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Add-on Loop Shaping via Youla Parameterization for Precision Motion Control

Authors:
ADD-ON LOOP SHAPING VIA YOULA PARAMETERIZATION FOR
PRECISION MOTION CONTROL
Xu Chen1, and Masayoshi Tomizuka1
1Department of Mechanical Engineering
University of California, Berkeley
Berkeley, California, USA
INTRODUCTION
Advances in manufacturing are urging the inno-
vation of new hardware and software in precision
motion control systems. In the year of 2011, the
manufacturing sector generated 12.2% of total
U.S. GDP [1]. This percentage is additionally pro-
jected to increase greatly in future [1]. As a result,
the continuously updated requirement of higher
speed and higher accuracy has placed new chal-
lenges for servo design, where standard feedback
control techniques alone (such as PID and H
control) are commonly not sufficient to achieve
the performance requirements [2].
In this paper, we discuss add-on loop-shaping
ideas via Youla parameterization, aka all stabi-
lizing controller parameterization, for improved
servo performance. Loop shaping here refers
to the frequency-domain servo design concept
about shaping the closed-loop dynamic behavior.
We present the design of flexible Youla parame-
terization to address common control challenges
in precision motion control. Specifically, we show
that the important problems of repetitive control,
active vibration rejection, and bandwidth adjust-
ment, can be uniformly formulated in the same
loop-shaping scheme via Youla parameterization.
One particular advantage of such a controller for-
mulation is that stability and servo performance
can be approximately separated, yielding an intu-
itive and performance-orientated design.
The discussed algorithms are best suited for pre-
cision control systems where the system dynam-
ics are linear time invariant, and an accurate sys-
tem model is available (from, e.g., system iden-
tification and finite element analysis). High sam-
pling rate, accurate sensors, and precision actua-
tors are common features of these systems. One
example is the wafer scanner for lithography in
the semiconductor industry. We will use an ex-
perimental setup of such a system for algorithm
verification later in the paper. The proposed se-
lection of Youla structure has also been success-
fully applied to hard disk drives [4], active sus-
pensions [3], and electrical power steering in au-
tomotive vehicles [6]. The unified analysis for dif-
ferent loop-shaping schemes and the detailed im-
plementation steps on the wafer scanner however
have not been discussed before.
THE DESIRED LOOP SHAPE
To motivate Youla parameterization, consider a
general feedback closed loop as shown in Fig.
1. To let the output yfollow the reference r
while rejecting the disturbances dand do, the
controller Cis designed such that Gry, the
transfer function from yto r, approximates unity;
and the transfer functions from the disturbances
to yare maintained small. The two problems
about reference tracking and disturbance regu-
lation are connected by the sensitivity function
S,1/(1 + P C)–the transfer function from do
to y. Standard PID and Hdesign can read-
ily achieve a magnitude response of Sas shown
in Fig. 2: it has small gains below the band-
width ωc, where disturbances are attenuated and
Gry=P C/(1 + P C )=1S1.
+-
C
P
+
d
o
+
e
y
r
+
+
FIGURE 1. Standard feedback design structure
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Magnitude (dB)
Frequency (Hz)
ωc
FIGURE 2. Magnitude response of a standard
sensitivity function
Due to imperfections in mechanical components
and the operation environments, one single con-
troller is commonly not sufficient for all tasks in
an actual system. For micro/nano scale precision
servo, feedback design has to be customized as
much as possible by considering characteristics
of the disturbance and the control task. Fig. 3
shows an example loop shape that is customized
for enhanced servo performance at a local fre-
quency region. The solid line is from a standard
design. The dashed/dotted lines are the modified
versions. Following the preceding discussions,
in the notch-shape region around 900 Hz, distur-
bances will be strongly attenuated and reference
components will be followed at an improved ac-
curacy. In this example we have just one notch
shape. More may be required for multiple dis-
turbance rejection and reference enhancement.
Such a concept is not unknown in control engi-
neering. However, the levels of design intuition,
achievable performances,stability requirements,
and algorithm flexibility differ greatly in different
controller constructions.
