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Transmission of Signal Nonsmoothness and
Transient Improvement in Add-on Servo Control
Tianyu Jiang and Xu Chen†
Abstract—Plug-in or add-on control is integral for high-
performance control in modern precision systems. Despite the
capability of greatly enhancing the steady-state performance,
add-on compensation can introduce output discontinuity and
significant transient response. Motivated by the vast application
and the practical importance of add-on control designs, this paper
identifies and investigates how general nonsmoothness in signals
transmits through linear control systems. We explain the jump
of system states in the presence of nonsmooth inputs in add-
on servo enhancement, and derive formulas to mathematically
characterize the transmission of the nonsmoothness. The results
are then applied to devise fast transient responses over the
traditional choice of add-on design at the input of the plant.
Application examples to a manufacturing control system are
conducted, with simulation and experimental results that validate
the developed theoretical tools.
Index Terms—nonsmooth inputs, transient control, disturbance
rejection
I. INTRODUCTION
Plug-in or add-on control design is central for servo en-
hancements in control engineering. In order to provide a
storage capacity in the tera-byte scale, a modern hard disk
drive (HDD) contains more than 900,000 data tracks within
one inch of the disk. Correspondingly, the width of each
track, called track pitch (TP), can easily fall 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
plug-in control commands. As another example, in high-speed
wafer scanning for semiconductor manufacturing, [1] showed
that 99.97% of the force commands in the positioning system
are contributions of add-on feedforward control.
In feedback algorithms, add-on servo is central for a large
class of design schemes that require a baseline feedback
controller. Two examples are: disturbance observers [2] and
Youla-parameterization-based loop shaping [3], [4]. Either for
general low-frequency enhancement [5]–[7], or for the exten-
sions to structured disturbance rejection [8]–[10], disturbance
observers usually update the commands at the input side of the
plant. Youla parameterization can be parameterized either as
an add-on compensation at the plant input side [11], [12], or a
combined compensation at the plant input and controller input
[13], [14]. In feedforward-related control, adaptive or sensor-
based feedforward compensation [15]–[17] can be configured
Tianyu Jiang (email: tianyu.jiang@uconn.edu) and Xu Chen (email:
xchen@uconn.edu) are with the Department of Mechanical Engineering,
University of Connecticut, Storrs, CT, 06269, USA. Corresponding author: †
as add-on algorithms either at the plant input or at the reference
input (see more details in Section III).
Fundamentally, add-on control brings servo enhancement
by introducing new dynamic properties in closed-loop signals.
Such a process induces certain degrees of nonsmoothness in
the signals. For meeting future demands in high-precision
systems, it is essential to understand what types of systems and
add-on changes create large transient, and what are the math-
ematical relationships between the signal nonsmoothness and
the induced transient. The importance of such considerations
is verified in simulation and experiments in [18], [19], which
compared the transient performance in different feedforward
control algorithms. Still, a full theoretical solution to the prob-
lem is intrinsically nontrivial, except for simple discontinuities
such as step and ramp signals. Despite the rich literature on
designs to achieve the desired steady-state performance, sparse
investigations on the transient in add-on compensation are
available, and a full understanding of the theoretical add-on
transient remains missing. This paper targets to bridge this gap.
The focuses are two-fold. First, we develop theoretical results
about input-to-output discontinuity and reveal its practical
importance for the transient performance in control design.
Second, new investigations are made to examine the transient
characteristics in different add-on control designs. We derive
an exact mathematical formula for computing the changes
in system outputs when the input and/or its derivatives have
discontinuities, and provide computation of the associated
transient response. One central result we obtain is that, the
common choice of performing add-on control at the input side
of the plant yields undesired long transients, if there are delays
during turning on the compensation. Solution of the problem is
discussed in details and verified on a precision motion control
platform in semiconductor manufacturing.
