Discrete-Time Reduced-Complexity Youla Parameterization for Dual-Input Single-Output Systems

Abstract

A controller design is presented for directly shaping the sensitivity function in the feedback control of dual-input single-output (DISO) systems. We provide special forms of discrete-time Youla-Kucera (YK) parameterization to obtain stabilizing controllers for DISO systems, and discuss concepts to achieve a reduced-complexity formulation. The proposed YK parameterization is based on an inverse of the loop transfer function, which gives benefits such as clearer design intuition and reduced complexity in construction. The algorithm is verified by simulations and experiments on hard disk drive systems, with application to wideband vibration and harmonic disturbance rejections.

Full text

Discrete-time Reduced-complexity Youla Parameterization for
Dual-input Single-output Systems
Xu Chen†and Masayoshi Tomizuka
Abstract— A controller design is presented for directly
shaping the sensitivity function in feedback control of dual-
input single-output (DISO) systems. We provide special
forms of discrete-time Youla-Kucera (YK) parameterization
to obtain stabilizing controllers for DISO systems, and
discuss concepts to achieve a reduced-complexity formu-
lation. The proposed YK parameterization is based on an
inverse of the loop transfer function, which gives benefits
such as clearer design intuition and reduced complexity in
construction. The algorithm is verified by simulations and
experiments on hard disk drive systems, with application to
wide-band vibration and harmonic disturbance rejections.
Index Terms— precision control, loop shaping, vibration
rejection, discrete-time Youla parameterization, dual-stage
actuation, dual-stage hard disk drives, multi-input single-
output systems
I. Introduction
Dual-stage actuation is a hardware solution to exceed
the performance limit of single-actuator control sys-
tems. In this mechanical configuration, two actuators
are combined for enhanced positioning: a coarse actu-
ator with a long movement range, and a fine actuator,
using e.g. piezoelectric materials, with a short stroke
but a higher achievable accuracy. The fine actuator is
mounted at the end of the coarse actuator to form a
second-stage actuation, which commonly has enhanced
mechanical performance at high frequencies, providing
the capacity to greatly increase the servo bandwidth.
Example applications of the technology include but are
not limited to: hard disk drives (HDDs) [1], machine
tools [2], [3], precision positioning tables and nanopo-
sitioners [4], [5].
In this study we investigate add-on loop-shaping
designs via Youla parameterization for dual-stage sys-
tems. Youla parameterization [6], [7], aka Youla-Kucera
(YK) parameterization and the parameterization for all
stabilizing controllers, offers a parameterization of all
stabilizing controllers, if a good model of the plant is
available. Such a construction guarantees the closed-
loop stability and renders loop shaping to simply the
design of a stable Q filter. For single-input single-
output (SISO) systems, YK parameterization has at-
tracted lots of research attentions in controller tuning,
†: corresponding author. This work was supported by a re-
search grant from Western Digital Corporation. Xu Chen is with
the Department of Mechanical Engineering, University of Con-
necticut, Storrs, CT, 06269, USA (email: xchen@engr.uconn.edu).
Masayoshi Tomizuka is with the Department of Mechanical Engi-
neering, University of California, Berkeley, CA, 94720, USA (email:
tomizuka@berkeley.edu)
adaptive control, and disturbance rejection [8]–[11].
Various challenges still exist for YK parameterization
in dual-input single-output (DISO) systems. Reference
[12] applied continuous-time multi-input multi-output
(MIMO) YK formulations to the DISO system, and
constructed the controller parameterization with two
design filters Qand R. [12] also investigated the solv-
ability of Qand R, and revealed the fundamental chal-
lenge of the two-degree-of-freedom design in standard
YK parameterization. In [13], another standard state-
space YK construction is adopted as a servo-enhancing
element. Here, instead of a full DISO design, a SISO
finite-impulse-response Qis used for the control of the
secondary actuator.
We focus on the special class of DISO systems and
discuss new investigations about YK parameterization
for add-on feedback design. Add-on feedback refers to
the design of additional features on a baseline control
system. We show that for DISO [and more generally
multi-input single-output (MISO) systems], we can for-
mulate a reduced-complexity YK parameterization and
obtain simpler realizations for loop shaping. Discrete-
time YK design is known to have differences com-
pared to the continuous-time version of the problem.
A second contribution of this study is the development
of a special DISO discrete-time coprime parameteriza-
tion scheme, which makes the Q-filter design greatly
simplified and approximately separated from the plant
characteristics.
