LQG control and linear policies for noisy communication links with synchronized side information at the decoder

Abstract

We consider a multidimensional closed-loop control problem where a stochastic linear time-invariant plant is connected via a communication system that includes an encoder, vector additive Gaussian noise (AGN) channel, decoder to a controller. For this setup, we first examine the impact of the communication link when causal side information (CSI) that is correlated to the plant is (possibly) allowed at the decoder and the performance criterion is the classical linear quadratic (LQ) cost. The CSI is modeled as a time-invariant vector-valued Gaussian process and when available at the decoder, it happens concurrently to the signal obtained from the vector AGN channel. Under the assumption that our encoder is linear, we leverage the separation principle of control and estimation, to obtain the following new results. (i) For multivariate Gaussian processes, we completely characterize a lower bound on the communication cost term of the optimal LQ Gaussian (LQG) cost in the form of a convex optimization problem and identify the best linear policies that achieve this bound. (ii) For scalar-valued Gaussian processes, we derive an analytical solution to the lower bound of the optimal communication cost term and show that for linear policies, CSI at the encoder does not improve the specific bound. (iii) For scalar-valued Gaussian processes, we also derive a linear time-invariant low delay joint source-channel coding (JSCC) scheme without bandwidth constraints, when CSI is available at the encoder, which demonstrates the case at which the specific lower bound communication cost expression is achievable. Further, we give a suboptimal linear time-invariant JSCC scheme when CSI is available only at the decoder. We supplement our results with various simulation studies.