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Inputtooutput Discontinuity and Transient Improvement in Addon Controls
Tianyu Jiang, Paul Hanrahan, and Xu Chen†
Abstract— Addon control is commonly used in control
systems for disturbance rejection and servo enhancements.
Despite the capability of greatly enhancing the steadystate
performance, even an ideal addon compensation can introduce
signiﬁcant output discontinuity and transient response. This
paper presents an exact formula for computing the output
discontinuity during addon servo, and discusses its inﬂuence
on transient performance in different control designs. We show
that adding the compensation signal as an update of the
reference yields much more desirable transient response than
the traditional choice of addon design at the input of the
plant. Evaluation of the results is performed on a semiconductor
manufacturing system.
Index Terms— addon control, inputtooutput discontinuity,
transient response, disturbance rejection
I. INTRODUCTION
Plugin or addon control design is central for servo
enhancements in control engineering. The term refers to
a large class of design schemes where additional control
commands are injected into the closed loop, on top of a
baseline feedback controller (Fig. 1). For example, the width
of a track in modern hard disk drives, called track pitch
(TP), is below 30nm. During read/write operations, servo
control must maintain a tracking error that is below 10%
TP while strong external disturbances can induce tracking
errors that are as large as 70% TP. Such large errors can
only be attenuated by adding plugin control commands. As
another example, in highspeed wafer scanning for semicon
ductor manufacturing, 99.97% of the force commands in the
positioning system are contributions of addon feedforward
control [1].
From the algorithm viewpoint, in feedback control, two
examples of addon design are: disturbance observers [2]
and Youlaparameterizationbased loop shaping [3], [4]. Dis
turbance observers usually update the control input via
feedback compensation [5]–[7]. Youla parameterization can
be parameterized either as an addon compensation at the
plant input side [8], [9], or a combined compensation at the
plant input and controller input [10], [11]. In feedforward
related control, iterative learning control (ILC) [12]–[16] and
adaptive or sensorbased feedforward compensation [17]–
[19] can be conﬁguraed as addon algorithms either at the
plant input or at the reference input.
Despite the signiﬁcant importance of addon control, lit
erature has mostly focused on designing its steadystate
response; much less is understood about controlling the re
sulting transient response. Among the related investigations,
†: Correspondence to: xchen@engr.uconn.edu. The authors are with
the Department of Mechanical Engineering, University of Connecticut.
Emails: {tianyu.jiang,paul.hanrahan,xchen}@uconn.edu
[20], [21] compared the transient performance in different
feedforward control algorithms, and showed that feedforward
injection at the reference side commonly provides faster
transient than that before the plant input. [15] compared
the difference of several ILC algorithms, and suggested to
choose reference update in inversemodel based ILC.
The focuses of this paper are twofold. First, we develop
theoretical results about inputtooutput discontinuity and
reveal its practical importance for the transient performance
in control design. Second, new investigations are made
to examine the transient differences in different addon
control designs. We derive the exact mathematical formula
for computing the changes in system outputs due to input
discontinuities, and provide computation of the associated
transient response. One central result we obtain is that the
common choice of performing addon control at the input
side of the plant yields undesired long transients, if there are
delays while turning on the compensation. Solution of the
problem is discussed in details and veriﬁed via simulation
and experiments on a wafer scanner prototype, a central
element for photolithography in semiconductor industry.
Notations: Gd→yddenotes the timedomain output of
the system Gd→ywith respect to the input d. For a closed
loop system consisting of a plant with transfer function
P, and a controller (in a negative feedback loop) C,T=
PC/(1+PC)denotes the complementary sensitivity function
(the transfer function from the reference to the plant output);
S=1/(1+PC)is the sensitivity function that deﬁnes the
dynamics from the disturbance to the plant output.
