2 Notations and Basic Assumptions

Abstract

In this chapter, we will adopt the analog erasure model to describe the communication channel present inside a control loop. This model, also referred to as the packet erasure or packet loss model, can be described as follows [8]. The channel operates in discrete time steps. At every time step, the channel accepts as input a finite dimensional real vector r(k). The value of the output of the channel y(k) is chosen according to an erasure process. At every time step, the erasure process assumes either the value T or the value R. If the value at time k is T , y(k + 1) = r(k) and a successful transmission is said to have occurred. Otherwise, y(k + 1) = φ and an erasure event, or a packet loss, is said to have occurred at time k. The symbol φ denotes that the receiver does not receive any data; however, the receiver is aware that an erasure event has occurred at that time. Note that we have assumed that the channel introduces a constant delay of one time step. The analog erasure model aims to capture the data loss effect due to a communication channel. Due to effects such as interference and fading in wireless channels, congestion in shared networks, or even overload and interrupts at a micro-controller, various parts of a control loop between the sensor and controller, or the controller and actuator, may exhibit information loss. By considering the idealization in which every successful transmission results in the communication of a real vector of a bounded dimension, such situations can be modeled using an analog erasure representation. While an analog erasure model has an infinite capacity in an information theoretic sense, it is often a useful representation for the cases when the communication protocols allow for large data packets to be transmitted at every time step. For instance, the minimum size of an ethernet data packet is 72 bytes. This is much more space for carrying information than usually required inside a control loop. If the data packets allow for transmission of control and sensing data to a high fidelity, the quantization effects are often ignored and an analog erasure model adopted. Various descriptions of the erasure process are possible. The following two models are the most popular: