An adaptive notch filter for suppressing mechanical resonance in high track density disk drives

Abstract

In disk drive servo system, a notch filter is usually used to suppress mechanical resonance of an actuator. However, the resonance frequency differs from drive to drive due to manufacturing tolerance and varies with temperature even within a single drive. In this paper, we present an adaptive digital notch filter that can identify the resonance frequency of the actuator quickly and can adjust automatically its center frequency. For the fast identification of resonance frequency, a new estimation filter and its tuning algorithm are developed. Also, we give a rigorous analysis for the convergence of our tuning algorithm. We furthermore present some experimental results using a commercially available hard disk drive in order to demonstrate the practical use of our work.

Appendix

1.1 < Proof of (15) and (16) >

We first prove Eq. 15. By Eq. 13, there exist positive integers k _i , $i = 1,2, \cdots ,N$ such that

$$ \omega _i = k_i \omega _{\text{d}} ,\quad i = 1,2, \cdots ,N $$

(62)

Define $c_i ,\phi _i ,$ $i = 1,2, \cdots ,N$ by

$$ c_i = a_i \left| {H\left( {{\text{e}}^{j\omega _i T_{\text{s}} } } \right)} \right|,\quad \phi _i = \theta _i + \angle H\left( {{\text{e}}^{j\omega _i T_{\text{s}} } } \right). $$

(63)

Then, by Eqs. 12, 14, 62, and Eq. 63,

$$ \begin{aligned} y_{k + L} & = \sum\limits_{i = 1}^N {c_i \cos \left( {\omega _i T_{\text{s}} (k + L) + \phi _i } \right)} = \sum\limits_{i = 1}^N {c_i \cos \left( {\omega _i T_{\text{s}} k + 2\pi \omega _i /\omega _{\text{d}} + \phi _i } \right)} \\ & = \sum\limits_{i = 1}^N {c_i \cos \left( {\omega _i T_{\text{s}} k + 2\pi k_i + \phi _i } \right)} = \sum\limits_{i = 1}^N {c_i \cos \left( {\omega _i T_{\text{s}} k + \phi _i } \right)} = y_k . \\ \end{aligned} $$

(64)

This completes the proof of Eq. 15.

Next, we prove Eq. 16. By Eqs. 12 and 63,

$$ |y_k |^2 = \sum\limits_{i = 1}^N {|c_i |^2 \cos ^2 \left( {\omega _i T_{\text{s}} k + \phi _i } \right)} + \sum\limits_{j = 1}^N {\sum\limits_{\begin{subarray}{l} i = 1 \\ i \ne j \end{subarray}}^N {c_i c_j \cos \left( {\omega _i T_{\text{s}} k + \phi _i } \right)\cos \left( {\omega _j T_{\text{s}} k + \phi _j } \right).} } $$

(65)

Summing both sides of Eq. 65 over one period L, we can obtain

$$ \sum\limits_{k = 1}^L {|y_k |^2 } = \sum\limits_{i = 1}^N {|c_i |^2 \sum\limits_{k = 1}^L {\cos ^2 \left( {\omega _i T_{\text{s}} k + \phi _i } \right)} } + \sum\limits_{j = 1}^N {\sum\limits_{\begin{subarray}{l} i = 1 \\ i \ne j \end{subarray}}^N {c_i c_j } } \sum\limits_{k = 1}^L {\cos \left( {\omega _i T_{\text{s}} k + \phi _i } \right)\cos \left( {\omega _j T_{\text{s}} k + \phi _j } \right)} $$

(66)

From trigonometric properties, we find that

$$ \sum\limits_{k = 1}^L {\cos ^2 \left( {\omega _i T_{\text{s}} k + \phi _i } \right)} = \frac{L} {2} + \sum\limits_{k = 1}^L {\cos \left( {2\omega _i T_{\text{s}} k + 2\phi _i } \right)} $$

(67)

$$ \begin{aligned} & \sum\limits_{k = 1}^L {\cos \left( {\omega _i T_{\text{s}} k + \phi _i } \right)\cos \left( {\omega _j T_{\text{s}} k + \phi _j } \right)} \\ & = \frac{1} {2}\sum\limits_{k = 1}^L {\left\{ {\cos \left( {(\omega _i - \omega _j )T_{\text{s}} k + \phi _i - \phi _j } \right) + \cos \left( {\left( {\omega _i + \omega _j } \right)T_{\text{s}} k + \phi _i + \phi _j } \right)} \right\}} . \\ \end{aligned} $$

(68)

From Eq. 14 and Eq. 62, we see that the functions cos(2ω _i T_sk+2φ _i ), cos((ω _i −ω _j )T _s k+φ _i −φ _j ), cos((ω _i +ω _j )T_sk+φ _i +φ _j ) are all periodic functions with period L. Hence, it follows that the sum of each function over one period L is zero. Hence, by Eq. 67 and Eq. 68, we have

$$ \sum\limits_{k = 1}^L {\cos ^2 \left( {\omega _i T_{\text{s}} k + \phi _i } \right)} = \frac{L} {2},\quad \sum\limits_{k = 1}^L {\cos \left( {\omega _i T_{\text{s}} k + \phi _i } \right)\cos \left( {\omega _j T_{\text{s}} k + \phi _j } \right)} = 0. $$

(69)

Hence, we readily see from Eq. 66 and Eq. 69 that

$$\sum\limits_{k = 1}^L {|y_k |^2} \;\;\; = \;\;\;\frac{L}{2}\,\,\sum\limits_{i = 1}^N {{\kern 1pt} |c_i |^2 {\kern 1pt} {\kern 1pt}} $$

(70)

This along with Eq. 63 completes the proof of Eq. 16.