Finite Gain lp Stabilization Is Impossible by Bit-Rate Constrained Feedback
In this paper, we show that the finite gain (FG) l p stabilization, with 1 ≤ p ≤ ∞, of a discrete-time, linear and time-invariant unstable plant is impossible by bit rate constrained feedback. In addition, we show that, under bit rate constrained feedback, weaker (local) versions of FG l p stability are also impossible. These facts are not obvious, since recent results have shown that input to state stabilization (ISS) is viable by bit-rate constrained control. We establish a comparison with existing work, leading to two conclusions: (1) in spite of ISS stability being attainable under bit rate constrained feedback, small changes in the amplitude of the external excitation may cause, in relative terms, a large increase in the amplitude of the state (2) FG l p stabilization requires logarithmic precision around zero, implying that even without bit-rate constraints FG l p stabilization is impossible in practice. Since our conclusions hold with no assumptions on the feedback structure, they cannot be derived from existing results. We adopt an information theoretic viewpoint, which also brings new insights into the problem of stabilization.
KeywordsExternal Excitation Quantization Scheme Uniform Quantizer Small Gain Theorem Logarithmic Quantizer
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