UNKNOWN MULTIPLE NARROW-BAND DISTURBANCE REJECTION IN HARD DISK
DRIVES–AN ADAPTIVE NOTCH FILTER AND PERFECT DISTURBANCE
Department of Mechanical Engineering
University of California, Berkeley
Berkeley, California 94720
Department of Mechanical Engineering
University of California, Berkeley
Berkeley, California 94720
In this paper, an adaptive control scheme is developed to re-
ject unknown multiple narrow-band disturbances in a hard disk
drive. An adaptive notch ﬁlter is developed to efﬁciently esti-
mate the frequencies of the disturbance. Based on the correctly
estimated parameters, a disturbance observer with a newly de-
signed multiple band-pass ﬁlter is constructed to achieve asymp-
totic perfect rejection of the disturbance. Evaluation of the con-
trol scheme is performed on a benchmark problem for HDD track
The track following control in a hard disk drive (HDD) sys-
tem aims at precisely regulating the read/write head on the data
track and reducing the Track Mis-Registration (TMR). One im-
portant source of TMR is the vibration/ﬂutter of disks at their
resonant frequencies. This behavior is driven by the turbulent
air ﬂow within the hard disk assembly, and creates the so-called
non-repeatable multiple narrow-band disturbances [1–3], whose
energy is highly concentrated at several unknown frequencies.
These disturbances differ from track to track and disk to disk
in nature, and may appear at frequencies higher than the open
loop servo bandwidth [1,2]. The above characteristics of narrow-
band disturbances lead to the difﬁculty of rejecting them using
traditional feedback control methods. With the rapid growth in
HDD’s storage density, compensation of the non-repeatable mul-
tiple narrow-band disturbances is becoming more and more an
Improving the mechanical design can help to reduce the
narrow-band disturbances but will signiﬁcantly increase the
product cost. Investigation of this problem in the community of
systems and controls has therefore been popular. For example,
Zheng and Tomizuka [4–6] suggested direct and indirect adap-
tive disturbance observer (DOB) schemes to estimate and cancel
the disturbance; Kim et al.  proposed a parallel add-on peak
ﬁlter to shape the open loop frequency response; Landau  used
Youla-Kucera parametrization to achieve adaptive narrow-band
disturbance rejection on an active suspension. These previous
methods have been focusing on rejecting disturbance with one
narrow-band. Yet, the ﬁeld of multiple narrow-band disturbance
rejection was seldom entered before.
This paper focuses on developing an adaptive control
scheme that compensates any number of unknown narrow-band
disturbances. The main techniques proposed are the minimal pa-
rameter adaptive notch ﬁlter (ANF) for frequency identiﬁcation
and the disturbance observer (DOB) for disturbance rejection.
The minimal parameter ANF was developed in the signal pro-
cessing community, rooted back to , reﬁned ﬁrst in  and
then in . It has the advantage of high computational efﬁ-
ciency, good numerical robustness, and fast convergence. This
paper proposes a computationally simpliﬁed version of , to
ﬁt the algorithm for our controller structure. The accurately
identiﬁed frequencies are then applied to construct a new type
of DOB, which can asymptotically achieve perfect narrow-band
disturbance rejection. Advantages of the proposed algorithm are:
it inherits the excellent properties of the minimal parameter ANF
in the frequency identiﬁcation; it is an add-on compensator and
can be simply integrated to the existing baseline controller struc-
ture; it rejects the narrow-band disturbance with minor inﬂuence
to the system sensitivity function at other frequencies; it can
achieve asymptotic perfect disturbance rejection.