We assume all signals are causal throughout this paper,
i.e., their values are zero when time t<0. g(i)(t)denotes
dig(t)/dti, the generalized ith order derivative of g(t).
u(t+
0) = limt→t+
ou(t),u(t−
0) = limt→t−
ou(t).
II. TRANSIENT IN ADD ON COMPENSATION
A. Example and Practical Importance
The inﬂuence of transient performance is signiﬁcant in
the ﬁnal achievable control accuracy. Consider an example
in Fig. 1. Assuming that the signals r,uur and uue are all
zero ﬁrst, we aim at regulating the output yin the presence
of the disturbance d. Here, the baseline feedback controller
Cis best tuned for regular servo performance and system
robustness; and uuc is the additional control injected at the
plant input side to compensate d.
If uuc =−d, certainly the disturbance is perfectly rejected.
This is ideally the goal for all observer or feedforward based
disturbance attenuations. However, in practice
1), strong external disturbances may not always
present, and uuc is turned on only when external
P

+
C+
d
y
u
uc
+
r
+
+
+
+
u
ur
e
+
+
u
ue
Fig. 1. Addon control designs in a feedback block diagram
disturbance reaches the threshold, at which the error
tolerance is violated;
2), the control system is usually subjected to different
tasks, where different disturbance properties require
different addon designs.
Hence, for disturbance injection, practically a switch is used
to turn on or off the addon compensation uuc.
B. Idealcase Addon Compensation
Recall Fig. 1. The location of addon compensation can
be at the reference input or the plant input; and the require
ment of servo enhancement may come from regulation or
tracking controls. We now add these additional conditions
and formally introduce the signals uur,uue , and uuc, which
are the added servoenhancement signals for updated refer
ence (UR), updated error (UE), and updated control (UC),
respectively.
Let G(s)be the closedloop transfer function from the
addon control to the plant output; and assume zero initial
conditions at t=0, i.e. y(i)(0)=diy(0)/dti=0. We focus
ﬁrst on attenuating the disturbance d, namely, we aim at
achieving
[G]uaddon +Gd→yd=0 (1)
where uadd−on is uuc,uur or uud ;Gd→y=P/(1+PC)is
the transfer function from dto y. From Fig. 1, Gequals
P/(1+PC)in UC. In UE and UR, the dynamics between
uadd−on and yequal the complementary sensitivity function,
namely, G=T=PC/(1+PC). Hence, under zero initial
conditions, the sufﬁcient conditions to satisfy (1) in updated
control and updated reference/error are, respectively, uuc =
−dand [C]uur/ue =−d.
C. Transient in ideal addon UC control
Without loss of generality, suppose the actual disturbance
dis as shown in the top subplot in Fig. 2, where at time t0
the plugin servo enhancement is turned on. The idealcase
UC command uuc is the solid line in the second subplot of
Fig. 2, which perfectly cancels the disturbance after time t0.
Let
G(s) = B(s)
A(s)=bnsn+bn−1sn−1+· · · +b1s+b0
sn+an−1sn−1+· · · +a1s+a0
(2)
and consider the response of G(s)to the combined input u=
d+uuc. Directly solving the associated ODE is not feasible
as the derivatives of uare not well deﬁned at t0. We will show
ƚϬ
ƵƵĐ
ƚ
ƚ
ĚнƵƵĐ
Ϭ
Ϭ
Ě
Fig. 2. Input discontinuity in updated control
how this input discontinuity creates abrupt changes in y(t)
and its derivatives. Notice that unlike the discontinuity in step
responses, the solution to this inputtooutput discontinuity
problem is nontrivial.
Recall (2) and note that u(t)equals zero ∀t>t0in Fig. 2.
The transient response y(t)in t∈(t0,∞)satisﬁes:
y(n)(t) + an−1y(n−1)(t) + ·· · +a0y(t) = 0
with the initial condition: ny(i)(t+
0)on−1
i=0(3)
i.e., the transient is the natural response of the system starting
with the initial condition {y(i)(t+
0)}n−1
i=0.