One challenging problem that suits for application
of the proposed algorithm is the rejection of external
vibrations in dual-stage HDD systems. Modern HDDs
are applied to multimedia applications where servo
control confronts much greater challenges than ever
before. For instance, in all-in-one personal computers
and digital TV accessories, HDDs are placed close to
high-power speakers that generate a significant level
of audio vibrations. Such external disturbances, after
passing through the mechanical components, translate
to structural vibrations of the HDD system, and excite
relative movements between the read/write head and
the track centers. Different from conventional narrow-
band vibrations (e.g., internal disk flutters, fan noise)
[14], audio vibrations are much more challenging to re-
ject due to their intrinsic properties of having multiple
resonances and very wide spectral peaks [15]. As the
magnitudes of such external vibrations depend on the
operation environment, flexible add-on servo design is

essential for external-vibration rejection.
A short version of the results was presented in
[16]. This study extends the results with full proofs of
theorems, generalizations to broader systems, as well
as full simulation and experimental results.
II. Standard YK Parameterization
Define Sas the set of stable, proper, and rational transfer
functions. For a SISO discrete-time system P(z−1), YK
parameterization starts with a coprime factorization of
P(z−1) to N(z−1)/D(z−1), where N(z−1)(∈ S) and D(z−1)(∈
S) are coprime over S, i.e., there exist U(z−1) and
V(z−1) in Ssuch that U(z−1)N(z−1)+V(z−1)D(z−1)=
1. If P(z−1) can be stabilized by a negative-feedback
controller C(z−1)=X(z−1)/Y(z−1), with X(z−1) and Y(z−1)
being coprime factorizations of C(z−1) over S, then YK
parameterization [6], [7] provides that any stabilizing
feedback controller can be parameterized as
Call(z−1)=X(z−1)+D(z−1)Q(z−1)
Y(z−1)−N(z−1)Q(z−1),Q(z−1)∈ S (1)
1Two important concepts are implied by the above
result. First, a controller parameterized as (1) is guaran-
teed to generate a stable closed loop, as long as Q(z−1) is
stable. Second, any stabilizing controller can be realized
in the form of (1).
A main benefit of (1) is that the closed-loop transfer
functions become affine in Q(z−1). Of particular interest
is the sensitivity function (the closed-loop transfer func-
tion from the output disturbance to the plant output):
S(z−1)=1
1+P(z−1)Call(z−1)(2)
=1
1+N(z−1)
D(z−1)
X(z−1)
Y(z−1)"1−N(z−1)
Y(z−1)Q(z−1)#(3)
where n1+[N(z−1)X(z−1)]/[D(z−1)Y(z−1)]o−1on the right
side of (3) equals So(z−1),1/(1+P(z−1)C(z−1))—the sen-
sitivity function of the baseline closed loop consisting
of P(z−1) and C(z−1).
For general MIMO systems, dimensions of transfer
functions play important roles. Let Snu×nydenote the
set of stable, proper, and rational transfer functions
with nuinputs and nyoutputs. Then:
Theorem 1: Consider an nu-input ny-output plant P.
Let Pbe stabilized by a controller C(in a negative feed-
back loop), with C=XY−1(X∈ Snu×ny,Y∈ Sny×ny) and
P=ND−1(N∈ Sny×nu,D∈ Snu×nu) being, respectively,
right-coprime factorizations of Cand P. Then the set
of all stabilizing controllers for Pis given by
{(X+DQ)(Y−NQ)−1:Q∈ Snu×ny}(4)
or in the left-coprime format
{(¯
Y−¯
Q¯
N)−1(¯
X+¯
Q¯
D) : ¯
Q∈ Snu×ny}(5)
1A mild condition that Y(z=∞)−N(z=∞)Q(z=∞),0 is needed
for the closed loop transfer functions to be proper and rational.
where P=¯
D−1¯
N(¯
N∈ Sny×nu,¯
D∈ Sny×ny) and C=¯
Y−1¯
X
(¯
X∈ Snu×ny,¯
Y∈ Snu×nu) are, respectively, left-coprime
factorizations of Pand C.
For simplicity, we have omitted the index (z−1) in
the above theorem, and will adopt this format for long
equations in the remaining analysis.
III. Proposed Algorithm for DISO Systems
Consider now the parameterization for dual-input
single-output systems, with the general control struc-
ture shown in Fig. 1. Here P(z−1)=[P1(z−1),P2(z−1)] is
the DISO plant; C(z−1)=[C1(z−1),C2(z−1)]Tis the single-
input dual-output (SIDO) baseline controller.