The remainder of this paper is organized as follows. Sec-
tion 2 formally deﬁnes the problem. The proposed DOB struc-
ture, and the design of a multiple narrow band-pass Q-ﬁlter are
presented in Section 3. Section 4 provides the adaptive ANF al-
gorithm for Q-ﬁlter parameter identiﬁcation. The efﬁcacy of the
proposed method is provided in Section 5, with an example of
rejecting two narrow-band disturbances. Section 6 concludes the
2 PROBLEM DEFINITION
Figure 1 shows the baseline feedback control loop for HDD
track following and the proposed compensation scheme to re-
ject the multiple narrow-band disturbances. The full-order plant
model Gpz−1contains the dynamics of the HDD servo sys-
tem including the power ampliﬁer, the voice-coil motor, and the
actuator mechanics. Throughout the paper we use the well for-
mulated open-source HDD benchmark simulation package 
as a demonstration tool. The dashed line in Fig. 2 shows
the frequency response of Gpz−1, which is a fourteenth-order
transfer function with several high frequency resonances. The
low order nominal plant model z−mGnz−1matches the low-
frequency dynamics of Gpz−1up to about 2000 Hz, as shown
in Fig. 2. The baseline feedback controller CFB z−1is a com-
monly used PID controller cascaded with three notch ﬁlters. Fig-
ure 3 shows the frequency response of the open loop system
CFB z−1Gpz−1, which has a gain margin of 5.45 dB, a phase
margin of 38.2 deg, and an open loop servo bandwidth of 1.19
The signals r(k),d(k),u(k), and n(k), are respectively the
reference, the input disturbance, the control input, and the out-
put disturbance. PES is the position error signal. The multi-
ple narrow-band disturbances of interest are assumed to be con-
tained in d(k), and have unknown frequencies between 300 Hz
and 2000 Hz [1, 2].
Figure 1. Structure of the proposed scheme for multiple narrow-band
We observe from Fig. 3, that the general baseline open loop
system has a decreasing high gain in the low frequency region,
Figure 2. Frequency responses of the full order plant and its nominal
Bode plot: Baseline open loop response
Figure 3. Frequency response of the open loop system with the baseline
controller CFB z−1.
which indicates that the closed loop system suppresses low fre-
quency disturbance but has less and less attenuation capacity as
the frequency increases. When special attenuation is needed at
several narrow frequency regions (say around 500 Hz and 1000
Hz), the baseline controller would fail to maintain the desired
servo performance. In this paper, we aim at providing a solution
to this important problem.
3 DOB FOR MULTIPLE NARROW-BAND DISTUR-
The proposed control scheme for multiple narrow-band dis-
turbance rejection is summarized in the dash-dotted box in Fig.
1. The add-on compensator is a disturbance observer (DOB) 
with a specially designed adaptive Q-ﬁlter Qz−1. As the name
of DOB suggests, the multiple narrow-band disturbance is ’ob-
served’ by the controller, and fed back to cancel itself. To see
this point, notice ﬁrst that the signal ˆ
d(k)is expressed by, in the
d(k) = Gn(z−1)−1[Gp(z−1)(u(k) + d(k))+ n(k)] −z−mu(k)
A stable inverse model Gnz−1−1is needed in the above
signal processing. If Gnz−1has minimal phase, its inverse can
directly be assigned, if not, stable inversion techniques such as
the ZPET method  should be applied.
Noting that Gnz−1−1=z−mz−mGnz−1−1, we can
transform Eq. (1) to
Notice that z−mGnz−1≈Gpz−1under 2000 Hz, which
gives hz−mGnz−1−1Gpz−1iu(k)−u(k)≈0. If in addi-
tion the output disturbance n(k)is small, then Eq. (2) becomes
d(k)≈z−md(k) = d(k−m).(3)
The above equation implies that ˆ
d(k)is a good estimate of
d(k), i.e., the disturbance is well ’observed’ by DOB.
In reality, the inﬂuence of n(k)and other noise can not
be omitted. To compensate only the narrow-band disturbance
without inﬂuencing other components in d(k), it is proposed to
d(k)through a multiple band-pass ﬁlter Qz−1, whose
pass-bands are sharply located at the narrow-band frequencies.
The compensation signal c(k)formed by ﬁltering ˆ
Qz−1, therefore contains only the multiple narrow-band dis-
turbance. Adding the negative of c(k)to the control input, we
achieve the compensation.
