Content uploaded by Xu Chen
Author content
All content in this area was uploaded by Xu Chen on Nov 25, 2015
Content may be subject to copyright.
Discretetime Reducedcomplexity Youla Parameterization for
Dualinput Singleoutput Systems
Xu Chen†and Masayoshi Tomizuka
Abstract— A controller design is presented for directly
shaping the sensitivity function in feedback control of dual
input singleoutput (DISO) systems. We provide special
forms of discretetime YoulaKucera (YK) parameterization
to obtain stabilizing controllers for DISO systems, and
discuss concepts to achieve a reducedcomplexity 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
wideband vibration and harmonic disturbance rejections.
Index Terms— precision control, loop shaping, vibration
rejection, discretetime Youla parameterization, dualstage
actuation, dualstage hard disk drives, multiinput single
output systems
I. Introduction
Dualstage actuation is a hardware solution to exceed
the performance limit of singleactuator 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
secondstage 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 addon loopshaping
designs via Youla parameterization for dualstage sys
tems. Youla parameterization [6], [7], aka YoulaKucera
(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 singleinput 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 dualinput singleoutput (DISO) systems. Reference
[12] applied continuoustime multiinput multioutput
(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 twodegreeoffreedom design in standard
YK parameterization. In [13], another standard state
space YK construction is adopted as a servoenhancing
element. Here, instead of a full DISO design, a SISO
ﬁniteimpulseresponse 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 addon feedback design. Addon feedback refers to
the design of additional features on a baseline control
system. We show that for DISO [and more generally
multiinput singleoutput (MISO) systems], we can for
mulate a reducedcomplexity YK parameterization and
obtain simpler realizations for loop shaping. Discrete
time YK design is known to have diﬀerences com
pared to the continuoustime version of the problem.
A second contribution of this study is the development
of a special DISO discretetime coprime parameteriza
tion scheme, which makes the Qﬁlter 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 dualstage HDD systems. Modern HDDs
are applied to multimedia applications where servo
control confronts much greater challenges than ever
before. For instance, in allinone personal computers
and digital TV accessories, HDDs are placed close to
highpower 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 addon servo design is
essential for externalvibration 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 discretetime 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 negativefeedback
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 closedloop transfer
functions become aﬃne in Q(z−1). Of particular interest
is the sensitivity function (the closedloop 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 nuinput nyoutput 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,
rightcoprime 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 leftcoprime 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, leftcoprime
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 dualinput
singleoutput 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 dualoutput (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 twoby
one SIDO transferfunction 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. Reducedcomplexity 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 dualinput system but the open
loop transfer function from e(k) to y(k) in Fig. 1 is
always singleinput singleoutput:
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 addon 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 closedloop
system is guaranteed if Q(z−1) is stable and the baseline
feedback loop (without the addon YK parameteriza
tion box) is stable. The closedloop 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 closedloop 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 servoenhancement scheme
From (9), the closedloop 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 servoenhancement design: after
creating a baseline controller that provides basic feed
back performance and robustness, addon 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
closedloop 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 decoupledsensitivity
(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 addon 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 highorder stable L(z−1), an optimal inverse
design based on H∞minimization is provided in [21].
101102103104
−20
0
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 IIIB, 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 discretetime YK parametrization usually
applies an unstructured ﬁniteimpulseresponse (FIR)
ﬁlter [8]–[10]. In the continuoustime 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 reducedcomplexity design in (9), we
would have S(e−jω)=So(e−jω)1−N(e−jω)Q(e−jω). The
Qﬁlter 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 addon 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
detI+˜
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 loopshaping 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 addon 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 lowfrequency models (recall Fig. 4)
while Pcan contain various complex dynamics.
VI. Qfilter 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 dualstage 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
continuoustime models are sampled at a sampling
time of Ts=0.04 ms. The disturbance data is from actual
measurements in audiovibration tests on HDDs.
101102103104
−50
0
50
Gain (dB)
101102103104
−180
−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 resonancecompensated plants are then treated
as P1(=PvNv)and P2(=PmNm)for the modelbased
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 inversebased SISO YK parame
terization (recall Section IIIB).