Solutions to the ODEs can be obtained using Laplace
transforms or direct computation via Calculus. It is however
central to recognize that the initial condition y(i)(t+
0)does not
equal y(i)(t−
0), i.e., the actual transient does not simply equal
the natural transient response for y(i)(t−
0)—the system states
right before application of addon compensation—to decay
to zero. This is due to the input discontinuity of u(i)(t−
0)
jumping to u(i)(t+
0)(= 0)in Fig. 2. Next we obtain the
formula of {y(i)(t+
0)}n−1
i=0based on u(i)(t−
0)and the dynamics
of G(s), then analyze the resulting transient performance.
III. THE IN PUTTO OUT PUT DISCONTINUITY
We introduce ﬁrst a matrix representation of discontinuous
signals (with a discontinuity of the ﬁrst kind) using the
Dirac delta function δ(t). Consider a function g(t)that is
discontinuous at t0with g(t−
0)6=g(t+
0)(= g(t0)). We can write
g(t) = e0µ(t−t0) + f0(t)(4)
where f0(t)is continuous at t0;e0=g(t+
0)−g(t−
0); and
µ(t−t0)is a unit step function.
If, furthermore, f0(t)is not differentiable at t0, we can
similarly write ˙
f0(t) = e1µ(t−t0) + f1(t), where f1(t)is
continuous at t0. The derivative of (4) thus must satisfy
˙g(t) = e0δ(t−t0) + e1µ(t−t0) + f1(t)(5)
where e1µ(t−t0)gives the ﬁrstorder discontinuity ˙g(t+
0)6=
˙g(t−
0). Further differentiation yields
¨g(t) = e0˙
δ(t−t0) + e1δ(t−t0) + e2µ(t−t0) + f2(t)
.
.
.
g(n)(t) = e0δ(n−1)(t−t0) + ·· · +enµ(t−t0) + fn(t)(6)
where
e0
e1
.
.
.
en
=
g(t+
0)−g(t−
0)
˙g(t+
0)−˙g(t−
0)
.
.
.
g(n)(t+
0)−g(n)(t−
0)
(7)
Equations (4)(6) can be compactly represented as
g(t)
g(1)(t)
g(2)(t)
.
.
.
g(n)(t)
T
=
f0(t)
f1(t)
f2(t)
.
.
.
fn(t)
T
+
µ(t−t0)
δ(t−t0)
δ(1)(t−t0)
.
.
.
δ(n−1)(t−t0)
T
e0e1... en
0.......
.
.
.
.
.......e1
0... 0e0
(8)
The results are applied next to characterize the input and
output discontinuities for establishing the main theorem:
Theorem 1: For the system in (2), if the input u(t)and
its derivatives have the following discontinuity at time t0
ut+
0−ut−
0
˙ut+
0−˙ut−
0
.
.
.
u(n)t+
0−u(n)t−
0
=
eu,0
eu,1
.
.
.
eu,n
then the output discontinuity ey,i:=y(i)(t+
0)−y(i)(t−
0)satisﬁes
1 0 ... 0
an−1
.......
.
.
.
.
.......0
a1... an−11
ey,0
ey,1
.
.
.
ey,n−1
=
bn0... 0
bn−1
.......
.
.
.
.
.......0
b1... bn−1bn
eu,0
eu,1
.
.
.
eu,n−1
(9)
Proof: Assuming zero initial conditions and ap
plying (8) to y(t)and u(t)in the ODE representation
of (2): y(n)(t) + an−1y(n−1)(t) + ·· · +a0y(t) = bnu(n)(t) +
bn−1u(n−1)(t) + · ·· +b1˙u(t) + b0u(t), we have
fy,0(t)
fy,1(t)
.
.
.
fy,n(t)
T
a0
.
.
.
an−1
1
−
fu,0(t)
fu,1(t)
.
.
.
fu,n(t)
T
b0
.