P
1
(z
¡
1
)
P
2
(z
¡
1
)
d
(k
)
y(k
)
u
1
(k
)
u
2
(k
)
e
(k
)
-
r(k
)
C
2
(z
¡
1
)
C
1
(z
¡
1
)
+
+++
+
P(z
-1
)
Fig. 1. General control structure for DISO systems
For this special class of MIMO system, the general-
ized (and more complicated) YK parameterization in
Theorem 1 certainly works, however at the expense of
reduced tuning intuitions and increased computation.
More specifically, as nu=2 and ny=1 in Fig. 1, the
Q parameter in both (4) and (5) will be a two-by-
one SIDO transfer-function matrix. Compared to the
SISO versions (1) and (3), both the controller and the
sensitivity function will be more complex to implement
and less intuitive to design.
A. Reduced-complexity formulation for DISO systems
To simplify the aforementioned difficulty, we discuss
next a special YK parameterization for DISO systems.
Notice that P(z−1) is a dual-input system but the open-
loop transfer function from e(k) to y(k) in Fig. 1 is
always single-input single-output:
L(z−1)=P1(z−1)C1(z−1)+P2(z−1)C2(z−1).(6)
If we treat L(z−1) as a plant, then the baseline closed
loop is composed of just two SISO elements: L(z−1) and
an identity feedback controller, as shown in Fig. 2.
+-
d
(k
)
r(k
)
++
y(k
)
L
(z
¡
1
)
1
Fig. 2. SISO viewpoint of DISO feedback systems
We propose to perform YK parameterization for the
fictitious plant L(z−1). Consider a coprime factorization
L(z−1)=N(z−1)/D(z−1).(7)

For Fig. 2 we can choose X(z−1)=Y(z−1)=1 for the
identity feedback block, and obtain a simplified form
of (1) for the DISO system:
˜
C(z−1)=1+D(z−1)Q(z−1)
1−N(z−1)Q(z−1),Q(z−1)∈ S.(8)
The controller in (8) can be realized by the dashed
box in Fig. 3, which is an add-on scheme that can be
switched on or off(by enabling or disabling the output
of Q(z−1)), depending on the operation environments.
P
1
(z
¡
1
)
P
2
(z
¡
1
)
d
(k
)
y(k
)
u
1
(k
)
u
2
(k
)
e
(k
)
-
r(k
)
C
2
(z
¡
1
)
C
1
(z
¡
1
)
+
+++
+
Q
(z
¡
1
)
+ +
+
+
a
d
d
-
o
n
Y
o
u
l
a
p
a
r
a
m
e
t
e
r
i
z
a
t
i
o
n
L(z
-1
)
D
(z
¡
1
)
N
(z
¡
1
)
Fig. 3. Proposed YK parameterization for DISO systems
Theorem 2: Let L(z−1)=N(z−1)/D(z−1) be the coprime
factorization of the fictitious plant in Fig. 3. Let the
controller be designed as (8). Stability of the closed-loop
system is guaranteed if Q(z−1) is stable and the baseline
feedback loop (without the add-on YK parameteriza-
tion box) is stable. The closed-loop poles under (8) are
composed of poles of the baseline feedback loop, and
the poles of Q(z−1), N(z−1), and D(z−1). Furthermore,
the sensitivity function is
S(z−1)=So(z−1)(1 −N(z−1)Q(z−1)) (9)
where So(z−1)=1/(1 +N(z−1)D−1(z−1)) =1/(1 +
P(z−1)C(z−1)) is the baseline sensitivity function.
Proof: Let N=BN/AN,D=BD/AD,Q=BQ/AQ,
where B{·}and A{·}denote, respectively, the numerator
and the denominator of the transfer functions. Then
based on (7) and (8),
L=N
D=BNAD
ANBD
,˜
C=ADAQAN+BDBQAN
ADAQAN− ADBNBQ
(10)
The closed-loop characteristic equation in Fig. 3 comes
from
1+L˜
C=0⇔ ADAQAN(BNAD+BDAN)=0 (11)
where BNAD+BDANis the characteristic polynomial
of the baseline feedback loop (obtained from 1+L=0).
The distribution of poles follows from (11).2
Finally, (9) is obtained by a substitution of X(z−1)=
Y(z−1)=1 to (3).
2Notice that the full order of the closed loop is the summation of
the orders of L,D,N, and Qfrom Fig. 3. The stable poles from AD
and ANare canceled between Land ˜
Cin (10). The output response
will only reveal the dynamics of the poles from AQ(BNAD+BDAN).
