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 <inline-formula> <tex-math notation="LaTeX">$X_{n+1}=A_{n}X_{n}+W_{n}-U_{n}$ </tex-math></inline-formula>, where the <inline-formula> <tex-math notation="LaTeX">$A_{n}$ </tex-math></inline-formula>’s are drawn independently at random at each time <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> from a known distribution with unbounded support, and where the controller receives at most <inline-formula> <tex-math notation="LaTeX">$R$ </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">$R$ </tex-math></inline-formula>. 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 <inline-formula> <tex-math notation="LaTeX">$A_{n}$ </tex-math></inline-formula> is typical, and an emergency mode (or zoom-out), where the realization of <inline-formula> <tex-math notation="LaTeX">$A_{n}$ </tex-math></inline-formula> is exceptionally large. To analyze the performance of the scheme we construct an auxiliary sequence that bounds the state <inline-formula> <tex-math notation="LaTeX">$X_{n}$ </tex-math></inline-formula>, and then bound auxiliary sequence in both the zoom-in and zoom-out modes.

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