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1. A Lower-Bound for Variable-Length Source Coding in Linear-Quadratic-Gaussian Control With Shared Randomness

   Cuvelier, T.C.; Tanaka, T.; & Heath, R.W. (June 2022, IEEE control systems letters)

   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. 

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2. Finite-horizon covariance control of linear time-varying systems

   https://doi.org/10.1109/CDC.2017.8264189  

   Goldshtein, Maxim; Tsiotras, Panagiotis (December 2017, Conference on Decision and Control)

   We consider the problem of finite-horizon optimal control of a discrete linear time-varying system subject to a stochastic disturbance and fully observable state. The initial state of the system is drawn from a known Gaussian distribution, and the final state distribution is required to reach a given target Gaussian distribution, while minimizing the expected value of the control effort. We derive the linear optimal control policy by first presenting an efficient solution for the diffusion-less case, and we then solve the case with diffusion by reformulating the system as a superposition of diffusion-less systems. We show that the resulting solution coincides with a LQG problem with particular terminal cost weight matrix. 

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3. Time-invariant prefix-free source coding for MIMO LQG control

   Cuvelier, T.C.; Tanaka, T.; & Heath, R.W. (November 2022, IEEE International Mediterranean Conference on Communications and Networking (MeditCom))

   In this work we consider discrete-time multiple-input multiple-output (MIMO) linear-quadratic-Gaussian (LQG) control where the feedback consists of variable length binary codewords. To simplify the decoder architecture, we enforce a strict prefix constraint on the codewords. We develop a data compression architecture that provably achieves a near minimum time-average expected bitrate for a fixed constraint on the LQG performance. The architecture conforms to the strict prefix constraint and does not require time-varying lossless source coding, in contrast to the prior art. 

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4. Heavy-traffic queue length behavior in a switch under Markovian arrivals

   https://doi.org/10.1017/apr.2023.60  

   Mou, Shancong; Maguluri, Siva Theja (March 2024, Advances in Applied Probability)

   This paper studies the input-queued switch operating under the MaxWeight algorithm when the arrivals are according to a Markovian process. We exactly characterize the heavy-traffic scaled mean sum queue length in the heavy-traffic limit, and show that it is within a factor of less than 2 from a universal lower bound. Moreover, we obtain lower and upper bounds that are applicable in all traffic regimes and become tight in the heavy-traffic regime. We obtain these results by generalizing the drift method recently developed for the case of independent and identically distributed arrivals to the case of Markovian arrivals. We illustrate this generalization by first obtaining the heavy-traffic mean queue length and its distribution in a single-server queue under Markovian arrivals and then applying it to the case of an input-queued switch. The key idea is to exploit the geometric mixing of finite-state Markov chains, and to work with a time horizon that is chosen so that the error due to mixing depends on the heavy-traffic parameter. 

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5. Online variable-length source coding for minimum bitrate LQG control

   Cuvelier, T. C.; Tanaka, T.; & Heath, R. W. (April 2023, 2023 IEEE Conference on Decision and Control)

   We propose an adaptive coding approach to achieve linear-quadratic-Gaussian (LQG) control with near- minimum bitrate prefix-free feedback. Our approach combines a recent analysis of a quantizer design for minimum rate LQG control with work on universal lossless source coding for sources on countable alphabets. In the aforementioned quantizer design, it was established that the quantizer outputs are an asymp- totically stationary, ergodic process. To enable LQG control with provably near-minimum bitrate, the quantizer outputs must be encoded into binary codewords efficiently. This is possible given knowledge of the probability distributions of the quantizer outputs, or of their limiting distribution. Obtaining such knowledge is challenging; the distributions do not readily admit closed form descriptions. This motivates the application of universal source coding. Our main theoretical contribution in this work is a proof that (after an invertible transformation), the quantizer outputs are random variables that fall within an exponential or power-law envelope class (depending on the plant dimension). Using ideas from universal coding on envelope classes, we develop a practical, zero-delay version of these algorithms that operates with fixed precision arithmetic. We evaluate the performance of this algorithm numerically, and demonstrate competitive results with respect to fundamental tradeoffs between bitrate and LQG control performance. 

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