Design and implementation as an servo-enhancement scheme
From (9), the closed-loop sensitivity function is de-
composed to the product of So(z−1)—the sensitivity
function of the baseline system—and the Q parameter-
ization term 1−N(z−1)Q(z−1). This makes the proposed
algorithm a tool for servo-enhancement design: after
creating a baseline controller that provides basic feed-
back performance and robustness, add-on features can
be directly introduced by designing 1 −N(z−1)Q(z−1),
which is affine in the design parameter Q(z−1). To
reduce the gain of the new sensitivity function S(z−1)
(i.e., enhanced servo performance) in certain frequency
region, we just need to design N(z−1)Q(z−1) to approx-
imate one in the same region. In addition, the baseline
closed-loop poles are always reserved, and new poles
(for different servo requirements) can be introduced by
designing the stable Q(z−1).
Remark: for designing C1(z−1) and C2(z−1) in Fig. 3,
many tools are available, such as decoupled-sensitivity
(see, e.g., [17]), PQ method [18], direct parallel design
[19], and MIMO design methods such as LQG and
H∞/H2control (see [20] and the references therein). In
this article, we focus on add-on servo enhancement,
and refer interested readers to the aforementioned lit-
erature about design of the baseline feedback loop.
B. Coprime parameterization for the fictitious plant
One ideal case for (7) is that N(z−1)=1. This occurs if
L(z−1)=1/L−1(z−1) is a valid coprime factorization, and
will provide a beneficial result of S(z−1)=So(z−1)(1 −
Q(z−1)), i.e., the Q design is completely separated from
the dynamics of L(z−1). If L−1(z−1) is not causal itself,
the ideal factorization can be approximated by
L(z−1)=z−m/L−1
m(z−1),z−m[z−mL−1(z−1)]−1.(12)
Namely, we add delays so that N(z−1)=z−m, and that
D(z−1)=z−mL−1(z−1) is proper/realizable.
The effect of delays: It is common for practical plants
to have input delays. In a DISO setting, the proposed
fictitious plant L(z−1)=P1(z−1)C1(z−1)+P2(z−1)C2(z−1)
has the advantage of reduced influence of delays. This
is one main benefit compared to performing SISO
YK parameterizations to P1(z−1) and P2(z−1) separately.
Let P1(z−1)=z−m1P1m(z−1), P2(z−1)=z−m2P2m(z−1); and
consider the example where m1>m2. Without loss of
generality, we assume that the controllers do not intro-
duce additional separate steps of delays. Then L(z−1)=
z−m2[zm2−m1P1m(z−1)C1(z−1)+P2m(z−1)C2(z−1)]. For the
term in the square bracket, the intermediate delay
zm2−m1will be absorbed in the transfer function. The
total delay for L(z−1) is hence m2=min{m1,m2}—the
minimum of the delay steps among all actuators.
We now apply the concept of (12) to form an inverse-
based coprime factorization of L(z−1) for Fig. 3. If
L−1(z−1) is stable, then L(z−1)=z−m[z−mL−1(z−1)]−1can
be directly used. If not, we approximate it and form a

robust YK parameterization scheme. Denote ˆ
L−1(z−1) as
the nominal stable inverse for L(z−1). Notice that general
feedback commonly aims at achieving a loop shape
similar to that in Fig. 4. For such simple cases we
can construct ˆ
L−1(z−1) by manually choosing poles and
zeros to match the frequency response of L(z−1). For
a complex high-order stable L(z−1), an optimal inverse
design based on H∞minimization is provided in [21].
101102103104
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20
40
60
Gain (dB)
Frequency (Hz)
constrain
mag.
accurate model match
Fig. 4. General loop shape of L(z−1) in servo design
C. The final implementation form and design intuitions
With the discussions in Section III-B, an approximate
coprime factorization for L(z−1) is
L(z−1)=N(z−1)
D(z−1)≈z−m
L−1
m(z−1)=z−m
z−mˆ
L−1(z−1)(13)
and (9) becomes, in the frequency domain,
S(e−jω)≈So(e−jω)(1 −e−mjωQ(e−jω)) (14)
which is obtained by letting z=ejωand ω=2πΩHzTs
(ΩHz is the frequency in Hz; Tsis the sampling
time). At frequency ω,S(e−jω) will have a small gain
if e−mjωQ(ejω) is close to unity; if |Q(e−jω)| ≈ 0 then
S(e−jω)≈So(e−jω), i.e., S(e−jω) remains unchanged at
frequencies where |Q(e−jω)| ≈ 0.