The remainder of the paper is organized as follows. Section
II describes the wafer scanner hardware on which verification
of the algorithm is performed. Section III reveals the transient
problem in add-on compensation, following which Sections
IV and V solve the mathematical problem. Simulation and
experimental results are provided in Section VI. Section VII
concludes the paper.
Notations and Assumptions: All signals and systems are as-
sumed to be causal and have real-valued coefficients. L{·}and
L−1{·}are, respectively, the operators of Laplace and inverse
Laplace transforms. For practical purposes, we exclusively
consider the first-kind or jump discontinuities in signals and
their derivatives; and denote u(t+
0)and u(t−
0), respectively, as
the right- and the left-hand limits of a signal u(t)at t0.f(i)(t)
denotes dif(t)/dti, the generalized i-th order derivative of a
function f(t).f(t)is said to have a kth-order discontinuity at
t0if y(k)(t+
0)6=y(k)(t−
0)—in other words, f(t)is of (differen-
tiability) class Ck−1but not of class Ckat t0.Gd→yddenotes
the time-domain output of the system with 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 output disturbance to the plant output.
Remark 1.We focus on analysis and control of the transient
behavior, and assume that the discontinuous change of input
properties does not yield system instability (which can be
guaranteed by, for instance, a sufficiently long dwell time
during switching [20]). For additional information on stability
of such switched systems, we refer readers to [21] [22] [23]
[24].
II. HA RDWARE DESCRIPTION AND NOTATIONS
The developed algorithm in this paper is verified via simu-
lation and experiments on a wafer scanner prototype, a central
element for photolithography in the advanced manufacturing
of integrated circuits for semiconductor industry. The precision
control here synchronizes the motions of a wafer stage and a
reticle stage. The motion control allows patterns on integrated
circuits to 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 pro-
vided in [8]. To achieve the nm-scale precision requirement,
high-performance actuation and measurement tools including
air bearings, epoxy-core linear permanent magnet motors
(LPMM), and laser interferometers are used. The control
commands are executed on a LabVIEW real-time system
with field-programmable gate array (FPGA). Fig. 1 shows the
frequency response, from the voltage input of the LPMM to
the position of the reticle stage.
102103
−160
−140
−120
−100
−80
Magnitude (dB)
102103
−200
−100
0
100
200
Phase (degree)
Frequency (Hz)
Fig. 1: Frequency response of the reticle stage.
III. 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.
2. Assuming first that the signals r,uur,uue are all zero, 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 to compensate d.
P
-+
C+
d
y
u
uc
+
r
+
+
+
+
u
ur
e
+
+
u
ue
Fig. 2: Add-on control designs in a feedback block diagram.
If uuc =−d, certainly the disturbance is perfectly rejected.
This is ideally the goal for all observer or feedforward based
disturbance attenuation designs, if injection of control com-
mand is at the plant input side. However, in practice:
1), strong external disturbances may not always present,
and uuc is turned on only when external disturbance
reaches the threshold, at which the error tolerance is
violated;1
2), the control system is usually subjected to differ-
ent tasks, where different disturbance properties re-
quire different add-on designs (indeed, if the add-on
scheme is universal for all situations, it should be
absorbed as part of the baseline controller).
Hence, for rejecting external disturbances, practically a switch
is used for turning on or off the compensation uuc. Consider
the case where dis a scaled step signal that occurred at 0.12
sec. If the add-on compensation is delayed by 2.4 sec (i.e.,
uuc is added at 2.52 sec), even with the “perfect” rejection
condition uuc =−d, significant transient response can happen
as shown in Fig. 3—the experimental results on the wafer
scanner system.
Certainly, the above example is for demonstration of the
problem, and provides only an extreme case where the add-on
compensation is turned on when an integrator in the baseline
controller Chas already greatly compensated the disturbance,
and dis simple enough to be perfectly rejected by simple
feedback. These simplifying conditions will be dropped in the
remainder of the paper, where the general problem of add-on
transient is addressed.
