By Germán A. Ramos
The tracking/rejection of periodic indications constitutes a large box of analysis within the keep an eye on conception and functions zone. Repetitive keep an eye on has confirmed to be a good solution to face this subject. despite the fact that, in a few functions the frequency of the reference/disturbance sign is time-varying or doubtful. This reasons a huge functionality degradation within the usual Repetitive regulate scheme. This publication offers a few suggestions to use Repetitive keep watch over in various frequency stipulations with no loosing steady-state functionality. it is usually a whole theoretical improvement and experimental ends up in consultant structures. The provided options are prepared in complementary branches: various sampling interval Repetitive keep watch over and excessive Order Repetitive regulate. the 1st process permits facing huge variety frequency adaptations whereas the second one permits facing small diversity frequency adaptations. The e-book additionally provides purposes of the defined thoughts to a Rota-magnet plant and to an influence lively filter out device.
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Extra info for Digital Repetitive Control under Varying Frequency Conditions
Sample text
7) Proof. It follows from the stability hypothesis that all the eigenvalues of Φ (T¯ ) are within the unit circle, which yields item 1 [10]. 2 is also necessary for a discrete-time LTI system. 2 taking into account the continuity of the matrix elements of Φ (Tk ) with respect to Tk . 4) for all sampling periods Tk ∈ T . 3. Let also {τ0 , . . , τq }, with τi+1 > τi , be a sufficiently fine grid of T . Then, one has to check that Lτi (PN ) < 0, ∀i = 0, . . 2. In case that there exists at least a single τi such that Lτi (PN ) ≥ 0, the gridding procedure proposed in [11] may be carried out as follows.
Let T¯ be a nominal sampling period and define θk = Tk − T¯ . 10) and Γ ∈ Ξ , with Ξ := Γ ; Γ Δ˜ (θ ) = Δ˜ (θ ) = Γ1 0 Γ2 I . 11) 22 3 Stability Analysis Methods uk vk Σ Δ (Tk − T¯ ) Fig. 4 The resulting feedback system Proof. 12) on the other hand, B p (Tk ) = Tk 0 eAr Bdr = T¯ 0 eAr Bdr + Tk T¯ eAr Bdr = B p (T¯ ) + = B p (T¯ ) + Δ (θk )A p (T¯ )B. 3). 4. 1. Assume that T¯ is a nominal sampling period. 15), and let also T ⊂ R+ be compact. 2) is uniformly BIBO stable in T . 4 Robust Analysis 23 Proof.
It can be seen that the maximum resulting gain at interharmonic frequencies is equal in both cases. However, robustness improvement against frequency variations/uncertainty at odd-harmonic frequencies is larger for the odd-harmonic IM presented here. 3 Selection of the Gain kr Besides the repetitive controller speed convergence, the parameter kr affects the magnitude response of the system sensitivity function at the inter-harmonic frequency intervals. This is an important fact since HORC increases the gain at those intervals and some alleviation of that problem can be attained by a proper selection of kr .