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Published November 2019 | Submitted
Journal Article Open

Rate-Cost Tradeoffs in Control

Abstract

Consider a control problem with a communication channel connecting the observer of a linear stochastic system to the controller. The goal of the controller is to minimize a quadratic cost function in the state variables and control signal, known as the linear quadratic regulator (LQR). We study the fundamental tradeoff between the communication rate r bits/sec and the expected cost b. We obtain a lower bound on a certain rate-cost function, which quantifies the minimum directed mutual information between the channel input and output that is compatible with a target LQR cost. The rate-cost function has operational significance in multiple scenarios of interest: among others, it allows us to lower-bound the minimum communication rate for fixed and variable length quantization, and for control over noisy channels. We derive an explicit lower bound to the rate-cost function, which applies to the vector, non-Gaussian, and partially observed systems, thereby extending and generalizing an earlier explicit expression for the scalar Gaussian system, due to Tatikonda el al. [2]. The bound applies as long as the differential entropy of the system noise is not −∞ . It can be closely approached by a simple lattice quantization scheme that only quantizes the innovation, that is, the difference between the controller's belief about the current state and the true state. Via a separation principle between control and communication, similar results hold for causal lossy compression of additive noise Markov sources. Apart from standard dynamic programming arguments, our technical approach leverages the Shannon lower bound, develops new estimates for data compression with coding memory, and uses some recent results on high resolution variablelength vector quantization to prove that the new converse bounds are tight.

Additional Information

© 2019 IEEE. Manuscript received October 18, 2017; revised August 17, 2018, August 27, 2018, and December 12, 2018; accepted January 11, 2019. Date of publication April 19, 2019; date of current version October 30, 2019. The work of V. Kostina was supported in part by the National Science Foundation (NSF) under Grant CCF-1566567 and Grant CCF-1751356. The work of B. Hassibi was supported in part by the NSF under Grant CNS-0932428, Grant CCF-1018927, Grant CCF-1423663, and Grant CCF-1409204, in part by the Grant from Qualcomm Inc., by NASA's Jet Propulsion Laboratory through the President and Director's Fund, and in part by the King Abdullah University of Science and Technology. This paper was presented in part at the 54th Annual Allerton Conference on Communication, Control and Computing, Monticello, IL, USA, October 2016 [1].

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