3.1 Q-ﬁlter Design for Perfect Disturbance Rejection
Assume that the multiple narrow-band frequencies are at
Ω1,...,Ωn(in Hz). Estimation of these frequencies will be dis-
cussed in the next section. In this section, we present the struc-
tural design of Qz−1, and assume Ωi’s are known.
To have sharp pass-bands at Ωi’s, poles of Qz−1need to
be located close to these frequencies. The second constraint on
Qz−1comes from Eq. (3), that the estimated disturbance ˆ
is a m-step delayed version of the true signal d(k). This translates
to the need on Qz−1to be able to provide a m-step forward
action to ˆ
Suppose the basis Q-ﬁlter for single narrow-band distur-
bance rejection has the form Q0z−1=NQz−1/DQz−1,
where NQz−1and DQz−1are polynomials of z−1. Us-
ing the results in the last subsection, that ˆ
c(k) = Qz−1ˆ
d(k), we have
When d(k)is a narrow-band sinusoidal signal with fre-
quency Ωo(in Hz) and sampling time Ts(in sec), by direct ex-
pansion, one can show that 1−2cos(ωo)z−1+z−2d(k) = 01,
where ωo=2πΩoTs. Therefore, if DQz−1is stable in Eq. (4),
DQ(z−1)−z−mNQ(z−1) = J(z−1)1−2cos(ωo)z−1+z−2,
where Jz−1is a polynomial of z−1, we can achieve asymptotic
perfect disturbance rejection, i.e.,
k→∞(d(k)−c(k)) = 0.(6)
In our nominal model, m=1. Assign Jz−1=1, and let
DQz−1be given by
where the shaping coefﬁcient αis a real number smaller than but
close to 1, such that Qz−1is stable and has two poles αe±jω0
at the narrow-band frequency.
Substituting Eq. (7) to Eq. (5), and applying the method of
undetermined coefﬁcients, we get
NQz−1=2(1−α)cos(ω0) + α2−1z−1.(8)
Introduce now the shorthand notation θ0=cos(ω0). The
ideal Q-ﬁlter for single narrow-band disturbance rejection is thus
When d(k)contains nnarrow-band components, the above
equation can be extended to
11/1−2cos(ωo)z−1+z−2is also known as the internal model of the nar-
row band disturbance d(k).
Figure 4 shows the frequency response of the above
Qz−1with α1=α2=0.998, Ts=3.788 ×10−5sec, θ1=
cos(2π×500Ts), and θ2=cos(2π×1200Ts). Notice that at the
central frequencies, the magnitude and the phase of Qz−1are
1 (0 dB) and 0 deg, respectively. Therefore, passing a broad band
d(k)through Qz−1, one ﬁlters out other compo-
nents and gets only the multiple narrow-band signals at 500 Hz
and 1200 Hz.
Frequence Response of Q-filter
Figure 4. Frequency response of a Q-ﬁlter with two narrow pass-bands.
The error rejection property of a control system is com-
monly evaluated by its sensitivity function Sz−1, which is the
transfer function from the output disturbance to the feedback er-
ror signal. Figure 5 shows the frequency response of Sz−1
for the proposed overall control structure when Qz−1is ﬁxed.
With the add-on compensation scheme, PES at 500 Hz and 1200
Hz can get greatly attenuated, due to the deep notches in the
magnitude response at the corresponding frequencies, while the
inﬂuence to the sensitivity at other frequencies is neglectable.
Stability of DOB (see [15, 16]) requires the nominal model
z−mGnz−1to have no zeros outside the unit circle and that
where ∆(z−1) = Gp(z−1)−z−mGn(z−1)/z−mGn(z−1)repre-
sents the multiplicative model mismatch. Plotting the magnitude
responses of 1/∆z−1and Qz−1in Fig. 6, we see that the
proposed DOB is stable as long as the narrow-band frequency is
less than 3000 Hz.