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Frequency (Hz)
Magnitude (dB)
Frequency response of the sensitivity function
FIGURE 3. A sample customized loop shape
Among the set of all controllers that stabilize the
plant, it is desired to choose the best possible de-
sign for implementation. Youla parameterization
provides a convenient way to realize this concept.
Theorem 1 Define the set S:={stable, proper,
and rational transfer functions}. A single-input-
single-output system P(z1)can be parame-
terized as P(z1) = N(z1)/D(z1), where
N(z1)and D(z1)are coprime over S, mean-
ing there exists U(z1),V(z1)in Ssuch that
U(z1)N(z1) + V(z1)D(z1)=1. If P(z1)
can be stabilized by a negative feedback con-
troller C(z1) = X(z1)/Y (z1), with X(z1)
and Y(z1)coprime over S, then any stabilizing
feedback controller can be parameterized as
X(z1) + D(z1)Q(z1)
Y(z1)N(z1)Q(z1), Q(z1)S.(1)
This powerful concept suggests us the following
design concept for precision motion control: (i)
design first a baseline controller that satisfies the
performance requirement under basic operations
(e.g., one that achieves a common loop shape as
shown in Fig. 2); (ii) apply Youla parameterization
and introduce a customized Q(z1)to modify the
servo loop under different servo requirements.
CUSTOMIZED YOULA PARAMETERIZATION
FOR PRECISION MOTION CONTROL
By adopting the concept of Youla parameteriza-
tion, we obtain very simple stability requirements
(just need Qto be stable) and strong algorithm
flexibility (all controllers can be parameterized by
(1)). This section discusses further customiza-
tions to extend the design intuition and achiev-
able performance. A realization of the control
scheme using controller (1) is shown in Fig. 4.
Three sets of elements need to be constructed
to close the loop: the design of the baseline con-
troller C(z1), the plant parameterization, and the
choice of Q(z1)in (1). In the context of precision
motion control, we propose the following design
steps:
+-X P
N
Q
Y
-1
D+ +
+
+++
FIGURE 4. Block diagram of the feedback control
system with Youla parameterization
step 1: use standard loop-shaping techniques
such as PID, lead-lag, or Hcontrol to design
a stable C(z1)for a baseline servo loop. The
controller coprime factorization can then be sim-
ply chosen as X(z1) = C(z1)and Y(z1) = 1
in Fig. 4. Besides its simplicity, such a factor-
ization brings increased design and tuning intu-
itions. Noting that Y(z1)=1, if we lump all dis-
turbances at the plant input, then the purpose of
the customized Youla parameterization is to use
the output of Q(the signal denoted by cin Fig. 4)
to approximate dfor disturbance cancellation.
The output of Qcan thus serve as an observed
disturbance signal for algorithm verification and
tuning. For instance, when testing the system of-
fline via simulation, we can compare cand dto
see if they match before closing the switch after
the Qblock. In the reference-tracking case, when
there is imperfect tracking, the signal ccan serve
to explain the equivalent disturbance for identify-
ing the critical errors.
step 2: factorize the discrete-time plant model by
P(z1) = zmPn(z1) = zm/P 1
n(z1)(2)
so that N(z1) = zmand D(z1) = P1
n(z1)
in Fig. 4. Here P1
n(z1)should be stable for a
valid coprime factorization. This is usually easy to
satisfy for motion control systems. If the inverse
plant is indeed unstable, we can use a stable ver-
sion to approximate it. As a design example that
will be used in the case-study section, if
P(z1) = z23.4766 ×107(1 + 0.8z1)
(1 z1)2(3)
then P1
n(z1) = (1 z1)2/3.4766 ×107/(1 +
0.8z1), whose poles are already inside the unit
circle. Hence we can choose the following co-
prime factorization N(z1) = z2, D(z1) =
(1 z1)2/[3.4766 ×107(1 + 0.8z1)].
