Download Adaptive Control by Edited by: Kwanho You PDF
By Edited by: Kwanho You
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Extra info for Adaptive Control
Example text
In general cases, it can be extended to the set of possible instances of θ (and/or f ) which do not contradict with the data at step k. We will see an example involving unknown f in next section. 1 Information-concentration estimator has the following properties: (i) Monotonicity: C0 ⊇ C1 ⊇ C2 ⊇ L ∞ (ii) Convergence: Sequence {Ck} has a limit set C∞ = ∩ C k ; k =1 must be a non-empty set (iii) If the system model and the a priori knowledge are correct, then with property θ and any element of can match the data and the model; (iv) If C∞ = ∅ , then the data {φk , z k } cannot be generated by the system model used by the IC estimator under the specified a priori knowledge.
13) Therefore, we always have ~ ~ θ k'2 ≥ θ k2 . This completes the proof. 2 Given a bounded sequence X k ∈ R m . 1. Its proof can be found in [Ma06]. 3 (Key Technical Lemma)Let of vectors such that {st } be a sequence of real numbers and {σ t } be a sequence Adaptive Estimation and Control for Systems with Parametric and Nonparametric Uncertainties 57 Assume that where α1 > 0, α 2 > 0 . Then || σ t || is bounded. Proof: This lemma can be found in [AW89, GS84]. 1 Define parameter estimate errors we have and .
1 reveals again the well-known result in a new way, where the adaptive controller is defined by Eq. 13) together with Eqs. 12). 2 If b = 1, ML 3 < + 2 , c = w = 0 , then the adaptive controller defined by Eqs. e. 4]. It can be proved by argument of contradiction. 2 Assume that L ∈ (0, 3 + 2 ), d ≥ 0, n0 ≥ 0 . 3]. 1: We divide the proof into four steps. In Step 1, we deduce the basic relation between yt+1 and , and then a key inequality describing the upper bound of | yt − yit | is established in Step 2.