Bit Rate Conditions to Stabilize a Continuous-Time Scalar Linear System Based on Event Triggering

This technical note is mainly concerned with bit rate conditions to stabilize a continuous-time scalar linear control system which transmits its feedback information through digital communication networks. It shows that when a packet is received, not only the bits inside that packet but also the receiving time instant of that packet carry information. The latter could carry a huge amount of information and can be obtained through simply receiving a packet without additional communication cost. Therefore it is an efficient way to encode information into the receiving time instant of a packet. The receiving time instant, however, cannot be accurately controlled by the sensor/sampler which can determine only sampling time instants. The difference between the sampling and receiving time instants of a packet is up to two delays, including the delay to process and encode the packet and the network delay to transmit that packet. These delays cannot be precisely known and have negative effects on the amount of information carried by receiving time instants. Under the boundedness assumption of these delays, this technical note provides some bit rate conditions to guarantee the system's input-to-state stability by making a good use of receiving time instants. Through event-triggering schemes, we realize these conditions which may require much lower stabilizing bit rates than periodic sampling schemes. Simulations are done to confirm the effectiveness of the obtained stabilizing bit rate conditions.

I. Introduction

Nowadays networked control systems (NCS) have caught more and more attention [1] . In such systems, feedback information is transmitted through digital communication networks, which break the perfect feedback condition due to many issues, such as quantization error, network delay and dropouts. The loss of perfect feedback information can degrade the performance of control systems and even destabilize them. To quantitatively analyze such issues is one active research area [1], [2]. Among these issues, this technical note is mainly concerned with quantization error and network delay. Quantization error results from the digital nature of feedback communication networks which quantize all feedback signals into a finite number of bits before transmission. Although a higher bandwidth of the feedback network, denoted as (bits per unit time), yields smaller quantization error, it occupies more network resources. So a balance regarding is necessary. Here we are interested in finding the minimum to guarantee the stability of the concerned system. This minimum question is well studied and often referred to as the data rate theorem question [2]. Before reviewing the related literature, we describe the main quantization procedure in Fig. 1. Fig. 1.

A typical architecture of a quantized continuous-time linear system.