Robust ℓp stability and robust ℓ∞ stabilization over a rate-limited digital channel

This paper is concerned with robust stabilization of an uncertain system over a rate-limited digital channel from the viewpoint of input-output stability. Because of the quantization in the communication channel, it is impossible to apply the traditional small gain theorem to establish robust stability of the feedback system. To overcome this difficulty, we introduce a new notion of small ℓp signal ℓp stability, and derive a sufficient robust stability condition against ℓp gain-bounded uncertainty based on this notion. Furthermore, for the case of p = ∞, a sufficient data rate for the existence of a robustly stabilizing encoder-controller pair is explicitly given.