Remark: Conventionally, the design of the Q filter in
YK parameterization does not have a commonly agreed
rule. General discrete-time YK parametrization usually
applies an unstructured finite-impulse-response (FIR)
filter [8]–[10]. In the continuous-time case, discussions
on using a linear combination of some basis transfer
functions [11], [22] have been explored. A more deeper
cause of the diverse Q designs is perhaps the infinite
choice of the plant parameterizations. Indeed, even
for the general reduced-complexity design in (9), we
would have S(e−jω)=So(e−jω)1−N(e−jω)Q(e−jω). The
Q-filter design hence is directly dependent on the
desired servo performance as well as the frequency
response of N(z−1). The fundamental principle of Q
design remains unchanged—in other words, to make
S(e−jω) small for enhanced servo at a particular fre-
quency, 1 −N(e−jω)Q(e−jω) should be small; to keep the
baseline performance at a frequency ωo,Q(e−jωo) should
be designed to approximate zero.
IV. Robustness Against Model Mismatch
This section considers the robustness of the proposed
control scheme. Assume that the plant is perturbed to
˜
Pi(z−1)=Pi(z−1)(1 +Wi(z−1)∆i(z−1)),i=1,2 (15)
with |∆i(e−jω)| ≤ 1∀ωand Wi(z−1) being the uncertainty-
weighting function. The overall feedback controller,
consisting of the add-on YK control and the baseline
C1(z−1) and C2(z−1) in Fig. 3, is given by
¯
C(z−1)="¯
C1(z−1)
¯
C2(z−1)#=1+D(z−1)Q(z−1)
1−N(z−1)Q(z−1)"C1(z−1)
C2(z−1)#.(16)
Robust stability analysis in robust control theory
seeks to find the minimum perturbation such that
detI+˜
P(e−jω)¯
C(e−jω)=1+˜
P(e−jω)¯
C(e−jω)=0.(17)
Theorem 3: For full perturbations where ∆i(e−jω) can
take any complex value satisfying |∆i(e−jω)| ≤ 1, the pro-
posed scheme is robustly stable if and only if nominal
(i.e., when Wi(e−jω)=0) stability holds and
|P1(e−jω)W1(e−jω)¯
C1(e−jω)|+|P2(e−jω)W2(e−jω)¯
C2(e−jω)|
|1+P1(e−jω)¯
C1(e−jω)+P2(e−jω)¯
C2(e−jω)|<1
(18)
where ¯
Ciis from (16).
Proof: See Appendix A.
Discussions: Certainly, there is always a tradeoffbe-
tween performance and robustness. Despite the natural
increase of stability requirement under heavy plant
perturbations, from the performance viewpoint, the al-
gorithm maintains the loop-shaping property for servo
enhancement. To see this, let Lpert ,[˜
P1,˜
P2]¯
Cbe the
perturbed loop transfer function and ∆ = (P1C1W1∆1+
P2C2W2∆2)/(P1C1+P2C2) be the normalized perturba-
tion w.r.t. L. After substituting (15) and (16) in Lpert and
separating the terms about L=P1C1+P2C2=N/D, the
perturbed sensitivity function is
Spert =1
1+Lpert
=1−NQ
1+N
D+(1+DQ)N
D∆(19)
where 1 −NQ is the term for the add-on performance
enhancement. This effect does not change in the pres-
ence of plant uncertainties. .
For the term due to system uncertainty in (19), by
controlling the magnitude of Q, we have the freedom
to make (1+DQ)(N/D)∆small. As a special case, when
Q=0, (19) simplifies to the baseline perturbed sensitiv-
ity function 1/[1 +L(1 + ∆)].
V. Generalization to SISO and MISO Systems
Recalling the proposed SISO viewpoint of loop shap-
ing in Fig. 2, we observe that the block diagram is
not limited to DISO plants where L=P1C1+P2C2.
For SISO and general MISO systems, the loop transfer
function Lobeys L=PnP
i=1PiCi(nPis the number of
inputs). As long as the plant has only one output, the

loop transfer function Lis always SISO and can be
treated as the fictitious plant in Fig. 2. Results in the
preceding analysis thus are directly applicable for SISO
and general MISO systems.
For SISO plants with complex dynamics, one useful
property of the proposed scheme is the capability of
simplified factorization for Lcompared to P, as the
general shape of Lis relatively standard and can be
approximated by low-frequency models (recall Fig. 4)
while Pcan contain various complex dynamics.
VI. Q-filter Design, Simulation,and Experiments
With S(e−jω)≈So(e−jω)(1−e−m jωQ(e−jω)) in (14), loop-
shaping design can simply concentrate on the add-
on element 1 −z−mQ(z−1). This section provides two
application examples in hard disk drive systems, one
about repetitive tracking and regulation, another about
rejecting disturbances including the audio vibrations
described in Section I.
