Coded Kalman Filtering over MIMO Gaussian Channels with Feedback

We consider the problem of remotely stabilizing a dynamical system. A sensor (encoder) co-located with the system communicates with a controller (decoder), whose goal is to stabilize the system, over a noisy communication channel with feedback. To accomplish this, the controller must estimate the system state with finite mean squared error (MSE). The vector-valued dynamical system state follows a Gauss-Markov law with additive control. The channel is a multiple-input multiple-output (MIMO) additive white Gaussian noise (AWGN) channel with feedback. For such a source, a linear encoder, and a MIMO AWGN channel, the minimal MSE decoder is a Kalman filter. The parameters of the Kalman filter and the linear encoder can be jointly optimized, under a power constraint at the channel input. We term the resulting encoder-decoder pair a coded Kalman filter. We establish sufficient and necessary conditions for the coded Kalman filter to achieve a finite MSE in the real-time estimation of the state. For sufficiency, we introduce a coding scheme where each unstable mode of the state is estimated using the channel outputs of a single sub-channel. We prove a coinciding necessity condition when either the source or channel is scalar and present a matrix-algebraic condition which implies the condition is necessary in general. Finally, we provide a new counter-example demonstrating that linear codes are generally sub-optimal for coding over MIMO channels.