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arXiv · 1604.01227

Rate of Prefix-free Codes in LQG Control Systems

Takashi Tanaka·Karl Henrik Johansson·Tobias Oechtering·Henrik Sandberg·Mikael Skoglund
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Abstract

In this paper, we consider a discrete time linear quadratic Gaussian (LQG) control problem in which state information of the plant is encoded in a variable-length binary codeword at every time step, and a control input is determined based on the codewords generated in the past. We derive a lower bound of the rate achievable by the class of prefix-free codes attaining the required LQG control performance. This lower bound coincides with the infimum of a certain directed information expression, and is computable by semidefinite programming (SDP). Based on a technique by Silva et al., we also provide an upper bound of the best achievable rate by constructing a controller equipped with a uniform quantizer with subtractive dither and Shannon-Fano coding. The gap between the obtained lower and upper bounds is less than $0.754r+1$ bits per time step regardless of the required LQG control performance, where $r$ is the rank of a signal-to-noise ratio matrix obtained by SDP, which is no greater than the dimension of the state.

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Takashi Tanaka, Karl Henrik Johansson, Tobias Oechtering, Henrik Sandberg, Mikael Skoglund. 2016-04-05. Rate of Prefix-free Codes in LQG Control Systems. https://doi.org/10.1109/isit.2016.7541729

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