.
.
bn−1
bn
=
µ(t−t0)
δ(t−t0)
.
.
.
δ(n−1)(t−t0)
T
eu,0eu,1... eu,n
eu,0
....
.
.
...eu,1
eu,0
b0
.
.
.
bn−1
bn
−
µ(t−t0)
δ(t−t0)
.
.
.
δ(n−1)(t−t0)
T
ey,0ey,1... ey,n
ey,0
....
.
.
...ey,1
ey,0
a0
.
.
.
an−1
1
(10)
where fy,i(t)and fu,i(t)are continuous at t=t0;ey,iand eu,i
are, respectively, the coefﬁcients of the ith order output and
input discontinuities satisfying the structure of (7).
To solve for ey,i’s in the output discontinuity, note that (10)
is equivalent to:
fy,0(t)
fy,1(t)
.
.
.
fy,n(t)
T
a0
.
.
.
an−1
1
−
fu,0(t)
fu,1(t)
.
.
.
fu,n(t)
T
b0
.
.
.
bn−1
bn
=
µ(t−t0)
δ(t−t0)
.
.
.
δ(n−1)(t−t0)
T
b0... bn−1bn
.
.
.......
bn−1...
bn
eu,0
eu,1
.
.
.
eu,n
−
µ(t−t0)
δ(t−t0)
.
.
.
δ(n−1)(t−t0)
T
a0... an−11
.
.
.......
an−1...
1
ey,0
ey,1
.
.
.
ey,n
After some algebra, matching coefﬁcients of the delta func
tion and its derivatives gives
a1... an−110
.
.
........
.
.
an−1....
.
.
10
ey,0
ey,1
.
.
.
ey,n
=
b1... bn−1bn0
.
.
........
.
.
bn−1....
.
.
bn0
eu,0
eu,1
.
.
.
eu,n
(11)
Rearranging the rows of (11), and noting that ey,nand eu,n
can be dropped as they have zero coefﬁcients, we get (9).
Theorem 1 fully characterizes the inputtooutput discontinu
ity. A unique solution always exists for obtaining {ey,i}n−1
i=0
in (9), as the matrix on the left is always nonsingular.
Veriﬁcation and direct system property analysis: Consider
the example of a ﬁrstorder system subjected to a rampto
step signal:
G(s) = 1
s+a,u(t) = (αt:t∈[0,t0)
αt0:t≥t0
(12)
We have u(t+
0) = u(t−
0), ˙u(t+
0)6=˙u(t−
0). Convolution or
inverse Laplace analysis gives
yt+
0=yt−
0
˙yt+
0=˙yt−
0(13)
¨yt+
0=¨yt−
0+˙ut+
0−˙ut−
0
namely, the ﬁrstorder input discontinuity creates a second
order output discontinuity. On the other hand, applying
Theorem 1 gives
100
a1 0
0a1
ey,0
ey,1
ey,2
=
000
100
010
eu,0
eu,1
eu,2
=
0
eu,0
eu,1
and hence
yt+
0−yt−
0
˙yt+
0−˙yt−
0
¨yt+
0−¨yt−
0
=
ey,0
ey,1
ey,2
=
0
eu,0
eu,1−aeu,0
=
0
0
˙ut+
0−˙ut−
0
The result matches with that in (13). More important, the
proposed algorithm has avoided the computation of the full
timedomain solution via convolution or Laplace analysis,
which can be complex for highorder systems, and infeasible
for generalized functions.
The case for addon servo enhancement: Applying (9) to
u(t)in Fig. 2, and noting the input discontinuity of eu,i=
0−u(i)(t−
0),∀i>0, we have
1 0 ... 0
an−1
.......
.
.
.
.
.......0
a1... an−11
ey,0
ey,1
.
.
.
ey,n−1
=
−
bn0··· 0
bn−1
.......
.
.
.
.
