Exact Minimum Number of Bits to Stabilize a Linear System
summary
Proves the exact threshold for causal quantizers: the minimum number of quantization levels is minimum M = floor(A)+1 (so one bit suffices for 1<A<2) under IID disturbances with bounded α-moments (stability in any β<α). Achievability uses a time-varying zoom-in/zoom-out probe scheme (K normal steps, a probe test, and exponential emergency zoom-outs) and certainty-equivalence control; converse rules out any fixed finite-level time-invariant quantizer. Analysis uses tail bounds and a majorizing martingale.
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chapters (15)
- 0:00Start
- 1:03Stochastic linear control
- 5:04Remote control over a rate-limited channel
- 5:52A well researched area
- 6:21Motivation
- 7:52A difficult problem...
- 11:05Simple question
- 14:41Fundamental limit
- 15:04The answer
- 19:26No rate constraints (infinite capacity channel)
- 20:24Bounded disturbances: achievable scheme
- 24:54Unbounded disturbances
- 34:46Analysis
- 36:06Extensions
- 37:09What about the full rate-cost tradeoff?