B. Ideal-case Add-on Compensation
Recall Fig. 2. The location of add-on compensation can be
at the reference input or the plant input; and the requirement
1In industrial applications, it is common to run a fault detector to monitor
the system performance and switch on the compensation when the servo
performance is degraded to be below a pre-specified performance threshold.
0 0.5 1 1.5 2 2.5 3 3.5 4
-3
-2
-1
0
1
2
3x 10
-5
time(s)
position(m)
w/o compensation
w/ perfect disturbance rejection
Disturbance injection
"Perfect" disturbance rejection at plant input
Fig. 3: Demonstration of transient behavior on a wafer scanner.
of servo enhancement may come from regulation or tracking
controls. These additional considerations are now added to
form a general block diagram with different configurations of
add-on signals. We now formally introduce the signals uur,
uue, and uuc , which are the added servo-enhancement signals
for updated reference (UR), updated error (UE), and updated
control (UC), respectively.
Let G(s)be the closed-loop transfer function from the add-
on control to the plant output. Assume zero initial conditions
at t=0, i.e. y(i)(0)=diy(0)/dti=0, and focus first on
attenuating the disturbance d, namely, we aim at achieving
[G]uadd-on +Gd→yd=0,(1)
where uadd−on is uuc,uur or uue ;Gd→y=P/(1+PC)is
the transfer function from dto y. From Fig. 2, Gequals
P/(1+PC)in UC. In UE and UR, the dynamics between
uadd−on and yboth equal the complementary sensitivity func-
tion, namely, G=T=PC/(1+PC). Hence, regardless of the
design methods, to satisfy (1), the ideal conditions in updated
control and updated reference/error are, respectively, uuc =−d
and [C]uur/ue =−d.
C. Transient in ideal add-on UC control
Without loss of generality, suppose the actual disturbance d
is as shown in the top subplot in Fig. 4, where at time t0the
plug-in servo enhancement is turned on. The ideal-case UC
command uuc is the solid line in the second subplot of Fig. 4,
which perfectly cancels the disturbance after time t0.
Let
G(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
uue ,d+uuc. Directly solving the associated ODE is not
feasible as derivatives of uare not well defined at time t0.
We will show 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 input-to-
output discontinuity problem is nontrivial.
For a general system, we derive next the exact mathematical
result of the transient after t=t0for uin Fig. 4. Recall (2)
d+uuc
d
uuc
0
0
t0
t
t
Fig. 4: Input discontinuity in updated control.
and note that u(t)equals zero ∀t>t0in Fig. 4. 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 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 under y(i)(t−
0)—the system states
right before the application of add-on compensation (recall the
examble in Fig. 3). This is due to the input discontinuity of
u(i)(t−
0)jumping to u(i)(t+
0)(= 0)in Fig. 4. 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.
IV. THE IN PU T-TO-O UT PU T DISCONTINUITY
Theorem 2. Let u(t)and y(t)be the input and the output of a
finite-dimensional real-coefficient linear system G, satisfying
y(n)(t) + an−1y(n−1)(t) + ·· · +a1˙y(t) + a0y(t)
=bnu(n)(t) + bn−1u(n−1)(t) + ·· · +b1˙u(t) + b0u(t)(4)
at time t0. If u (t)and/or its derivatives have discontinuities:
u(t+
0)−u(t−
0) = eu,0,˙u(t+
0)−˙u(t−
0) = eu,1,... , u(n)(t+
0)−
u(n)(t−
0) = eu,n, then y(t)and/or its derivatives contain dis-
continuities that satisfy
1 0 ... 0
an−1
.......
.
.
.
.
.......0
a1... an−11
y(t+
0)−y(t−
0)
˙y(t+
0)−˙y(t−
0)
.
.
.
y(n−1)(t+
0)−y(n−1)(t−
0)
=
bn0... 0
bn−1
.......
.
.
.
.
.......0
b1... bn−1bn
eu,0
eu,1
.