.......0
b1··· bn−1bn
u(t−
0)
˙u(t−
0)
.
.
.
u(n−1)(t−
0)
(14)
which solves the pursued problem of obtaining the initial
condition for the ODE (3).
IV. TIMEDO MAI N RESP ONS E AND TRANSIENT SP EED
For the response to the input in Fig. 2, although u(t) = 0
for t>t0, the transient is the natural response of (3) with
the initial conditions
ny(i)(t+
0) = y(i)(t−
0) + ey,ion−1
i=0(15)
i.e., it consists of the ”natural transient” for y(t−
0)to decay
to zero, plus effects of the input discontinuities from (14).
We derive in this section the inﬂuences of the inputto
output discontinuities (15) to (3), the ODE that characterizes
the transient response.
Corollary 1: Let the same assumptions in Theorem 1
hold. The transient response due to the input discontinuity
{eu,i}n
i=0has the following Laplace transform:
Y(s,ei) =
e−t0s
bnsn−1+· · · +b2s+b1
bnsn−2+· · · +b3s+b2
.
.
.
bns+bn−1
bn
T
eu,0
eu,1
eu,2
.
.
.
eu,n−1
A(s)(16)
Proof: Consider t+
0as the initial time for analyz
ing the output behavior in t∈(t0,∞). In the Laplace do
main we have L{y(t)}=Y(s),L{˙y(t)}=sY (s)−yt+
0,
and L{y(n)(t)}=snY(s)−sn−1yt+
0−sn−2˙yt+
0− ··· −
y(n−1)t+
0, or more compactly:
L
y(t)
˙y(t)
.
.
.
y(n)(t)
=
1
s
.
.
.
sn
Y(s)−
0... ... 0
1 0 .
.
.
s1....
.
.
.
.
.......0
sn−1... s1
yt+
0
˙yt+
0
.
.
.
y(n−1)t+
0
e−t0s
(17)
Writing the Laplace transform of (3) as
[a0,a1,...,an−1,1]L[y(t),˙y(t),...,y(n−1)(t),y(n)(t)]T=0
and using (17), we can solve for Y(s), which equals
e−t0s
A(s)
sn−1
sn−2
.
.
.
1
T
1 0 .. . 0
an−1
.......
.
.
.
.
.......0
a1.. . an−11
y(t+
0)
˙y(t+
0)
.
.
.
y(n−1)(t+
0)
(18)
Substituting in y(i)(t+
0) = y(i)(t−
0) + ey,iyields the de
composition Y(s) = Ys,t−
0+Y(s,ei), where Ys,t−
0and
Y(s,ei)are, respectively, the Laplace transforms of the
natural transient and the transient due to input discontinuity.
Using (9) gives
e−t0s
A(s)
sn−1
sn−2
.
.
.
1
T
1
an−1
...
.
.
.......
a1.. . an−11
ey,0
ey,1
.
.
.
ey,n−1
=
e−t0ssn−1,...,1
A(s)
bn
bn−1
...
.
.
.......
b1.. . bn−1bn
eu,0
eu,1
.
.
.
eu,n−1
(19)
which, after simpliﬁcations, is equivalent to (16).
To examine the transient speed in addon control, consider
ﬁrst the natural response Ys,t−
0, which is (18) with t+
0
replaced by t−
0, namely,
Ys,t−
0=e−t0s[G0(s)]yt−
0+ [G1(s)] ˙yt−
0+.. . (20)
where
G0(s) = sn−1+an−1sn−2+·· · +a1
A(s)=1
s−a0
A(s)
1
s(21)
G1(s) =
.
.
.
sn−2+an−1sn−3+· ·· +a2
A(s)=1
sG0(s)−a1
A(s)
1
s(22)
Gi(s) = 1
sGi−1(s)−ai
A(s)
1
s,∀i∈{1,2,...,n−1}(23)
In the time domain, from Final Value Theorem, all elements
L−1[Gi(s)]’s in (20) have zero steadystate values. Hence
the transient indeed eventually converges to zero. From (21),
L−1[G0(s)] is the difference between a unit step and the step
response of a0/A(s), whose transient duration depends on
the poles from A(s) = 0.
