Reducing the LQG Cost with Minimal Communication

We study the linear quadratic Gaussian (LQG) control problem, in which the controller's observation of the system state is insufficient to attain a desired quadratic control cost. To achieve the desired LQG cost, we introduce a communication link from the observer (encoder) to the controller (decoder). We investigate the fundamental tradeoff between a desired LQG cost and the required communication (information) resources, measured with the conditional directed information. The optimization domain is all encoding-decoding policies, and our first result is the optimality of memoryless encoders that only transmit Gaussian measurements of the current system state. Additionally, it is shown that even if the controller's measurements are made available to the encoder it does not reduce the minimal directed information. The main result is a semidefinite programming (SDP) formulation for that optimization problem, which applies to general time-varying linear dynamical systems in the finite-horizon scenario, and to time-invariant systems at infinite horizon. For the latter scenario of time-invariant systems, we show that it is unnecessary to consider time-varying policies as a simple and optimal time-invariant policy can be directly constructed from the SDP solution. Our results extend a seminal work by Tanaka et al. to the scenario where the controller has access to a noisy measurement of the system state. We demonstrate the viability of this extra resource by illustrating that even low-quality measurements may have a significant impact on the required communication resources.