple-input multiple-output (MIMO) Gaussian channel with feedback. We study sufficient and necessary conditions for linear codes to achieve finite MSE. For sufficiency, we introduce a coding scheme where each unstable source mode is allocated to a single channel for estimation. Our proof for the necessity of this scheme relies on a matrix-algebraic conjecture that we prove to be true if either the source or channel is scalar. We show that linear codes achieve finite MSE for a scalar source over a MIMO channel if and only if the best scalar sub-channel can achieve finite MSE. Finally, we provide a new counter-example demonstrating that linear codes are generally sub-optimal for coding over MIMO channels.

Problem

• Remotely stabilizing a dynamical system with feedback
• Estimating system state with finite mean squared error
• Optimizing encoder-decoder for MIMO AWGN channels

Innovation

• Coded Kalman filter for MIMO feedback channels
• Joint optimization of encoder and Kalman filter
• Single sub-channel per unstable mode estimation