Dynamic Quantizer

Overview

In networked control systems, control signals and sensor data must be transmitted over digital communication channels with limited capacity. Before transmission, continuous-valued signals are quantized into a finite set of discrete levels. Coarser quantization reduces the required data rate but introduces larger quantization errors, which can degrade control performance or even destabilize the system. Managing this trade-off between data compression and control performance is a central challenge in networked control.

A dynamic quantizer addresses this challenge by incorporating internal feedback into the quantization process. Unlike a static quantizer that simply rounds each sample independently, a dynamic quantizer — such as a delta-sigma modulator — uses a linear filter and feedback of the quantizer output to shape the quantization error over time. This noise-shaping mechanism pushes the quantization error energy away from the frequency bands critical for control, enabling effective signal compression with minimal impact on control performance.

Our research focuses on the comprehensive design of dynamic quantizers under communication rate constraints. When the number of quantization levels is limited, the quantizer's filter parameters, quantization interval (step size), and the number of output levels are tightly coupled: changing one affects the feasibility and optimality of the others. We established the fundamental relationship between the minimum quantization interval, the number of levels, the signal range, and the internal signal amplification of the filter — characterized by the l1-norm of its impulse response. This relationship reveals that the filter must be designed so that its internal dynamics fit within the budget of available quantization levels; otherwise, the internal state may saturate and performance breaks down.

Based on this analysis, we developed a design method that simultaneously optimizes all quantizer parameters using a combination of invariant set analysis (via LMI conditions) and particle swarm optimization (PSO). This hybrid approach yields high-quality initial solutions through control-theoretic analysis and refines them through global search.

Our work on dynamic quantizers has been extended to several directions, including MIMO systems, unilateral control under one-way communication, periodically time-varying quantizer structures, and cross-domain applications such as image color quantization for DLP projectors and pre-filter design for AD/DA conversion.

For a detailed explanation, see the blog article: Compressing Control Signals by Design: Dynamic Quantizer Optimization for Level-Constrained Communication (Qiita)

Key paper: H. Okajima, K. Sawada and N. Matsunaga, "Dynamic Quantizer Design Under Communication Rate Constraints," IEEE Transactions on Automatic Control, Vol. 61, No. 10, pp. 3190–3196, 2016. (IEEE Xplore) (PDF)

MATLAB code: GitHub