In this dissertation, we are interested in channels that are discrete-time and packet-based in nature. At every discrete time step, a packet of data is created and transmitted over the channel. The receiver tries to estimate the data that was transmitted using the information it receives over the channel. A general model for a communication channel used in, for instance information theory, is as a probability matrix [39]. If the channel supports m input symbols and n output symbols, then the channel is completely described by an m×n matrix whose (i, j)-th element denotes the probability that the channel will output the j-th output symbol given that the i-th symbol was input1. While this characterization is appropriate for many information-theoretic purposes, for the
purpose of analyzing the effect of communication links on estimation and control, this model can introduce too many details that can cloud the picture. An alternative is to summarize the effect of the communication link (and the concomitant physical layer information transmission and reception mechanism) in terms of various metrics or effects that the communication channels introduce into the system. Some of these effects are:
1. Time delay: Before data is transmitted over a channel, it is usually buffered, quantized and coded. All these operations consume some time. After a propagation delay, the data is decoded at the receiver end. If the data is not received properly, the communication protocol may specify a re-transmission of the data. Thus, by the time the information is used by the receiver, a delay has been introduced. Usually this delay is random and the probability distribution of the delay may change over time.
2. Data loss: In most communication protocols, if the receiver does not receive a data packet within a specified time limit, the packet is assumed to be lost. This data loss can happen due to a variety of reasons. For instance, in a shared multiple access medium such as the wireless channel, simultaneous transmission by two transmitters may lead to loss of data from both the transmitters. If transmission occurs over a network of communication channels, overflow of buffers can also lead to packet loss. Finally, if the data is extremely degraded by the time it reaches the receiver, the communication protocol may call for the packet to be dropped. 3. Quantization: Many communication channels and protocols are digital in nature. Thus, any
data that needs to be transmitted over a channel needs to be quantized. The number of bits that can be transmitted at every time step is usually upper-bounded.
1
4. Data corruption: The signal that the receiver regenerates may not be identical to the signal that the transmitter wished to communicate due to noise or attenuation introduced by the channel. While most communication protocols specify error detection and error correction codes, they might not be sufficient to provide immunity to particularly bad channel use instances. Modeling a channel in terms of these metrics rather than in terms of a physical layer model that specifies the coding, transmission method, channel structure and so on simplifies the analysis and design of an estimation and control loop. It also separates the communication and control design parts of the problem, thus leading to a layered design approach. The control engineer can then demand a certain quality of service from the communication layer in terms of such metrics. In turn, the communication engineer can specify the range of values that the metrics can assume and the control goals are then computed based on these values. This notion of a layered architecture, even though sub-optimal in general, leads to simplicity and tractability in design and is often credited with the success of the Internet (due to the OSI model), serial computation (due to the von-Neumann bridge) and so on [72].
In this dissertation, for the most part, we will concentrate on the stochastic packet dropping effect of the channel. The time line for the operation of such a link is as follows. At every time step k,
• A packet containing some function of the information that the transmitter has access to is created at the transmitter side of the link.
• The packet is sent across the link.
• At time step k+ 1, the packet is either received without error, or dropped, probabilistically. The information set that the transmitter has access to may have to satisfy some constraints. As an example, there is usually a limit on the memory at the transmitter that limits the amount of data that can be stored. We also impose the constraint that the function communicated over the link should be a finite vector. The packet dropping is a random process. If the packet drops are independent from one time step to the next and occur with the same probability at every time step, they are said to have occurred in an independent and identically distributed (i.i.d.) fashion. In some channels, drops are correlated from one time step to the next. This correlation can be captured by a more sophisticated model such as a Markov chain. In the classic Gilbert-Elliot channel model [56, 67] shown in Figure 3.6, the channel is assumed to exist in two states. The ‘bad’ state corresponds to the channel dropping packets and the ‘good’ state corresponds to a successful transmission. The channel switches between the two states according to a Markov chain. This model can capture the effect displayed, e.g., by a wireless channel in which packet drops occur in a bursty fashion. More sophisticated models comprising of multiple Markov states, each corresponding to a different
Figure 3.6: The classical Gilbert-Elliot channel model.
probability of packet drop are also available (see, e.g., [196, 207]). In any case, the packet dropping is a random process. We refer to individual (i.e. deterministic) realizations of this random process as packet drop sequences. A packet drop sequence is a binary sequence {λ(k)}∞k=0 in which λ(k) takes the value “received” if the link delivers the packet at time stepk, and “dropped” if the packet is dropped.
This model is referred to as thepacket erasure modelof the channel. We assume sufficient bits per data packet and a high enough data rate so that quantization error is negligible. This assumption merely means that a sufficient number of bits are available so that the effect of the quantization error is swamped by the effect of the process and the measurement noises. We do not assume an infinite number of bits, so that strategies based on interleaving of bits to transmit an infinite amount of data are not admissible2. We also assume that enough error-correction coding is done within the
packets so that the packets are either dropped or received without error. Finally, we will nominally consider the delays, if any, introduced by the channel to be less than one time step according to which the discrete-time dynamical process evolves. We will, however, revisit the issue of delays larger than one time step later in the chapter. We will, sometimes, also make an assumption of a one bit acknowledgement being available to the transmitter corresponding to whether or not the packet was received over the channel at the previous time step. Since the acknowledgement is just one-bit, it can be transmitted with a much higher reliability. We will assume that when an acknowledgement mechanism is present, acknowledgements are not dropped. Note that an implicit assumption in the model is that the receiver knows that a packet has been erased. Thus, it does not come up with a
2
Moreover, the optimal encoder / decoder algorithms we present will achieve the same performance as if we were indeed transmitting an infinite amount of data.
faulty estimate of the data transmitted over the link.