The simulation uses the dual-stage HDD benchmark
system on Page 195 of [20]:
Pv(s)=2.04 ×1021
s+3.14 ×104
s2+2073s+4.3×108
s2+1508s+3.55 ×108
×1
(s2+301.6s+2.58 ×105)(s2+1244s+1.7×109)
Pm(s)=20 ×5.45 ×107
s2+2450s+1.7×109
s2+4524s+2.08 ×109
s2+6032s+3.64 ×109
where the subscripts vand mdenote, respectively,
voice coil motor (VCM) and micro actuator (MA)—
the main components of the two actuators. The plant
models, whose magnitude responses are shown in Fig.
5, are obtained from an actual test drive system. The
continuous-time models are sampled at a sampling
time of Ts=0.04 ms. The disturbance data is from actual
measurements in audio-vibration tests on HDDs.
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Gain (dB)
101102103104
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−90
0
90
180
Phase (degree)
Frequency (Hz)
Pv
Pm
Fig. 5. Frequency responses of the plant
A set of notch filters are designed first to compensate
the resonances:
Nv(z−1)=1
1.2356
z2−1.416z+0.9415
z2−1.297z+0.92
z2+0.1525z+0.95
z2+0.4155z+0.0144
Nm(z−1)=1
2.0669
z2+1.32z+0.7856
z2+0.6232z−0.277
z2+0.148z+0.9
z2+0.439z+0.84
The resonance-compensated plants are then treated
as P1(=PvNv)and P2(=PmNm)for the model-based
controller design. Using this approach, we simplify the
nominal plant models to (see the verifications in Fig. 6)
ˆ
P1(z−1)=z−2(0.0247 +0.02444z−1+0.00374z−2)
1−2.272389z−1+1.5540584z−2−0.281376z−3
ˆ
P2(z−1)=0.3674868z−1
Notice that ˆ
P1(z−1) contains two steps of delays—less
convenient for direct inverse-based SISO YK parame-
terization (recall Section III-B).
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100
Gain (dB)
101102103104
−180
−90
0
90
180
Phase (degree)
Frequency (Hz)
PvNv
nominal P1
PmNm
nominal P2
Fig. 6. Frequency responses of P1,P2, and their nominal models
The baseline controllers use the previously men-
tioned decoupled-sensitivity scheme, with Cv(z−1)=
1.6×1.2356(1 −0.9875z−1)2/(1 −0.2846z−1)/(1 −0.99z−1),
C2(z−1)=1.227 ×2.0669(1 −0.081z−1)/(1 −0.9158z−1),
and C1(z−1)=Cv(z−1)(1 +C2(z−1)ˆ
P2(z−1)). The design
gives 1 +L(z−1)≈(1 +P1(z−1)Cv(z−1))(1 +P2(z−1)C2(z−1)),
so that So(z−1)=1/(1 +L(z−1)) ≈S1o(z−1)S2o(z−1)
where S1o(z−1)=1/(1 +P1(z−1)Cv(z−1)) and
S2o(z−1)=1/(1 +P2(z−1)C2(z−1)) are respectively the
decoupled sensitivities as if two independent feedback
loops about P1(z−1) and P2(z−1) are formed. Fig. 7
shows the magnitude responses of the decoupled
sensitivities, which corresponds to the example loop
shape L(z−1) previously shown in Fig. 4. The notch
filtering and baseline design offer to provide a smooth
magnitude response of L(z−1). Hence although the
order of L(z−1) is 24, a low-order nominal ˆ
L(z−1) can
be readily obtained. By minimizing ||L(z−1)−ˆ
L(z−1)||∞
with balanced model truncation via the square root
method in MATLAB, a fourth-order ˆ
L(z−1) is obtained.
N(z−1) and D−1(z−1) in (13) are, respectively, z−1(i.e.,
m=1) and Lm=0.639743(1 −0.9894z−1)(1−0.984z−1)(1−
0.71196z−1)/[(1 −0.99044z−1)(1 −0.918129z−1)(1 −
1.987587z−1+0.9879973z−2)], which is a minimum-
phase system.
