#### Sequential Source Coding for Stochastic Systems Subject to Finite Rate Constraints

##### Photios A. Stavrou, Mikael Skoglund, Takashi Tanaka

In this paper, we apply a sequential source coding framework to analyze fundamental performance limitations of stochastic control systems subject to feedback data-rate constraints. We first show that the characterization of the rate-distortion region obtained using sequential codes with a per-time average distortion constraint can be simplified for spatially IID $m$-order Markov sources and generalized to total (across time) average distortion constraints. Furthermore, we show that the corresponding minimum total-rate achieved by sequential codes is precisely the nonanticipative rate distortion function (NRDF) also known as sequential RDF. We use our findings to derive {\it analytical} non-asymptotic, finite-dimensional bounds on the minimum achievable performance in two control-related application examples. (a) A parallel time-varying Gauss-Markov process with identically distributed spatial components that is quantized and transmitted with an instantaneous data-rate, obtained though the solution of a dynamic reverse-waterfilling algorithm, through a noiseless channel to a minimum mean-squared error (MMSE) decoder. For this example, we derive non-asymptotic lower and upper bounds (per dimension) on the minimum achievable total-rate. (b) A time-varying quantized LQG closed-loop control system, with identically distributed spatial components and with a random resource allocation. For this example, we apply the results obtained from the quantized state estimation problem to derive analogous bounds on the non-asymptotic total-cost of control.

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