.
.
eu,n−1
.(5)
Theorem 2 fully characterizes the input-to-output discon-
tinuity. The matrix on the left side of (5) is nonsingular.
Therefore a unique solution exists for obtaining {ey,i}n−1
i=0:=
{y(i)(t+
0)−y(i)(t−
0)}n−1
i=0. No knowledge of u(t)is required
except at t0, the instance of discontinuity. More specifically,
solutions of ey,i’s can be obtained by forward substitution after
solving the matrix equality:
ey,0=bneu,0(6a)
ey,1=bn−1eu,0+bneu,1−an−1ey,0(6b)
.
.
.
ey,n−1=
n−1
∑
j=0
bj+1eu,j−
n−2
∑
j=0
aj+1ey,j.(6c)
Remark 3.(5) provides up to the (n−1)-th order output
discontinuity. If the value of ey,nis of interest, Theorem 2 can
be applied to the augmented system G(s) = (bnsn+1+bn−1sn+
··· +b1s2+b0s)/(sn+1+an−1sn+· · · +a1s2+a0s). Similar
procedures can provide other higher-order discontinuities.
Numerical Verification: Consider the response of a first-
order system to a ramp-to-step signal:
G(s) = 1
s+a,u(t) = (αt:t∈[0,t0)
αt0:t≥t0
.(7)
In this example, we have u(t+
0) = u(t−
0), ˙u(t+
0)6=˙u(t−
0).
Convolution or inverse Laplace analysis which gives y(t) =
α
at+α
a2e−at −α
a2, if t∈[0,t0);y(t) = α
at0+α
a2[e−at −e−a(t−t0)],
if t≥t0; ˙y(t) = α
a−α
ae−at ,if t∈[0,t0); ˙y(t) = −α
a[e−at −
e−a(t−t0)],if t≥t0; ¨y(t) = αe−at,if t∈[0,t0); and ¨y(t) =
α[e−at −e−a(t−t0)],if t≥t0. Then
yt+
0=yt−
0
˙yt+
0=˙yt−
0(8)
¨yt+
0=¨yt−
0+˙ut+
0−˙ut−
0,
namely, the first-order input discontinuity creates a second-
order output discontinuity.
Alternatively, apply Theorem 2 and Remark 3 to the same
system. Noticing that G(s) = 1/(s+a) = s/(s2+as),we have:
100
a1 0
0a1
ey,0
ey,1
ey,2
=
000
100
010
eu,0
eu,1
eu,2
=
0
eu,0
eu,1
.
Hence y(t+
0)−y(t−
0) = 0, ˙y(t+
0)−˙y(t−
0) = eu,0=0, and ¨y(t+
0)−
¨y(t−
0) = eu,1−aeu,0=˙u(t+
0)−˙u(t−
0). The result matches with
that in (8). More important, the computation here removes
the necessity to compute the full time-domain solution, which
is not only long and complex for high-order systems, but
also infeasible for general signals without given time-domain
models.
A. Proof and Analysis
We introduce first a representation of discontinuous signals
using Dirac delta functions. In the remainder of the texts, we
will use µ(t)to denote the unit step signal, i.e. µ(t) = 1∀t≥0
and µ(t) = 0∀t<0; and denote δ(t)as the Dirac delta function
that satisfies Rt
0δ(τ)dτ=µ(t)and R∞
0δ(τ−T)g(τ)dτ=g(T)
if g(t)is continuous. As a distribution (a.k.a. generalized
function), δ(t)satisfies dµ(t)/dt=δ(t).
Consider a piecewise continuous function g(t)with a first-
kind/jump discontinuity at t0. We can write
g(t) = eg,0µ(t−t0) + f0(t),(9)
where f0(t)is continuous at t0;eg,0=g(t+
0)−g(t−
0); and
µ(t−t0)creates the jump discontinuity at t=t0, as shown
in the example in Fig. 5.
t
t0
f0(t)
g(t)
Fig. 5: Decomposition of discontinuity in g(t)at t0.
