Rate-cost tradeoffs in control. Part I: lower bounds

Consider a distributed 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 limsup of the expected cost b. We obtain a lower bound on a certain cost function, which quantifies the minimum mutual information between the channel input and output, given the past, that is compatible with a target LQR cost. The rate-cost function has operational significance in multiple scenarios of interest: among other, it allows us to lower bound the minimum communication rate for fixed and variable length quantization, and for control over a noisy channel. Our results extend and generalize an earlier explicit expression, due to Tatikonda el al., for the scalar Gaussian case to the vector, non-Gaussian, and partially observed one. The bound applies as long as the system noise has a probability density function. Apart from standard dynamic programming arguments, our proof technique leverages the Shannon lower bound on the rate-distortion function and proposes new estimates for information measures of linear combinations of random vectors.