Combining the decoupled-sensitivity design and
the proposed YK scheme, the new sensitivity func-
tion becomes [recall (14)], S(z−1)≈S1o(z−1)S2o(z−1)(1 −

101102103104
−50
−40
−30
−20
−10
0
10
Gain (dB)
Frequency (Hz)
VCM stage
MA stage
Overall
Fig. 7. Magnitude responses in the decoupled-sensitivity design
z−mQ(z−1))—the cascade of three independent compo-
nents. With the form of 1−z−mQ(z−1), the Q-filter design
falls into the same class of problem as that in [21], [23]–
[25].3For m=1, the following band-pass filter [21], [23]
Q(z−1)=(α−1)a+(α2−1)z−1
1+aαz−1+α2z−2,(20)
or more generally, Q(z−1)=BQ(z−1)/AQ(z−1) with
AQ(z−1)=1+
n−1
X
i=1
ai(αiz−i+α2n−iz−2n+i)+anαnz−n+α2nz−2n
BQ(z−1)=
2n
X
i=1
(αi−1)aiz−i+1,ai=a2n−i(21)
can achieve sample loop shapes as shown in Figs. 8
and 9 [nin (21) is the number of bands in the figures].
101102103104
−100
−50
0
Frequency (Hz)
Magnitude (dB)
baseline
w/ proposed Youla scheme
Fig. 8. Enhanced sensitivity functions for n=3 narrow bands
As the sensitivity function is the output disturbance
rejection function, the reduced local gains in Fig. 8
make the design suitable for rejecting strong vibrations
at several close frequencies. Fig. 9 expands the range of
disturbance attenuation and suits for mixed structural
vibrations that come from a frequency-rich excitation
source. The change of notch width is controlled by
the coefficient α∈(0,1)in (20) and (21). It can be
observed that in both Figs. 8 and 9, S(z−1) has very
small gain at bands of frequencies, while amplifications
3[21], [23]–[25] discuss only SISO designs using the concept
of extended disturbance observers. The Q-design methodology can
however be applied to the problem in the present study.
101102103104
0
50
100
150
Frequency (Hz)
Magnitude (dB)
Overall open−loop response
101102103104
−150
−100
−50
0
Magnitude (dB)
Sensitivity functions
baseline
w/ proposed Youla scheme
Fig. 9. Loop shaping design for rejection of two wide-band vi-
brations: upper plot—sensitivity functions So(z−1) and S(z−1); lower
plot—corresponding open loop responses L(z−1) and L(z−1)˜
C(z−1)
at other frequencies are very small.4Fig. 9 also plots the
magnitude response of the open-loop transfer function,
which shows the equivalent effect of high-gain control
at the disturbance frequencies.
Fig. 10 shows the error spectra under audio vibra-
tions, with and without the design in Fig. 9. Due to the
deep notches in the sensitivity function, the strong fre-
quency components at around 1000 and 2300 Hz have
been successfully rejected without visual amplification
of other error components. Overall, the proposed algo-
rithm provides an 42.3 percent of 3σ(σis the standard
deviation) reduction in the position error signal (PES).
500 1000 1500 2000 2500 3000
0
1
2
3
normalized |FFT| (nm)
w/ baseline decoupled sensitivity
3σ = 44.25 nm
500 1000 1500 2000 2500 3000
0
1
2
3
Frequency (Hz)
w/ proposed add-on enhancement
normalized |FFT| (nm)
3σ = 25.53 nm
Fig. 10. Error spectra with and without compensation
4The coefficients ai’s in (20) and (21) determine the center fre-
quencies of the notches in Figs. 8 and 9. For (20), we have a=
−2cos(2πΩTs), where Ωis the desired notch frequency in Hz. For
the case with multiple bands, the filter parameter aiin (21) and the
notch frequency Ωisatisfy 1+Pn−1
i=1ai(αiz−i+α2n−iz−2n+i)+anαnz−n+
α2nz−2n=Qn
i=1(1 −2cos(2πΩiTs)αz−1+α2z−2).

Fig. 11 compares the proposed design with the pop-
ular peak/resonant filter algorithm5[26], [27] for HDD
band-limited disturbance rejection. Both algorithms are
configured to achieve similar disturbance rejection at
six wide frequency bands. Notice that the proposed
algorithm focuses more on the overall target frequency
range and has a larger effective attenuation range.
The peak filter algorithm on the other hand is a
combination of six “discrete” attenuation ranges, with
less consideration on the overall performance (at some
intermediate frequencies the disturbances are actually
amplified). Another hidden difference is the capability
of adaptive configurations. Interested readers can refer
to [10], [21] for details of adaptive Q parameterization
in YK schemes.
103104
−35
−30
−25
−20
−15
−10
−5
0
5
10
Frequency (Hz)
Magnitude (dB)
baseline | S |
| S | w/ optimal peak filter
| S | w/ proposed scheme
Fig. 11. Performance comparison with peak filter algorithm
The parameterization is additionally verified via ex-
periments on a Western Digital 2.5-inch test drive, for
rejection of repeatable runout (RRO) errors that come
from nm-scale imperfections in the data tracks and
disk rotations. Such errors are also typical in general
mechanical systems involving periodic motions.
