We consider a discrete-Time linear quadratic Gaussian networked control setting where the (full information) observer and controller are separated by a fixed-rate noiseless channel. The minimal rate required to stabilize such a system has been well studied. However, for a given fixed rate, how to quantize the states so as to optimize performance is an open question of great theoretical and practical significance. We concentrate on minimizing the control cost for first-order scalar systems. To that end, we use the Lloyd-Max algorithm and leverage properties of logarithmically-concave functions to construct the optimal quantizer that greedily minimizes the cost at every time instant. By connecting the globally optimal scheme to the problem of scalar successive refinement, we argue that its gain over the proposed greedy algorithm is negligible. This is significant since the globally optimal scheme is often computationally intractable. All the results are proven for the more general case of disturbances with logarithmically-concave distributions.
Bibliographical noteKAUST Repository Item: Exported on 2022-06-28
Acknowledgements: The work of A. Khina has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 708932. The work of Y. Nakahira was funded by grants from AFOSR and NSF, and gifts from Cisco, Huawei, and Google. The work of Y. Su was supported in part by NSF through AitF-1637598. The work of B. Hassibi was supported in part by the NSF under grants CNS-0932428, CCF-1018927, CCF-1423663 and CCF-1409204, by a grant from Qualcomm Inc., by NASA’s Jet Propulsion Laboratory through the President and Director’s Fund, by King Abdulaziz University, and by King Abdullah University of Science and Technology.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.