Similarly, if ˙
f0(t)is furthermore discontinuous at t0, we
have ˙
f0(t) = eg,1µ(t−t0)+ f1(t), where f1(t)is continuous at
t0. The derivative of (9) thus must satisfy
˙g(t) = eg,0δ(t−t0) + eg,1µ(t−t0)+ f1(t),(10)
where eg,1µ(t−t0)gives the first-order discontinuity ˙g(t+
0)6=
˙g(t−
0).
Further differentiation yields
¨g(t) = eg,0˙
δ(t−t0) + eg,1δ(t−t0) + eg,2µ(t−t0) + f2(t)
.
.
.
g(n)(t) = eg,0δ(n−1)(t−t0) + .. .
+eg,n−1δ(t−t0) + eg,nµ(t−t0)+ fn(t),(11)
where f2,..., fnare continuous at t0, and the nonsmoothness
of g(t)is characterized by
eg,0
eg,1
.
.
.
eg,n
=
g(t+
0)−g(t−
0)
˙g(t+
0)−˙g(t−
0)
.
.
.
g(n)(t+
0)−g(n)(t−
0)
.(12)
Equations (9)-(11) can be compactly written as:
g(t)
g(1)(t)
.
.
.
g(n)(t)
T
=
f0(t)
f1(t)
.
.
.
fn(t)
T
+
µ(t−t0)
δ(t−t0)
.
.
.
δ(n−1)(t−t0)
T
eg,0eg,1... eg,n
0.......
.
.
.
.
.......eg,1
0... 0eg,0
.(13)
For matrix-vector operations in the form of (13), the fol-
lowing result will appear to be useful:
Fact 4. The following is true:
e0e1... en
0e0... en−1
.
.
........
.
.
0... 0e0
a0
.
.
.
an−1
1
=
a0... an−11
.
.
.......0
an−1.......
.
.
1 0 ... 0
e0
.
.
.
en−1
en
.
We now formally prove Theorem 2.
Proof. Replacing {g,fi,eg,i}with {y,fy,i,ey,i}
and {u,fu,i,eu,i}, respectively in (13);
and applying the resulting equations to
[y(t)y(1)(t).. . y(n)(t)][ a0. . . an−11]T=
[u(t)u(1)(t).. . u(n)(t)][ b0. . . bn−1bn]T—the
vector form of (4)—give
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
0eu,0
....
.
.
.
.
.......eu,1
0... 0eu,0
b0
.
.
.
bn−1
bn
−
µ(t−t0)
δ(t−t0)
.
.
.
δ(n−1)(t−t0)
T
ey,0ey,1... ey,n
0ey,0
....
.
.
.
.
.......ey,1
0... 0ey,0
a0
.
.
.
an−1
1
,(14)
where fy,i(t)and fu,i(t)are continuous at t=t0;ey,i=
y(i)(t+
0)−y(i)(t−
0); and eu,i=u(i)(t+
0)−u(i)(t−
0).
To solve for ey,i’s, using Fact 4, we translate (14) 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
.
.
.......0
bn−1.......
.
.
bn0... 0
eu,0
eu,1
.
.
.
eu,n
−
µ(t−t0)
δ(t−t0)
.
.
.
δ(n−1)(t−t0)
T
a0... an−11
.
.
.......0
an−1.......
.
.
1 0 ... 0
ey,0
ey,1
.
.
.
ey,n
.
δ(t−t0),˙
δ(t−t0),...,δ(n−1)(t−t0)are linearly independent,
and cannot be expressed as linear combinations of the contin-
uous functions on the left side of the last equality. Hence, their
coefficients on the right side must be zero. This corresponds
to
a1... an−11
.
.
.......0
an−1.......
.
.
1 0 ... 0
ey,0
ey,1
.
.
.
ey,n−1
=
b1... bn−1bn
.