For the response due to input discontinuities, similar
construction gives that Y(s,ei)in (19) is
Y(s,ei) = −e−t0sGB,0(s)u(t−
0) + GB,1˙u(t−
0) + .. . (24)
GB,0(s) = bnsn−1+·· · +b2s+b1
A(s)=G(s)1
s−b0
A(s)
1
s(25)
GB,1(s) = bnsn−2+·· · +b3s+b2
A(s)=GB,0(s)1
s−b1
A(s)
1
s(26)
.
.
.
GB,i(s) = 1
sGB,i−1(s)−bi
A(s)
1
s,∀i∈{1,2,...,n−1}(27)
where we have used the fact that eu,i=0−u(i)(t−
0). Thus,
L−1[GB,0(s)] is the transient difference between the step
responses of G(s)and b0/A(s). In (25) and (21), the scaling
a0and b0change only the relative magnitude of the response.
For a fast overall transient response, (24) needs to match the
transient speed of (20). If the step response of G(s)is slow,
namely, L−1[G(s)/s]in (25) is slow compared to L−1[1/s]
in (21), then the transient of L−1[GB,0(s)] will be slower
than that of L−1[G0(s)].
Actually, to have the same speed of response, the ideal
case can be seen to be that G(s) = 1, i.e., the addon
compensation is directly applied on the output y(which is,
of course, not feasible in practice).
Recall in Fig. 1, that G(s)—the dynamics between the
addon control command and the plant output—is P/(1+
PC)in UC and PC/(1+PC)in UR/UE. Among the close
loop transfer functions in a general feedback block diagram
in Fig. 1, PC/(1+PC)has the dynamic response that is most
close to G(s) = 1, and hence will provide the fastest transient
response from addon injection perspective.
On the other hand, from the polezero point of view, UR
also has faster transient. The zeros of G(s)in UC and UR
contain, respectively, the poles of Cand the zeros of C. The
feedback controller C, if feasible, is preferred to be designed
to have stable zeros, as openloop unstable zeros slow down
the transient and will yield various fundamental limitations
in the steadystate performance of a feedback system. For
the poles of C, marginally stable poles or poles close to
the imaginary axis are needed, especially for mechanical
systems, for highgain feedback at low frequencies (consider,
e.g., the case of PID control).
As a robustness analysis, notice that Gi(s)in (23) and
GB,i(s)in (27) share a similar form. Should there be model
uncertainties, such a structural similarity remains unchanged.
Hence, the transient speed still follows the discussed trend. If
feasible, an online system identiﬁcation can be further used
for an adaptive transient computation.
V. EXAMPLE AND EX PE R IM ENT S
Let P(s) = 1/(0.2556s2+0.279s)and C(s) =
100001+21
s+0.012sin Fig. 1. P(s)here is the
nominal model of reticle stage of the waferscanner system.
The precision control here synchronizes the motions of
a wafer stage and a reticle stage, such that patterns on
integrated circuits can be precisely transformed from a
mask on the reticle stage to different locations of the
silicon wafer on the wafer stage. A picture of the physical
system is provided in [6]. To achieve the nmscale precision
demand, highperformance actuation and measurement tools
including air bearings, epoxycore linear permanent magnet
motors (LPMM), and laser interferometers are used, with
the control commands executed on a LabVIEW realtime
system with ﬁeldprogrammable gate array (FPGA). The
magnitude response of the reticle stage is provided in [22].
