554 research outputs found
Error Correcting Codes for Distributed Control
The problem of stabilizing an unstable plant over a noisy communication link
is an increasingly important one that arises in applications of networked
control systems. Although the work of Schulman and Sahai over the past two
decades, and their development of the notions of "tree codes"\phantom{} and
"anytime capacity", provides the theoretical framework for studying such
problems, there has been scant practical progress in this area because explicit
constructions of tree codes with efficient encoding and decoding did not exist.
To stabilize an unstable plant driven by bounded noise over a noisy channel one
needs real-time encoding and real-time decoding and a reliability which
increases exponentially with decoding delay, which is what tree codes
guarantee. We prove that linear tree codes occur with high probability and, for
erasure channels, give an explicit construction with an expected decoding
complexity that is constant per time instant. We give novel sufficient
conditions on the rate and reliability required of the tree codes to stabilize
vector plants and argue that they are asymptotically tight. This work takes an
important step towards controlling plants over noisy channels, and we
demonstrate the efficacy of the method through several examples.Comment: 39 page
Minimum Bitrate Neuromorphic Encoding for Continuous-Time Gauss-Markov Processes
In this work, we study minimum data rate tracking of a dynamical system under
a neuromorphic event-based sensing paradigm. We begin by bridging the gap
between continuous-time (CT) system dynamics and information theory's causal
rate distortion theory. We motivate the use of non-singular source codes to
quantify bitrates in event-based sampling schemes. This permits an analysis of
minimum bitrate event-based tracking using tools already established in the
control and information theory literature. We derive novel, nontrivial lower
bounds to event-based sensing, and compare the lower bound with the performance
of well-known schemes in the established literature
Directed Data-Processing Inequalities for Systems with Feedback
We present novel data-processing inequalities relating the mutual information
and the directed information in systems with feedback. The internal blocks
within such systems are restricted only to be causal mappings, but are allowed
to be non-linear, stochastic and time varying. These blocks can for example
represent source encoders, decoders or even communication channels. Moreover,
the involved signals can be arbitrarily distributed. Our first main result
relates mutual and directed informations and can be interpreted as a law of
conservation of information flow. Our second main result is a pair of
data-processing inequalities (one the conditional version of the other) between
nested pairs of random sequences entirely within the closed loop. Our third
main result is introducing and characterizing the notion of in-the-loop (ITL)
transmission rate for channel coding scenarios in which the messages are
internal to the loop. Interestingly, in this case the conventional notions of
transmission rate associated with the entropy of the messages and of channel
capacity based on maximizing the mutual information between the messages and
the output turn out to be inadequate. Instead, as we show, the ITL transmission
rate is the unique notion of rate for which a channel code attains zero error
probability if and only if such ITL rate does not exceed the corresponding
directed information rate from messages to decoded messages. We apply our
data-processing inequalities to show that the supremum of achievable (in the
usual channel coding sense) ITL transmission rates is upper bounded by the
supremum of the directed information rate across the communication channel.
Moreover, we present an example in which this upper bound is attained. Finally,
...Comment: Submitted to Entropy. arXiv admin note: substantial text overlap with
arXiv:1301.642
Tracking and Control of Gauss-Markov Processes over Packet-Drop Channels with Acknowledgments
We consider the problem of tracking the state of Gauss–Markov processes over rate-limited erasure-prone links. We concentrate first on the scenario in which several independent processes are seen by a single observer. The observer maps the processes into finite-rate packets that are sent over the erasure-prone links to a state estimator, and are acknowledged upon packet arrivals. The aim of the state estimator is to track the processes with zero delay and with minimum mean square error (MMSE). We show that, in the limit of many processes, greedy quantization with respect to the squared error distortion is optimal. That is, there is no tension between optimizing the MMSE of the process in the current time instant and that of future times. For the case of packet erasures with delayed acknowledgments, we connect the problem to that of compression with side information that is known at the observer and may be known at the state estimator—where the most recent packets serve as side information that may have been erased, and demonstrate that the loss due to a delay by one time unit is rather small. For the scenario where only one process is tracked by the observer–state estimator system, we further show that variable-length coding techniques are within a small gap of the many-process outer bound. We demonstrate the usefulness of the proposed approach for the simple setting of discrete-time scalar linear quadratic Gaussian control with a limited data-rate feedback that is susceptible to packet erasures
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