.
.......0
bn−1.......
.
.
bn0... 0
eu,0
eu,1
.
.
.
eu,n−1
.(15)
Re-arranging the rows gives (5).
The case for add-on servo enhancement: Applying (5) to
u(t)in Fig. 4, 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)
.(16)
B. Discussions
Rewriting (5) symbolically as Maey=Mbeuand applying
Taylor expansion to M−1
agive ey=M−1
aMbeu=∑∞
k=0(I−
Ma)kMbeu. Noting the lower-triangular form of Ma, we can
further simplify the expression to ey=∑n
k=0(I−Ma)kMbeu, as
I−Mais nilpotent and (I−Ma)kvanishes for k>n.
From the results, the generalized output discontinuity is a
linear and continuous function of the discontinuities in the
input and its derivatives. Bounded input discontinuities gener-
ate bounded discontinuities in the output. More specifically,
keykq≤ k∑n
k=0(I−Ma)kMbkp→qkeukpwhere p,q∈[1,∞);
k·kp→qis an induced matrix norm; k·kpand k·kqare vector
norms. The numeric value of the upper bound is problem-
dependent. The matrix ∑n
k=0(I−Ma)kMbhowever is always
lower-triangular and has easy-to-compute matrix norms, as
(I−Ma)kand Mbare both lower-triangular (with actually all
diagonal entries equal to 1, 0, or bn). Furthermore, as Ma
and Mbdo not contain a0and b0,k∑n
k=0(I−Ma)kMbkp→q
is independent of b0and a0. Discontinuities in {y(i)(t)}n−1
i=0
is therefore independent of the DC gain and not related
to the magnitude response of the system [see also (6a)-
(6c)]. Switched systems/signals can thus generate significant
transient responses while providing zero steady-state errors.
Influence of high-order input discontinuities: If the input is
continuous, i.e. eu,0=0, the resulting output is still continuous
but not necessarily smooth. For instance, if bnis large in (6b),
large discontinuity in ˙y(t)occurs even if there is only a small
change of ˙u(t). Notice that the result may appear counter to
intuitions and perceptions in conventional analysis, which may
lead to the assertion that b1—the scaling coefficient of ˙u(t)in
(4)—is the dominant factor.
Furthermore, high-order input discontinuities eu,i’s only
influence high-order derivatives in the output. More specifi-
cally, the ith-order output discontinuity ey,ionly depends on
{eu,j:j≤i}, based on the mathematical relations in (6a)-(6c).
Influence of the relative degree of the system: If bn6=0,
a discontinuous u(t)will render y(t)and all its derivatives
discontinuous. The direct implication is in line with the
conventional practice that jump discontinuities are undesired
in general switched control. In addition, from (6a), the jump
in the output is only linearly dependent on the jump in the
input and bn—the direct gain of the system.
In the case that the system is time-invariant, if the relative
degree of the transfer function associated to (4) is r—in other
words, bn, ..., bn−r+1all equal zero—then ey,0, ..., ey,r−1on
the left side of (5) must be zero, namely, y, ˙y, ... , y(r−1)are all
continuous at t0, and the input nonsmoothness can only cause
discontinuities in the higher-order derivatives y(r), ..., y(n−1),
etc.
The case for non-ideal add-on control: If the disturbance
estimation contains errors or there exists actuation delay tad,
the ideal compensation is no longer feasible, as shown in Fig.
6. Recall Fig. 4. In the case with non-ideal add-on control, the
condition that the augmented command u(t),d(t) + uuc(t)
equals zero ∀t>t0(i.e. u(i)(t+
0) = 0,∀i>0) will not hold.