Evaluating the transient properties such as the impulse
and the step responses reveals that PC/(1+PC)provides
much faster transient response than P/(1+PC)with respect
to rapid changing input signals. For actual disturbance rejec
tion,1abrupt step disturbances are ﬁrst injected to the plant at
around 0.12 sec. Fig. 3 shows the effect of addon compensa
tion in simulation. As the baseline controller already contains
an integrator, the step disturbance is asymptotically rejected
in the ﬁrst subplot. For verifying the transient performance,
addon compensation is turned on at t0=0.5 sec, using the
UC and UR conﬁgurations in Section IIB.
0 0.5 1 1.5 2 2.5 3 3.5
position (m)
#105
0
1
2
w/o addon compensation
time (sec)
0 0.5 1 1.5 2 2.5 3 3.5
position (m)
#105
1
0
1
2
w/ addon compensation
UC
UR
Fig. 3. Effects of addon compensation in simulation
Both addon compensation schemes affect the system in
the direction of reducing the error. For UC compensation,
however, there is a large undershoot due to the system
dynamics, yielding a much slower and worse transient com
pared to UR. Experimentation on the physical system also
strongly veriﬁes the results as shown in Fig. 4, where it is
observed that the simulation and experimental results almost
overlapped with each other, and UR provides almost zero
transient response in the actual system.
Of course, the transient problem exists not just for the
case of step disturbances but for any addon design with
inputdiscontinuities. Fig. 5 reveals the addon transient in
compensating frequencydependent disturbances (a 500 Hz
vibration). A similar superior performance of UR addon
design is observed. In this example, it is no longer possible
to straightforwardly tell the direction of the adverse transient
in UC, as the highfrequency input and its derivatives change
1For implementation, the pure differentiation action in the PID controller
is replaced by its causal version s/(1+τs)where 0 <τ<< 1.
0 0.5 1 1.5 2 2.5 3 3.5
position (m)
#105
0
1
2
w/o addon compensation
simulation
experiment
0 0.5 1 1.5 2 2.5 3 3.5
position (m)
#105
1
0
1
2
w/ addon UC compensation
simulation
experiment
time (sec)
0 0.5 1 1.5 2 2.5 3 3.5
position (m)
#105
1
0
1
2
w/ addon UR compensation
simulation
experiment
Fig. 4. Simulation and experimentation comparison of UC and UR
compensation
very rapidly with respect to time; and the obtained conclu
sions in the paper are increasingly important to avoid large
servo errors during controller implementation.
time(s)
0.1 0.2 0.3 0.4 0.5 0.6
position(m)
#106
6
4
2
0
w/o compensation
UC
UR
Fig. 5. Experimental comparison of addon vibration compensations:
compensation turned on at 0.1 sec, to attenuate a 500Hz external vibration
(the residual errors are from an internal 18 Hz motor force ripple)
Last but not least, we have shown the results for add
on compensation in regulation control. The same conclusion
holds in tracking control, which can be translated to a
regulation problem from the view point of feedback errors.
The transient properties of the related transfer functions
PC/(1+PC)and P/(1+PC)do not change.
VI. CONCLUSIONS
This paper addresses the general inputtooutput discon
tinuity problem and applies the results to the transient
improvement of addon control designs. Simulation and
experimental results are provided to show validity of the
theoretical analysis. Essentially, undesired transients occur as
long as there are input discontinuities acting upon a dynamic
system with poor transient properties. The obtained result
is important not only for addon compensation schemes,
but also for other applications such as switching between
multiple controllers.
REFERENCES
[1] M. Heertjes, D. Hennekens, and M. Steinbuch, “MIMO feedforward
design in wafer scanners using a gradient approximationbased algo
rithm,” Control Eng. Pract., vol. 18, no. 5, pp. 495 – 506, 2010.
[2] K. Ohnishi, “Robust motion control by disturbance observer,” Journal
of the Robotics Society of Japan, vol. 11, no. 4, pp. 486–493, 1993.
[3] D. Youla, J. Bongiorno Jr, 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.
[4] V. Kucera, “Stability of discrete linear feedback systems,” in Proc. 6th
IFAC World Congress, paper 44.1, vol. 1, 1975.