101102103104
−100
−50
0
50
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 decoupledsensitivity 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 loworder 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 fourthorder 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 decoupledsensitivity 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 decoupledsensitivity design
z−mQ(z−1))—the cascade of three independent compo
nents. With the form of 1−z−mQ(z−1), the Qﬁlter design
falls into the same class of problem as that in [21], [23]–
[25].3For m=1, the following bandpass 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 frequencyrich 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 Qdesign 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 wideband 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 openloop transfer function,
which shows the equivalent eﬀect of highgain 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 addon 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
bandlimited 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 2.5inch test drive, for
rejection of repeatable runout (RRO) errors that come
from nmscale imperfections in the data tracks and
disk rotations. Such errors are also typical in general
mechanical systems involving periodic motions.
The Qﬁlter 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 zerophase lowpass 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 Qﬁlter design [recall (14)]. An additional 17
5The peak ﬁlter is added in the VCM stage; the baseline design
uses the same decoupledsensitivity 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. Timedomain (top plot) and frequencydomain (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 repeatableerror rejection
baseline with proposed algorithm
Normalized 3σ100 44.99
VII. Conclusion
We have introduced a loopshaping 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
servoenhancement design is to build a baseline system
ﬁrst and then focus on the aﬃne addon Q parame
terization 1 −NQ. Several application examples have
demonstrated the validity of the algorithm and ex
plained its design intuitions. The loopshaping idea is
seen, via both simulation and experiments, to provide
strong ﬂexibility in precision servos.
References
[1] D. Y. Abramovitch and G. F. Franklin, “A brief history of disk
drive control,” IEEE Control Syst. Mag., vol. 22, no. 3, pp. 28–42,
2002.
[2] B.S. Kim, J. Li, and T.C. Tsao, “Twoparameter robust repetitive
control with application to a novel dualstage actuator for
noncircular machining,” IEEE/ASME Trans. Mechatronics, vol. 9,
no. 4, pp. 644–652, 2004.
[3] W. Dong, J. Tang, and Y. ElDeeb, “Design of a linearmotion
dualstage actuation system for precision control,” Smart Mate
rials and Structures, vol. 18, no. 9, p. 095035, 2009.
[4] S.K. Hung, E.T. Hwu, M.Y. Chen, and L.C. Fu, “Dualstage
piezoelectric nanopositioner utilizing a rangeextended optical
ﬁber fabryperot interferometer,” IEEE/ASME Trans. Mechatron
ics, vol. 12, no. 3, pp. 291–298, 2007.
[5] C.L. Chu and S.H. Fan, “A novel longtravel piezoelectric
driven linear nanopositioning stage,” Precision Engineering,
vol. 30, no. 1, pp. 85 – 95, 2006.
[6] D. Youla, J. J. Bongiorno, and H. Jabr, “Modern wiener–hopf
design of optimal controllers part i: The singleinputoutput
case,” IEEE Trans. Autom. Control, vol. 21, no. 1, pp. 3–13, 1976.
[7] V. Kucera, “Stability of discrete linear feedback systems,” in
Proc. 6th IFAC World Congress, paper 44.1, vol. 1, 1975.
[8] R. de Callafon and C. E. Kinney, “Robust estimation and
adaptive controller tuning for variance minimization in servo
systems,” Journal of Advanced Mechanical Design, Systems, and
Manufacturing, vol. 4, no. 1, pp. 130–142, 2010.
[9] F. B. Amara, P. T. Kabamba, and A. G. Ulsoy, “Adaptive si
nusoidal disturbance rejection in linear discretetime systems—
part i: Theory,” Journal of Dynamic Systems, Measurement, and
Control, vol. 121, no. 4, pp. 648–654, 1999.
[10] I. D. Landau, A. C. Silva, T.B. Airimitoaie, G. Buche, and
M. Noe, “Benchmark on adaptive regulation–rejection of
unknown/timevarying multiple narrow band disturbances,”
European Journal of Control, vol. 19, no. 4, pp. 237 – 252, 2013.
