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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 Gr→y, 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 H∞design 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

Gr→y=P C/(1 + P C )=1−S≈1.

+-

C

P

+

d

o

+

e

y

r

+

d

+

FIGURE 1. Standard feedback design structure

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0

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.

102103104

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0

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(z−1)can be parame-

terized as P(z−1) = N(z−1)/D(z−1), where

N(z−1)and D(z−1)are coprime over S, mean-

ing there exists U(z−1),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 con-

troller C(z−1) = X(z−1)/Y (z−1), with X(z−1)

and Y(z−1)coprime over S, then any stabilizing

feedback controller can be parameterized as

X(z−1) + D(z−1)Q(z−1)

Y(z−1)−N(z−1)Q(z−1), Q(z−1)∈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(z−1)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(z−1), the plant parameterization, and the

choice of Q(z−1)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 H∞control to design

a stable C(z−1)for a baseline servo loop. The

controller coprime factorization can then be sim-

ply chosen as X(z−1) = C(z−1)and Y(z−1) = 1

in Fig. 4. Besides its simplicity, such a factor-

ization brings increased design and tuning intu-

itions. Noting that Y(z−1)=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(z−1) = z−mPn(z−1) = z−m/P −1

n(z−1)(2)

so that N(z−1) = z−mand D(z−1) = P−1

n(z−1)

in Fig. 4. Here P−1

n(z−1)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(z−1) = z−23.4766 ×10−7(1 + 0.8z−1)

(1 −z−1)2(3)

then P−1

n(z−1) = (1 −z−1)2/3.4766 ×10−7/(1 +

0.8z−1), whose poles are already inside the unit

circle. Hence we can choose the following co-

prime factorization N(z−1) = z−2, D(z−1) =

(1 −z−1)2/[3.4766 ×10−7(1 + 0.8z−1)].

step 3: with the discussed choices of X(z−1),

Y(z−1),N(z−1), and D(z−1), the extended

feedback controller in (1) becomes ˜

C(z−1) =

[C(z−1)+ P−1

n(z−1)Q(z−1)]/[1−z−mQ(z−1)]. Us-

ing (2) and after some algebra, we can derive the

new sensitivity function

˜

S(z−1) = 1

1 + P(z−1)˜

C(z−1)=1−z−mQ(z−1)

1 + P(z−1)C(z−1)

(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(z−1) =

1/(1+P(z−1)C(z−1)) only by a multiplicative term

1−z−mQ(z−1). To introduce small magnitude re-

sponse in Fig. 2, we just need to design Q(z−1)

such that 1−e−jωm Q(e−jω )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(z−1)[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(z−1). 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 1−z−mQ(z−1)

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

1−z−mQ(z−1)have a comb shape in the magni-

tude response (see Fig. 6). This can be achieved

by careful pole-zero placement in 1−z−mQ(z−1),

via internal model principle [5].

(iii) Bandwidth extension: if the original baseline

design C(z−1)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(z−1), which will

make 1−z−mQ(z−1)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(z−1)2], where kis a constant. Due to

computation delays and signal processing, there

is an additional one-step delay for P(z−1). The

zero at −1makes the nominal inverse of P(z−1)

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 z−2and P−1

n(z−1)a valid co-

prime factorization of (3).

The baseline controller is a PID controller C=

10000[1 + 2Ts/(1 −z−1)+0.012(1 −z−1)/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]

Qz−1=1−αNz−(N−m−nq)

1−αNz−Nz−nqqz, z−1

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, z−1)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 1−z−mQ(z−1)(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)

100101102103

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0

Magnitude (dB)

Frequency (Hz)

Q(z−1)

FIGURE 6. A Q-design example

1000 2000 3000 4000 5000 6000

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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.