Stabilizing a System With an Unbounded Random Gain Using Only Finitely Many Bits

We study the stabilization of a linear control system with an unbounded random system gain where the controller must act based on a rate-limited observation of the state. More precisely, we consider the system X _n+1 =A _n X _n +W _n -U _n , where the A _n 's are drawn independently at random at each time n from a known distribution with unbounded support, and where the controller receives at most R bits about the system state at each time from an encoder. We provide a time-varying achievable strategy to stabilize the system in a second-moment sense with fixed, finite R. While our previous result provided a strategy to stabilize this system using a variable-rate code, this work provides an achievable strategy using a fixed-rate code. The strategy we employ to achieve this is time-varying and takes different actions depending on the value of the state. It proceeds in two modes: a normal mode (or zoom-in), where the realization of A _n is typical, and an emergency mode (or zoom-out), where the realization of A _n is exceptionally large. To analyze the performance of the scheme we construct an auxiliary sequence that bounds the state X _n , and then bound auxiliary sequence in both the zoom-in and zoom-out modes.