[5] C. J. Kempf and S. Kobayashi, “Disturbance observer and feedforward
design for a highspeed directdrive positioning table,” IEEE Trans.
Control Syst. Technol., vol. 7, no. 5, pp. 513–526, 1999.
[6] X. Chen and M. Tomizuka, “New repetitive control with improved
steadystate performance and accelerated transient,” IEEE Trans. Con
trol Syst. Technol., vol. 22, no. 2, pp. 664–675, March 2014.
[7] ——, “A minimum parameter adaptive approach for rejecting multiple
narrowband disturbances with application to hard disk drives,” IEEE
Trans. Control Syst. Technol., vol. 20, no. 2, pp. 408 –415, march
2012.
[8] I. D. Landau, A. C. Silva, T.B. Airimitoaie, G. Buche, and M. Noe,
“Benchmark on adaptive regulation—rejection of unknown/time
varying multiple narrow band disturbances,” European Journal of
Control, vol. 19, no. 4, pp. 237 – 252, 2013.
[9] X. Chen and M. Tomizuka, “Control methodologies for precision
positioning systems,” in Proceedings of American Control Conference,
June 2013, pp. 3710–3717.
[10] 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.
[11] B. D. Anderson, “From youla kucera to identiﬁcation, adaptive and
nonlinear control,” Automatica, vol. 34, no. 12, pp. 1485 – 1506, 1998.
[12] D. Bristow, M. Tharayil, and A. G. Alleyne, “A survey of iterative
learning control,” IEEE Control Syst. Mag., vol. 26, no. 3, pp. 96–
114, 2006.
[13] K. L. Moore, Y. Chen, and H.S. Ahn, “Iterative Learning Control: A
Tutorial and Big Picture View,” Proceedings of IEEE Conference on
Decision and Control, no. Ilc, pp. 2352–2357, 2006.
[14] H.S. Ahn, Y. Chen, and K. L. Moore, “Iterative Learning Control:
Brief Survey and Categorization,” IEEE Transactions on Systems, Man
and Cybernetics, Part C (Applications and Reviews), vol. 37, no. 6,
pp. 1099–1121, Nov. 2007.
[15] S. Mishra, “Fundamental issues in iterative learning controller design:
Convergence, robustness, and steady state performance,” Ph.D. disser
tation, 2009.
[16] N. Amann, D. H. Owens, and E. Rogers, “Iterative learning control
using optimal feedback and feedforward actions,” Int. J. Control,
vol. 69, no. 2, pp. 203–226, 1996.
[17] M. Bodson, a. Sacks, and P. Khosla, “Harmonic generation in adap
tive feedforward cancellation schemes,” IEEE Trans. Autom. Control,
vol. 39, no. 9, pp. 1939–1944, 1994.
[18] J. Zhang, R. Chen, G. Guo, and T.s. Low, “Modiﬁed adaptive
feedforward runout compensation for dualstage servo system,” IEEE
Trans. Magn., vol. 36, no. 5, pp. 3581–3584, 2000.
[19] B. Widrow and E. Walach, Adaptive Inverse Control, Reissue Edition:
A Signal Processing Approach. WileyIEEE Press, Nov. 2007.
[20] B. Rigney, L. Y. Pao, and D. Lawrence, “Settle time performance
comparisons of stable approximate model inversion techniques,” Pro
ceedings of American Control Conference, p. 6 pp., 2006.
[21] ——, “Nonminimum phase dynamic inversion for settle time appli
cations,” IEEE Trans. Control Syst. Technol., vol. 17, no. 5, pp. 989–
1005, Sept 2009.
[22] X. Chen, T. Jiang, and M. Tomizuka, “Pseudo youlakucera parame
terization with control of the waterbed effect for local loop shaping,”
Automatica, vol. 62, pp. 177 –183, Dec. 2015.