4 ADAPTIVE NOTCH FILTER FOR FREQUENCY IDEN-
The proposed Q-ﬁlter design requires knowledge of the fre-
quency information θi=cos(2πΩiTs), which is not priori avail-
able. In this section, we apply an adaptive notch ﬁlter (ANF)
Frequency response: error r ejection function/sensitivity func tion
w ith DOB
w ithout DOB
Figure 5. Frequency response of the closed loop sensitivity function.
Figure 6. Magnitude responses of 1/∆z−1and Qz−1.
algorithm to estimate these quantities. As has been discussed
d(k)contains the multiple narrow-band disturbance as
well as other noise components. A low-pass ﬁlter is thus ﬁrst
constructed to ﬁltered out the components in ˆ
d(k)that are out of
interest. The ﬁltered signal z(k)is ﬁnally a multiple narrow-band
signal with small noise-signal ratio, and can be identiﬁed using
the parameter estimation scheme in this section.
The intuition of the ANF algorithm comes from the fact that
energy of the narrow-band disturbance z(k)is highly concen-
trated at several frequencies Ω= [Ω1,...,Ωn]T(in Hz). If one
constructs a notch ﬁlter Hz−1with multiple center frequen-
cies at [Ω1,...,Ωn]T, and passes z(k)through Hz−1, the out-
put eo(k)should have the least energy.
Introduce the normalized frequency ωi=2πΩiTs(in radi-
ans). The transfer function of a qualifying notch ﬁlter  is
Hoωi,z−1has two poles at αe±jωiand two zeros at βe±jωi.
With the shaping coefﬁcients βchosen close to 1, and α<β<
1, the ﬁlter will have a strong attenuation to the input signal at
To ﬁnd the unknown ω0
is, an adaptive algorithm is neces-
sary. Considering the numerator and the denominator of Hz−1
as two entire sections2and estimating their coefﬁcients, Neho-
rai developed the algorithm in . Applying the cascaded se-
ries of second-order sections as shown in Eq. (12), and updat-
ing directly ωigave rise to , which was later reﬁned in .
The above algorithms identify only nparameters, which is the
minimum possible number for nnarrow-band components. The
main advantage of  over  is that cascaded ﬁlter structures
are numerically more efﬁcient and stable when the ﬁlter order is
We notice, however, from the control aspect of view, that
the value of ωiis not directly needed (cosωiis the term really
implemented in the controller). To directly estimate ωi, 
and  needed to calculate trigonometric functions within each
iteration, which can be an expensive task in HDD servo control,
since the microprocessor has limited ability in these computa-
tions. To localize the ANF method for HDD control, we intro-
duce θi=cos(ωi), and modify the algorithm in  as follows:
In Eq. (13), let
and introduce the unknown parameter vector θ= [θ1,θ2·· · θn]T,
Eq. (12) becomes
where we used the notation Hθ,z−1to emphasize that this is
a transfer function with unknown parameter θ. The objective of
ANF design is to ﬁnd the best parameter estimate, such that the
following cost function is minimized
2i.e., applying the digital ﬁlter in its direct-form structure.
where e0(k) = Hθ,z−1z(k)is the output error.
The transfer function Hθ,z−1is nonlinear in θ. To ﬁnd
the best estimation, the celebrated Gauss-Newton Recursive Pre-
diction Error Method (RPEM) (chapter 11 in ) suggests to
apply the following iterative formulas
θ(k) = ˆ
θ(k−1) + F(k−1)ψ(k−1)eo(k)
λ(k) + ψT(k−1)F(k−1)ψ(k−1),(17)
F(k) = 1
λ(k) + ψT(k−1)F(k−1)ψ(k−1),
where ψ(k−1)=[ψ1(k−1),...,ψn(k−1)]T,ψi(k−1) =
θi(k−1), and λ(k)is the forgetting factor.
The above modiﬁed algorithm has several nice properties:
1. stability of the Gauss-Newton RPEM is guaranteed if
θ(k),z−1)is stable during the adaptation , which can
be easily checked by monitoring if |ˆ
θi(k)|<1, due to our
cascaded construction of Eq. (15).
θ(k)unbiasedly converges to a local minimum .