[11] B. D. Anderson, “From youla kucera to identiﬁcation, adaptive
and nonlinear control,” Automatica, vol. 34, no. 12, pp. 1485 –
1506, 1998.
[12] J. Zheng, W. Su, and M. Fu, “Dualstage actuator control design
using a doubly coprime factorization approach,” IEEE/ASME
Trans. Mechatronics, vol. 15, no. 3, pp. 339–348, 2010.
[13] G. Guo, Q. Hao, and T.S. Low, “A dualstage control design
for high track per inch hard disk drives,” IEEE Trans. Magn.,
vol. 37, no. 2, pp. 860–865, Mar. 2001.
[14] L. Guo and Y.J. Chen, “Disk ﬂutter and its impact on hdd servo
performance,” IEEE Trans. Magn., vol. 37, no. 2, pp. 866 –870,
Mar. 2001.
[15] J. R. Deller, J. H. L. Hansen, and J. G. Proakis, DiscreteTime
Processing of Speech Signals. WileyIEEE Press, Sep. 1999.
[16] X. Chen and M. Tomizuka, “Reducedcomplexity and ro
bust youla parameterization for discretetime dualinputsingle
output systems,” in Proc. IEEE/ASME Int. Conf. on Adv. Intelligent
Mechatronics, Wollongong, Australla, July 912, 2013, pp. 490–497.
[17] R. Horowitz, Y. Li, K. Oldham, S. Kon, and X. Huang, “Dual
stage servo systems and vibration compensation in computer
hard disk drives,” Control Engineering Practice, vol. 15, no. 3,
pp. 291–305, 2007.
[18] S. Schroeck, W. Messner, and R. McNab, “On compensator de
sign for linear timeinvariant dualinput singleoutput systems,”
IEEE/ASME Trans. Mechatronics, vol. 6, no. 1, pp. 50–57, Mar.
2001.
[19] T. Semba, T. Hirano, J. Hong, and L.S. Fan, “Dualstage servo
controller for hdd using mems microactuator,” IEEE Trans.
Magn., vol. 35, no. 5, pp. 2271–2273, Sep 1999.
[20] A. Al Mamun, G. Guo, and C. Bi, Hard disk drive: mechatronics
and control. CRC Press, 2007.
[21] X. Chen and M. Tomizuka, “Selective model inversion and
adaptive disturbance observer for timevarying vibration rejec
tion on an activesuspension benchmark,” European J. of Control,
vol. 19, no. 4, pp. 300 – 312, 2013.
[22] J. C. Doyle, B. A. Francis, and A. Tannenbaum, Feedback control
theory. Macmillan, 1992, vol. 134.
[23] X. Chen and M. Tomizuka, “A minimum parameter adaptive
approach for rejecting multiple narrowband disturbances with
application to hard disk drives,” IEEE Trans. Control Syst. Tech
nol., vol. 20, no. 2, pp. 408 –415, march 2012.
[24] X. Chen, A. Oshima, and M. Tomizuka, “Inverse based local
loop shaping for vibration rejection in precision motion control,”
in Proc. 6th IFAC Symposium on Mechatronic Systems, Hangzhou,
China, April 1012, 2013, pp. 490–497.
[25] X. Chen and M. Tomizuka, “New repetitive control with im
proved steadystate performance and accelerated transient,”
IEEE Trans. Control Syst. Technol., vol. 22, no. 2, pp. 664–675,
March 2014.
[26] J. Zheng, G. Guo, Y. Wang, and W. Wong, “Optimal narrow
band disturbance ﬁlter for pztactuated head positioning control
on a spinstand,” IEEE Trans. Magn., vol. 42, no. 11, pp. 3745–
3751, 2006.
[27] T. Atsumi, A. Okuyama, and M. Kobayashi, “Trackfollowing
control using resonant ﬁlter in hard disk drives,” IEEE/ASME
Transactions on Mechatronics, vol. 12, no. 4, pp. 472–479, Aug
2007.
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 worstcase 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.