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By Eric Amar, R. Gay, T. V. Nguyen

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5). If x ∈ Λ, then Q[α(x(k))y(k)] is equal to Q[F Q(x(k))] for any α(x(k)), we let α(x(k)) = 0 for simplicity. 10). Last, α(x(k)) can be chosen as 0 when Q(F Q(x(k))) = 0. 2, now we will show that there exists a suitable d such that F x(k) − dQ(F Q(x(k))) = 0 if and only if x ∈ Λ. In the following, two-dimensional system is discussed for simplicity. Firstly, we consider supx∈Ωη F x and inf x∈Ωη F x, where η = [η1 η2 ]T and η1 , η2 ∈ {±vi , i ∈ Z≥0 }. Because F x is linear, supx∈Ωη F x and inf x∈Ωη F x should happen on the edge of Ωη .

10 and Fig. 11). 5 0 10 20 30 40 50 60 70 80 90 100 90 100 k Fig. 01. 5 0 10 20 30 40 50 60 70 80 k Fig. 01. P (k), increases (see Fig. 15). While L = 1000, the norm of covariance of estimated error, P (k), increases remarkably (see Fig. 18). It is worth to say that even if L is very large, for example L = 1000, the states of system are bounded although the bound is very large. 01, the error induced by quantization is small, and it is helpful to decrease the bound of estimated state x ˆ(k|k) after quantizing.

2 v2 Q(z) v1 v0  U 1 2 v1  U 1 2 0 v0  v v U 1 2 v0 U 1 2 z v1 -v0 -v1 -v2 Fig. 2 Logarithmic quantizer with dead-zone. The mapping Q : Rn → Rn is the vector quantization function satisfying: Q(x) = ξ, where ξi = Q(xi ). 1. Note that the logarithmic quantizer considered here is different from traditional quantizer used in [40] and [51], where quantization coarseness ρ < 1 and quantization levels decrease monotonously. But in this chapter, quantization levels increase monotonously according to ρ > 1.

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