Observability via Communication Channels

In this section, we consider a nonlinear, uncertain, discrete-time dynamic system of the form:

In our observability problem, a sensor measures the state x(t)and is connected to the controller that is at the remote location. Moreover, the only way of communicating information from the sensor to that remote location is via a digital communication channel that carries one discrete-valued symbol h(jT )at time jT , selected from a coding alphabet H of size l. Here T > = 1is a given integer period, and j =1, 2, 3,....

This restricted number l of codewords h(jT )is determined by the transmission data rate of the channel. For example, if µ is the number of bits that our channel can transmit, then l =2µ is the number of admissible codewords. We assume that the channel is a perfect noiseless channel and that there is no time delay. Let R > = 0 be a given constant. We consider the class CR of such channels with any period T satisfying the following transmission data rate constraint:

The rate R =0 corresponds to the case when the channel does not transmit data at all.

We consider the problem of estimation of the state x(t) via a digital communication channel with a bit-rate constraint. Our state estimator consists of two components. The first component is developed at the measurement location by taking the measured state x(·) and coding to the codeword h(jT ). This component will be called a “coder.” Then the codeword h(jT )is transmitted via a limited capacity communication channel to the second component, which is called a “decoder.” The second component developed at the remote location takes the codeword h(jT ) and produces the estimated states xˆ((j - 1)T +1),..., xˆ(jT - 1),xˆ(jT ). This situation is illustrated in Fig. 2.1 (where y = x now).

The coder and the decoder are of the following forms, respectively:

 

Furthermore, ║.║ denotes the standard Euclidean vector norm:

Definition 2.2.1. The system (2.2.1) is said to be observable in the communication channel class CR if for any e> 0, a period T > = 1 and a coder–decoder pair (2.2.3), (2.2.4) with a coding alphabet of size l satisfying the constraint (2.2.2) exist such that

for any solution of (2.2.1).