step 3: with the discussed choices of X(z1),
Y(z1),N(z1), and D(z1), the extended
feedback controller in (1) becomes ˜
C(z1) =
[C(z1)+ P1
n(z1)Q(z1)]/[1zmQ(z1)]. Us-
ing (2) and after some algebra, we can derive the
new sensitivity function
˜
S(z1) = 1
1 + P(z1)˜
C(z1)=1zmQ(z1)
1 + P(z1)C(z1)
(4)
Although multiple elements have been added to
the baseline loop, (4) is quite simple as it dif-
fers from the original sensitivity function S(z1) =
1/(1+P(z1)C(z1)) only by a multiplicative term
1zmQ(z1). To introduce small magnitude re-
sponse in Fig. 2, we just need to design Q(z1)
such that 1ejωm Q(e )has low gains at the
interested frequency region. This helps to pro-
vide intuitive designs that reach high achievable
performances. Well-formulated tools such as in-
ternal model principle, Diophantine/Bezout equa-
tion, and convex optimization can be applied to
design Q(z1)[3-6]. Relevant examples for pre-
cision motion control include:
(i) Active vibration rejection: although precision
systems commonly include vibration-absorbing
elements such as vibration isolation tables, pas-
sive damping and spring elements have a physi-
cal bandwidth above which they can not respond
fast enough for energy absorption. There are also
environmental disturbance that heavily depends
on the operation condition and can even be time-
varying. The loop shape in Fig. 3 suits for at-
tenuating these vibrations actively from the con-
trol perspective. The different attenuation levels
in the dash/dotted lines can be easily achieved
by configurations in Q(z1). Additionally, we can
observe that, when strongly attenuating distur-
bances at a local frequency region, the sensitiv-
ity function Sdid not have visible large amplifi-
cation at other frequency regions. The essential
design in this scheme is to assign 1zmQ(z1)
a notch-filter structure. Depending on the width of
the desired notch, we can classify the problem to
rejections of narrow-band disturbances [3,4] and
general band-limited vibrations [6].
(ii) Repetitive control: this is common in manufac-
turing process where the majority of operations
are repetitive. From Fourier series theory, any
periodic disturbance or reference can be decom-
posed to summations of sinusoidal components
at multiples of a fundamental frequency. For the
feedback loop to have enhanced servo perfor-
mance at these repetitive frequencies, we can let
1zmQ(z1)have a comb shape in the magni-
tude response (see Fig. 6). This can be achieved
by careful pole-zero placement in 1zmQ(z1),
via internal model principle [5].
(iii) Bandwidth extension: if the original baseline
design C(z1)is too conservative, the proposed
scheme can serve as a bandwidth-extension el-
ement. This can be readily done by assigning
a low-pass filter structure to Q(z1), which will
make 1zmQ(z1)a high-pass filter.
CASE STUDY ON A WAFER SCANNER
Successful implementations of vibration rejection,
repetitive disturbance cancellation, and periodic
trajectory tracking have been obtained on a wafer-
scanner testbed as shown in Fig. 5. This is an
essential equipment in the semiconductor indus-
try.1Due to limited space, we discuss below de-
tails about just the periodic tracking results on the
reticle stage.
The plant has a continuous-time model P(s) =
1/(0.2556s2+ 0.279s). After discretization at a
sampling time of Ts= 0.4ms, the zero-order-
hold equivalent of P(s)has the structure of k(z+
1)/[z(z1)2], where kis a constant. Due to
computation delays and signal processing, there
is an additional one-step delay for P(z1). The
zero at 1makes the nominal inverse of P(z1)
1There are two stages in the system: a wafer stage and a
reticle stage, both capable of two-DOF plane motions. In the
testbed one axis is set up in each stage for algorithm test.
FIGURE 5. A testbed of wafer scanner system
marginally stable. Replacing it with a stable one
at 0.8and adjusting kso that the system gain
matches that of P(s)at low frequencies, we ob-
tain the nominal plant model discussed in (3).
This slight modification of the system zero at fre-
quencies far above the system bandwidth does
not yield much modeling errors in the frequency
domain, and makes z2and P1
n(z1)a valid co-
prime factorization of (3).
The baseline controller is a PID controller C=
10000[1 + 2Ts/(1 z1)+0.012(1 z1)/Ts]. It
stabilizes the loop and provides a baseline sensi-
tivity function in the shape of Fig. 2. Letting the
wafer scanner repeatedly track a scanning trajec-
tory, we obtain the tracking error as shown in the
dashed line in Fig. 7, where we can directly ob-
serve the periodic pattern for the error signal.
