Optimal Sampling Rate and Quantization for Networked Control Systems

Sampling and quantization in Networked Control Systems (NCS) are addressed in this paper. The NCS studied in this paper consists of a continuous time plant, a sensor network and a discrete time controller. The amount of network induces delay to the control system is a function of the sampling rate of the control system. An upper bound for the delay will be found using network calculus which is a theory for deterministic queuing. From the control side of view, we consider a delayed sampled data system and propose a method to study the effects of sampling and delay in a unified framework. We define the quality-of-control in the sense of a ${\mathcal{H}_\infty }$ norm of the system and we propose a method to minimize this norm. According to our result, the optimal solution may not corresponds to the lowest sampling rate and it depends on the dynamics of the system and parameters of the communication link. We also extend the result to investigate the effect of quantization by proposing a new finite level quantizer. Similar to the sampling rate, we show that a larger number of allocated bits for quantization does not necessarily result in a better quality-of-control.

I. Introduction

In the past few years, control of networked systems has attracted a lot of attention. In NCSs, physical plants, communication links, and controllers are analyzed to achieve satisfactory quality-of-control. In these systems, the physical plant is connected to the controller via a shared network. Network calculus which is a theory of deterministic queuing was introduced in [1] to model various communication networks. There is much research devoted to developing network calculus [2], [3]. However, just a few papers addressed the network calculus for networked control systems. In [4] and [5] event-based controllers were designed for the discrete time case, and network calculus was used to model the network. In [6] the authors addressed a control problem in NCS and designed discrete time controllers for stabilization in the presence of network delays; they also used network calculus to model the communication network. By using network calculus, we can find an upper bound for the induced network delay in the control loop. To calculate the delay, we need to know an upper bound of input traffic to the network. We precisely know the input traffic to the network as we know the number of sensors and actuators, and the sampling frequency of the control system.