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Additional resources for CLP Training Guide : Lotus Notes, w. CD-ROM: Lotus Notes System Administration and Application Development
8) observable. Assume indeed that this set is not a dense one. Then there exists an ∂Φ (ξ) is lower than open subset W of IRn such that for every ξ ∈ W , the rank of ∂x or equal to n − 1. In particular there exists k ≤ n − 1 and diﬀerentiable functions k α1 , . . , αk , such that dLjf0 (h)(ξ) = i=1 αi (ξ)dLi−1 f0 (h)(ξ), for 1 ≤ j ≤ n. Now 0 0 taking any diﬀerent initial states ξ 0 , ξ such that dLjf0 (h)(ξ 0 ) = dLjf0 (h)(ξ ), for 0 ≤ j ≤ k − 1 and consider the associated trajectories ξ(t), ξ(t) of the non forced system (u = 0), it follows that h(ξ(t)) = h(ξ(t)).
As in ). 4 below for the case of two subsystems. 21) Clearly here one can consider the system as the interconnection of the following two subsystems: ⎧ ⎧ ⎨ x˙ 1 = x2 ⎨ x˙ 3 = x4 + ϕ(v) (Σ1 ) x˙ 2 = u1 and (Σ2 ) x˙ 4 = u2 ⎩ ⎩ y1 = x1 y = x3 where v = x2 deﬁnes the interconnection. It is also clear that (Σ1 ) being linear and observable, it admits an observer (say O1 ), as well as (Σ2 ) whenever v is considered as a known input for (Σ2 ) (let (say O2 (v)) denote the corresponding observer). An Overview on Observer Tools for Nonlinear Systems System Sub-systems Ȉ1 y u 23 u1 Ȉ1 u2 Ȉ2 Ȉ2 y1 y2 ?
In particular, assumption A1) implies that : ∂ϕki (u, z) ≡ 0 for j ≥ i + 1 and 1 ≤ i ≤ nk − 1. ∂zkj A2) the function ϕ is global Lipschitz with respect to z locally uniformly in u. Before stating our main theorem, let us analyse the meaning of the condition given by assumption A1) more closely. 40) Let (i, k) ∈ I and set also: I(i, k) = l; 1 ≤ l ≤ p for which ∃j, 2 ≤ j ≤ nl such that Notice that I and I(i, k) may be empty. 42) or equivalently, (2i − 1) (2j(i, l, k) − 3) δk − δl > 0. 44) σ , σ = (σ1 , .