3. it inherits most of the advantages of [9,11], such as fast con-
vergence, computational efﬁciency, and numerical robust-
ness. Moreover, it does not require computing sine and co-
Similar to , to obtain ﬁrst e0(k) = Hθ,z−1z(k), we
from which we have
xj(k) = 1−2βθiz−1+β2z−2
i.e., in the state-space representation
Zi(k+1) = h2αθi−α2
1 0 iZi(k) + h1
xi(k)=[2(α−β)θi,β2−α2]Zi(k) + xi−1(k).(22)
We can then iteratively get e0(k), with eo(k) = xn(k)and
x0(k) = z(k).
To get ψi(k−1) = −∂eo(k)/∂ˆ
θi(k−1), we notice that
Using Eq. (14) and Eq. (15), we get
where ∂Aoθi,γz−1/∂θi=−2γz−1,γ=α,β. Substituting the
above back to Eq. (23), and changing θito its estimated value
θi(k−1), we arrive at the following simple formula:
ψi(k−1) = 2[eFi(β,k)−eFi(α,k)],(25)
where eFi(γ,k) = γz−1/Aoˆ
which can again be calculated using a state-space realization
Wi(γ,k+1) = h2γˆ
1 0 iWi(γ,k) + h1
eFi(γ,k) = γ,0Wi(γ,k).(27)
We notice that the above result has a similar structure with
that in , but does not require to calculate sine or cosine func-
tions. Analogous to , the recursive parameter estimation is
ﬁnally summarized as follows:
Initialization: αo=0.8, αend =0.995, αrate =0.99,3β=
0.9999, Zi(0) = Wi(γ,0) = 0, F(0)≈100/E[eo]2·I,ˆ
initial guess of the parameters, λ(0) = λ0,λ(∞) = λend ,λrate =
Main loop: for k=1,2, . . .
step 1, prediction error computation: for i=1 : n
xi(k) = [2(α−β)ˆ
with x0(k) = z(k)and eo(k) = xn(k).
3αis designed to increase exponentially from αoto α∞, at the rate of αrate ,
such that the notches get sharper and sharper to better capture the narrow-band
step 2, regressor vector computation: for i=1 : n
eFi(γ,k) = γ,0Wi(γ,k),γ=α,β,(29)
ψi(k−1) = 2(eFi(β,k)−eFi(α,k)).(30)
step 3, parameter update using Eq. (17) and Eq. (18).
step 4, projection of unstable parameters: for i=1 : n, if
θi(k) = ˆ
step 5, a posteriori prediction error ¯e(k)computation and state
vector update: for i=1 : n
¯xi(k) = 2(α−β)ˆ
θi(k),β2−α2Zi(k) + ¯xi−1(k),
Zi(k+1) = h2αˆ
1 0 iZi(k) + h1
with ¯xo(k) = z(k)and ¯e(k) = ¯xn(k).
Wi(γ,k+1) = h2γˆ
1 0 iWi(γ,k) + h1
step 6, forgetting factor and notch ﬁlter shape coefﬁcient update:
replace αby αend −[αend −α]αrate , and
λ(k+1) = λend −[λend −λ(k)]λrate .(34)
Remark: (1) The above algorithm does not involve any sine or
cosine functions, but instead performs one additional simple step
to assure the stability of the ﬁlter. (2) The a posteriori informa-
tion is applied as in , to improve the estimation precision.
(3) As long as the initial parameter guesses are not too far away
from the true values, the estimation is unbiased even under the
presence of noise .
5 SIMULATION RESULT
The proposed adaptive compensator for multiple narrow-
band disturbance rejection was implemented in the HDD bench-
mark simulation tool . The baseline control system is as
shown in Section 2. All common disturbances in HDD, including
the torque disturbance, the disk ﬂutter disturbance, the repeatable
runout (RRO), and the measurement noise, are added in the sim-
ulation. The sampling time, the spindle rotation speed, and the
track density are respectively 3.788 ×10−5sec, 7200 rpm, and
100k Tracks Per Inch (TPI). The multiple narrow-band distur-
bance in NRRO was modeled as the sum of several sinusoidal
signals [1, 2], and injected at the input to the plant with center
frequencies 500 Hz and 1200 Hz.
