# Min-Max Design of Feedback Quantizers for Netorwked Control Systems

## TL;DR

This paper introduces a novel feedback quantizer design for networked control systems that avoids overloading by using an $l_{
abla}$ norm constraint, optimizing the error feedback filter to minimize bit usage or output error.

## Contribution

It proposes an overloading-free feedback quantizer based on a Delta-Sigma modulator with an $l_{
abla}$ norm constraint, optimizing the filter design for minimal bits or error.

## Key findings

- Designed an overloading-free feedback quantizer for control systems.
- Derived minimum bit requirements for given error constraints.
- Validated the approach with numerical examples.

## Abstract

In a networked control system, quantization is inevitable to transmit control and measurement signals. While uniform quantizers are often used in practical systems, the overloading, which is due to the limitation on the number of bits in the quantizer, may significantly degrade the control performance. In this paper, we design an overloading-free feedback quantizer based on a Delta-Sigma modulator,composed of an error feedback filter and a static quantizer. To guarantee no-overloading in the quantizer, we impose an $l_{\infty}$ norm constraint on the feedback signal in the quantizer. Then, for a prescribed $l_{\infty}$ norm constraint on the error at the system output induced by the quantizer, we design the error feedback filter that requires the minimum number of bits that achieves the constraint. Next, for a fixed number of bits for the quantizer, we investigate the achievable minimum $l_{\infty}$ norm of the error at the system output with an overloading-free quantizer. Numerical examples are provided to validate our analysis and synthesis.