Recall the loop-shaping idea in Fig. 3 and the
discussion in item (ii) in the last section. If we add
the shape in the top plot of Fig. 6 to the baseline
Fig. 2, we can reduce the periodic errors at multi-
ples of the fundamental frequency. The Q filter to
achieve Fig. 6 is [5]
Qz1=1αNz(Nmnq)
1αNzNznqqz, z1
which is a special periodic signal extractor as
shown in the bottom plot of Fig. 6. Here mis
the plant delay in Youla parameterization; Nis
the period of the trajectory; q(z, z1)is a zero-
phase low-pass filter with order nq; and αis a
design parameter that controls the width of the
comb shapes. An αcloser to 1gives a flatter
shape for 1zmQ(z1)(reduced amplification
of the non-repetitive disturbances). A trade off in
this case is that the algorithm requires more ac-
curate knowledge about the period of the distur-
bance/trajectory. After the proposed compensa-
tion scheme is turned on, the errors in Fig. 7 are
significantly reduced to be two-magnitude lower
than the original values.
ACKNOWLEDGMENT
100101102103
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Magnitude (dB)
1−z−mQ(z−1)
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Magnitude (dB)
Frequency (Hz)
Q(z−1)
FIGURE 6. A Q-design example
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0
2
4x 10−4
Sample
Position error (m)
w/o compensation
w/ compensation
FIGURE 7. Tracking errors
The authors thank the support from CML at UC
Berkeley, Nikon, Agilent Technologies, and Na-
tional Instruments.
REFERENCES
[1] Advanced Manufacturing Portal,
http://www.manufacturing.gov
[2] Tan K., et. al. Precision motion control: design
and implementation. Springer, 2008.
[3] Chen X. and Tomizuka M., Adaptive Model In-
version For Rejection of Time-varying Vibrations
On A Benchmark Problem, to appear in The Eu-
ropean Control Conf. 2013, Jul. 17-19, 2013.
[4] ——, A Minimum Parameter Adaptive Ap-
proach for Rejecting Multiple Narrow-Band Dis-
turbances with Application to Hard Disk Drives,
IEEE Trans. Control Syst. Technol., vol. 20, no.
2, pp. 408-415, Mar. 2012.
[5] ——, New Repetitive Control with Improved
Steady-state Performance and Accelerated Tran-
sient, to appear in IEEE Trans. Control Syst.
Technol. 2013.
[6] Chen X., Oshima A., and Tomizuka M., Inverse
Based Local Loop Shaping For Vibration Rejec-
tion In Precision Motion Control, to appear in The
6th IFAC Symposium on Mechatronic Syst., Apri.
10-12, 2013.
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Article
Full-text available
Many servo systems are subjected to narrow-band disturbances that generate vibrations at multiple frequencies. One example is the track-following control in a hard disk drive (HDD) system, where the airflow-excited disk and actuator vibrations introduce strong and uncertain spectral peaks to the position error signal. Such narrow-band vibrations differ in each product and can appear at frequencies above the bandwidth of the control system. We present a feedback control scheme that adaptively enhances the servo performance at multiple unknown frequencies, while maintaining the baseline servo loop shape. A minimum parameter model of the disturbance is first introduced, followed by the construction of a novel adaptive multiple narrow-band disturbance observer for selective disturbance cancellation. Evaluation of the proposed algorithm is performed on a simulated HDD benchmark problem.
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Precision Tracking Motion Control.- Automatic Tuning of Control Parameters.- Co-ordinated Motion Control of Gantry Systems.- Geometrical Error Compensation.- Electronic Interpolation Errors.- Vibration Monitoring and Control.- Other Engineering Aspects.
Adaptive Model Inversion For Rejection of Time-varying Vibrations On A Benchmark Problem, to appear in The European Control Conf
  • X Chen
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Chen X. and Tomizuka M., Adaptive Model Inversion For Rejection of Time-varying Vibrations On A Benchmark Problem, to appear in The European Control Conf. 2013, Jul. 17-19, 2013.
New Repetitive Control with Improved Steady-state Performance and Accelerated Transient
--, New Repetitive Control with Improved Steady-state Performance and Accelerated Transient, to appear in IEEE Trans. Control Syst. Technol. 2013.