Coding for Control and Connections with Information Theory
This chapter introduces policies and actions regarding the selection of quantizers and controllers in networked control. It exhibits the important differences between the real-time communication formulation and the traditional Shannon theoretic setup whic
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Coding for Control and Connections with Information Theory
5.1 Introduction In this chapter, we study quantizers and encoders. The notion of a quantizer was introduced formally in Sect. 4.7. We will discuss further properties of quantizers, and their performance, and bring a perspective where we view quantizers as decision variables. The chapter is also concerned with the derivation of fundamental bounds in connection with stabilizability of a linear system over a communication channel. The ideas and results presented here will be used throughout the rest of the book. The chapter introduces the notion of real-time coding and defines the selection of a quantizer function as a decision problem in Sect. 5.2. In Sect. 5.3, a review of basic operational definitions in information theory is presented. Section 5.4 highlights the subtle differences between the performances of optimal coding for a single random variable and the limit performance of a sequence of optimal codes for blocks of random variables as the block length becomes unbounded (a common view adopted in Shannon’s formulation of information theory). Performance bounds of quantizers for causal and noncausal coding of unstable processes are studied in Sect. 5.5. Fundamental lower bounds for stabilization are presented in Sect. 5.6.
5.2 Quantization and Real-Time Coding 5.2.1 Real-Time Coding In real-time applications such as remote control of time-sensitive processes, causality in encoding and decoding is a natural limitation. As discussed earlier, there is a natural causal ordering of events in a controlled process, consisting of measurement, estimation, and actuation. All these events need to take place in real time and not
S. Y¨uksel and T. Bas¸ar, Stochastic Networked Control Systems, Systems & Control: Foundations & Applications, DOI 10.1007/978-1-4614-7085-4 5, © Springer Science+Business Media New York 2013
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5 Coding for Control and Connections with Information Theory Bin size
−(K/2)Δ
(K/2)Δ
Overflow bin
Overflow bin
Fig. 5.1 A modified uniform quantizer. There is a single overflow bin
with significant delay. In the following, we provide a characterization of information structures in such systems and obtain fundamental bounds on transmission rates for stabilization of such systems. Essential in communication problems is the embedding of information into a finite set, possibly with loss of information. This is done through quantization, which is a mapping from a larger alphabet to a smaller alphabet. Before proceeding further, recall from Definition 4.7.1 that a quantizer Q is a (Borel-measurable) function from a topological space X to a finite index set M := {1, 2, . . . , M }. We define the bins or cells in such a quantizer as the sets Bi = {x ∈ X : Q(x) = i},
i ∈ M.
Thus, a quantizer partitions its domain set. Occasionally, quantization bins are represented by reconstruction values. Traditionally, in source-coding theory, a quantizer is also characterized by a collection of reconstruction values in addition to a set of partitions.
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