The Q-filter applies the enhanced repetitive con-
trol scheme recently developed in [25]: Q(z−1)=
z−nqq(z,z−1)[(1 −αN)z−(N−m−nq)]/[1 −αNz−N]. Here, Nis
the period of the first harmonic component; α(∈[0,1])
determines the width of attenuation region; q(z,z−1) is
a zero-phase low-pass filter for robustness; and nqis
the highest order of zin q(z,z−1).
Fig. 12 shows the time trace of the PES and its FFT
spectrum. In the illustrative example, N=310, m=2,
αN=0.99, and nq=1. The proposed algorithm is seen to
effectively reject the harmonic disturbances in both time
and frequency domains. In the top plot, the width of
one track is in the order of 100 nm. In the bottom plot,
it can be observed that the single Q filter compensates
multiple spectral peaks in a large range of frequencies:
nearly full rejection below 1000 Hz and partial rejection
up to around 3000 Hz. Such attenuation is directly
reflected in Fig. 13, which plots the magnitude response
of 1−z−mQ(z−1), i.e., changes in the sensitivity function
due to the Q-filter design [recall (14)]. An additional 17
5The peak filter is added in the VCM stage; the baseline design
uses the same decoupled-sensitivity loop shaping.
0 500 1000 1500
−10
0
10
Time/sampling time
PES %(Track)
102103104
0
1
2
3
4
5
Frequency (Hz)
normalized FFT amplitude (%Track)
baseline
w/ compensation
Fig. 12. Time-domain (top plot) and frequency-domain (bottom plot)
disturbance rejection result for harmonic errors.
10
2
10
3
10
4
-60
-50
-40
-30
-20
-10
0
Magnitude (dB)
Frequency (Hz)
Fig. 13. Magnitude response of 1−z−mQ(z−1) for RRO rejection
revolutions of data are collected and analyzed. The 3σ
values of the error signal are shown in Table I. Here
the baseline 3σvalue is normalized to 100, and the
proposed algorithm is seen to provide an overall 55.01
percent of performance improvement.
TABLE I
Algorithm performance in repeatable-error rejection
baseline with proposed algorithm
Normalized 3σ100 44.99
VII. Conclusion
We have introduced a loop-shaping concept for DISO
systems (with extension to SISO and MISO systems).
Mathematically, the proposed algorithm performs loop
shaping via formulating the sensitivity function as S≈
(1−NQ)/(1 +P1C1+P2C2). Instead of directly augment-
ing the controllers C1and C2, the idea of the proposed
servo-enhancement design is to build a baseline system
first and then focus on the affine add-on Q parame-
terization 1 −NQ. Several application examples have
demonstrated the validity of the algorithm and ex-
plained its design intuitions. The loop-shaping idea is

seen, via both simulation and experiments, to provide
strong flexibility in precision servos.
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Appendix A: Proof of Theorem 3
Proof: Substituting (16) into (17) gives
1+˜
P1(e−jω)¯
C1(e−jω)+˜
P2(e−jω)¯
C2(e−jω)=0.(22)
When nominal stability holds, the distance from
P(e−jω)¯
C(e−jω) to the (−1,0)point is always positive,
thus 1 +P1(e−jω)¯
C1(e−jω)+P2(e−jω)¯
C2(e−jω),0.Divid-
ing this quantity on both sides of (22) gives [1 +
˜
P1(e−jω)¯
C1(e−jω)+˜
P2(e−jω)¯
C2(e−jω)]/[1+P(e−jω)¯
C(e−jω)] =
0.Substituting in (15) and using P¯
C=P1¯
C1+P2¯
C2
yield 1 +P1¯
C1W1∆1/(1 +P¯
C)+P2¯
C2W2∆2/(1 +P¯
C)=0.
The worst-case minimum perturbation happens when
|∆1(e−jω)|=|∆2(e−jω)|=|∆o(e−jω)|and the perturbation
directions are such that
1−
P1¯
C1W1
1+P¯
C
|∆o| − 
P2¯
C2W2
1+P¯
C
|∆o|=0 (23)
In other words, |∆o|=|1+P1¯
C1+P2¯
C2|/(|P1¯
C1W1|+
|P2¯
C2W2|). If (18) is valid then |∆0|>1, which is not
possible as |∆i| ≤ 1 by definition; hence the system is
robustly stable. For the “only if” part of the proof, if
(18) is violated, then |∆o|<1. Letting |∆1|=|∆2|=|∆o|
with ∠∆i=−∠Pi¯
CiWi/(1 +P¯
C)achieves (23) and hence
system instability.