An Asymptotically Optimal Two-Part Fixed-Rate Coding Scheme for Networked Control

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Keeler, Jonathan
Mathematics , Probability Theory , Markov Chains , Stochastic Control , Networked Control
It is known that under fixed-rate information constraints, adaptive quantizers can be used to stabilize an open-loop-unstable linear system on $\mathbb{R}^n$ driven by unbounded noise. These adaptive schemes can be designed so that they have near-optimal rate, and the resulting system will be stable in the sense of having an invariant probability measure, or ergodicity, as well as boundedness of the state second moment. In addition, structural results and information theoretic bounds on the performance of encoders have also been studied. However, the performance of such adaptive quantizers beyond stabilization has not been addressed. In this thesis, we construct a two-part adaptive coding scheme that achieves state second moment convergence to the classical optimum (i.e., for the fully observed setting) under a mild moment condition on the noise process. The first part, as in prior work in this context, leads to ergodicity (via positive Harris recurrence) and the second part ensures that the state second moment converges to the classical optimum at high rates. These results are established using an intricate analysis which uses random-time Lyapunov drift conditions as a core tool.
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