Figure 7 shows the time trace of the position error signal
without compensation. It is observed that the peak values of
PES exceeded the standard PES upper bound of 15% Track Pitch
(TP). The dotted line in Fig. 9 presents the spectrum of the PES
without compensation. We can see that the PES had strong en-
ergy components at 500 Hz and 1200 Hz. Without compensa-
tion, the Track Mis-Registration (TMR), deﬁned as 3 times the
standard deviation of the PES, was 21.87% TP.
0 5 10 15 20
Figure 7. Time trace of the position error signal without compensation.
0 5 10 15 20
Figure 8. Time trace of the position error signal with compensation.
500 1000 1500 2000 2500
w/ compensation 3σ = 10.25 %TP
w/o compensation 3σ = 21.87 %TP
Figure 9. Spectra of the position error signals with and without the com-
With the same baseline controllers, the proposed add-on
compensation scheme was applied to improve servo perfor-
mance. The low-pass ﬁlter in Fig. 1 was designed using MAT-
LAB’s Filter Design Toolbox, to have a cut-off frequency of
2000 Hz.The parameter adaptation was initialized at ˆ
iTs), where Ωo
1=100 Hz and Ωo
2=1000 Hz, in view
of the fact that the multiple narrow-band disturbance of interest
lies between 300 Hz and 2000 Hz. Figure 10 shows the online
identiﬁcation of the parameters ˆ
θ2. Figure 11 shows the
equivalent online frequency estimation, via the transformation
θ2)/(2πTs). It is ob-
served that the parameters converged to their true values within
one revolution, i.e., 0.00833 sec.
0 1 2 3 4 5
Figure 10. Online parameter estimation of the two narrow-band signals.
0 1 2 3 4 5
Estimated frequency (Hz)
Figure 11. Equivalent frequency identiﬁcation of the two narrow-band
Recall the stability condition at the end of section 3, that
Ωi(k)should be lower than the DOB stability threshold 3200 Hz.
θi(k)should be larger than cos(2π×3200TS)
for implementation. Once the estimated parameters fell into this
region, the adaptive DOB was constructed to reject the multiple
Notice that the proposed Q-ﬁlter is given by
In the simulation example, n=2 and α1=α2=0.998. To gen-
erate the compensation signal c(k) = Qz−1ˆ
d(k)at each itera-
tion, Eq. (35) was realized by, in the state space form,
Si(k+1) = 2αθi−α2
1 0 Si(k) + 1
ci(k) = (1−α)4αθ2
i−1−α,−2α2θiSi(k) + 2θiˆ
with θireplaced by its latest stable estimate ˆ
θi(k), and c(k) =
d(k) = ∑n
Figure 8 shows the resulting PES time trace. It is seen that
after a transient response of about 1 revolution, the PES was re-
duced to within 10% TP. In the frequency domain, we observe
from Fig. 9, that the strong energy components at 500 Hz and
1200 Hz were greatly attenuated, while the spectrum of the PES
at other frequencies was almost identical to that without compen-
sation. The TMR was reduced to 10.25% TP, implying a 53.13%
In this paper, an adaptive control scheme was proposed for
rejecting multiple narrow-band disturbances in HDD track fol-
lowing. It consists of an adaptive notch ﬁlter to estimate the
frequencies of the disturbances and a disturbance observer with
a multiple band-pass ﬁlter tuned for the estimated frequencies.
Simulation results on a realistic open-source HDD benchmark
problem showed that the proposed algorithm signiﬁcantly re-
duced the PES and the TMR. The proposed method is suitable
in control systems that demand heavy disturbance attenuation at
This work was supported by the Computer Mechanics Lab-
oratory (CML) in the Department of Mechanical Engineering,
University of California at Berkeley. The authors gratefully ac-
knowledge Dr. Qixing Zheng’s useful discussions.
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