A Lower-bound for Variable-length Source Coding in Linear-Quadratic-Gaussian Control with Shared Randomness

In this letter, we consider a Linear Quadratic Gaussian (LQG) control system where feedback occurs over a noiseless binary channel and derive lower bounds on the minimum communication cost (quantified via the channel bitrate) required to attain a given control performance. We assume that at every ti...

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Published in:arXiv.org
Main Authors: Cuvelier, Travis C, Tanaka, Takashi, Heath, Robert W
Format: Paper
Language:English
Published: Ithaca Cornell University Library, arXiv.org 02.06.2022
Subjects:
Coders
Codes
Coding
Control systems
Counters
Feedback control
Linear quadratic Gaussian control
Lower bounds
Randomness
ISSN:2331-8422
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Abstract In this letter, we consider a Linear Quadratic Gaussian (LQG) control system where feedback occurs over a noiseless binary channel and derive lower bounds on the minimum communication cost (quantified via the channel bitrate) required to attain a given control performance. We assume that at every time step an encoder can convey a packet containing a variable number of bits over the channel to a decoder at the controller. Our system model provides for the possibility that the encoder and decoder have shared randomness, as is the case in systems using dithered quantizers. We define two extremal prefix-free requirements that may be imposed on the message packets; such constraints are useful in that they allow the decoder, and potentially other agents to uniquely identify the end of a transmission in an online fashion. We then derive a lower bound on the rate of prefix-free coding in terms of directed information; in particular we show that a previously known bound still holds in the case with shared randomness. We generalize the bound for when prefix constraints are relaxed, and conclude with a rate-distortion formulation.
AbstractList In this letter, we consider a Linear Quadratic Gaussian (LQG) control system where feedback occurs over a noiseless binary channel and derive lower bounds on the minimum communication cost (quantified via the channel bitrate) required to attain a given control performance. We assume that at every time step an encoder can convey a packet containing a variable number of bits over the channel to a decoder at the controller. Our system model provides for the possibility that the encoder and decoder have shared randomness, as is the case in systems using dithered quantizers. We define two extremal prefix-free requirements that may be imposed on the message packets; such constraints are useful in that they allow the decoder, and potentially other agents to uniquely identify the end of a transmission in an online fashion. We then derive a lower bound on the rate of prefix-free coding in terms of directed information; in particular we show that a previously known bound still holds in the case with shared randomness. We generalize the bound for when prefix constraints are relaxed, and conclude with a rate-distortion formulation.
Author Cuvelier, Travis C
Tanaka, Takashi
Heath, Robert W
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SubjectTerms Coders
Codes
Coding
Control systems
Counters
Feedback control
Linear quadratic Gaussian control
Lower bounds
Randomness
Title A Lower-bound for Variable-length Source Coding in Linear-Quadratic-Gaussian Control with Shared Randomness
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