This, however, does not constrain one to compute the transient
response due to input discontinuity. To be more specific, one
can write the input signal as:
u(t) = (d(t):t∈[0,t0)
d(t) + uuc(t) = ε(t)6=0 : t>t0
,(17)
where t0is the actual implementation time of uuc (i.e., it
may contain the actuation delay) and ε(t)is the residual
term characterizing the difference between the ideal add-
on control command and the actual command. The induced
input discontinuities at time t0can thus be written as eu,i=
u(i)(t+
0)−u(i)(t−
0) = d(i)(t+
0) + u(i)
uc (t+
0)−d(i)(t−
0)=[d(i)(t+
0)−
d(i)(t−
0)] + u(i)
uc (t+
0). When d(t)is of class Ckat t0, the term
in the square bracket equals zero ∀0≤i≤k, yielding the
formula eu,i=u(i)
uc (t+
0),∀0≤i≤k. Note that although t0is
not known when there is actuation delay, u(i)
uc (t+
0)(if exists)
is available since it is the initial condition of the actual add-
on signal. Moreover, (5) works without requiring the condition
that u(i)(t+
0) = 0,∀i>0. Based on the relations in (6a)-(6c), the
output discontinuity {ey,i: 0 ≤i≤k}can be derived without
the need to know d(t)and t0.
As an example, recall the system and the input in (7).
If we regard u(t) = αt,t≥0 and u(t) = αt0,t≥t0, re-
spectively, as the disturbance d(t)and residual term ε(t)in
(17), then the corresponding add-on signal (which we design)
is uuc(t) = ε(t)−d(t) = αt0−αt,t≥t0. One observes that
u(i)
uc (t+
0)is exactly the i-th order input discontinuity. Similar
to the application of Theorem 2 and Remark 3 to (7), this
information of add-on control uuc(t)is adequate and sufficient
to identify the induced output discontinuities.
d+uuc
d
uuc
0
0
t0
t
t
estimated uuc
delayed uuc
tad
Fig. 6: Non-ideal add-on UC control.
V. TIME-D OM AI N RES PO NS E AN D TRANSIENT SPE ED
Based on analysis in the last section, motion control systems
that are powered by motors with double integrator (inertia
system) or 1/(ms2+bs)types of nominal dynamics always
generate continuous outputs; systems with very fast input-
output dynamics, such as piezoelectric actuators (whose nom-
inal dynamics can be commonly modeled as a constant-gain
system), are sensitive to nonsmoothness in the input. On
the other hand, many servo controllers can be considerably
sensitive to input discontinuities. For instance, consider the
causal implementation of an PID controller CPID(s) = kp+
ki/s+kds/(εs+1) = [(kpε+kd)s2+ (kp+kiε)s+ki]/[s(εs+
1)], where ε1 is a small positive number. The direct gain
of the controller kp+kd/εcan be enormous in practice. Let
eand ube the input and output of CPID(s). Then (5) gives
u(t+
0)−u(t−
0) = (kp+kd
ε)[e(t+
0)−e(t−
0)], ˙u(t+
0)−˙u(t−
0) = (ki−
kd
ε2)[e(t+
0)−e(t−
0)] + (kp+kd
ε)[ ˙e(t+
0)−˙e(t−
0)]. Therefore, both
the controller output and its derivatives will exhibit significant
jump discontinuities when the input has only small disconti-
nuities. A lead-lag controller can reduce the amplification of
the input discontinuities, but still has a direct-gain component.
The induced large magnitude in the control command can
easily trigger the input saturation and the resulting nonlinear
effects in the feedback loop (and even instability and hardware
damage). Moreover, omitted in conventional transient analysis,
abnormal faults triggered by such discontinuity amplification
are much more difficult to detect and analyze (see also the
second example in Section VI).
The analysis in the preceding paragraphs focused on charac-
terizing the output at the switching instance. Together with the
quantitative analysis, the following qualitative measures can be
made about intrinsic properties of the system and signals that
will induce small/smooth transient: large relative degree in the
transfer function (LTI case), small coefficients for high-order
derivatives, and small low-order discontinuities in the input
signal. To further reveal characteristics of the transient after
switching, we derive next the exact time-domain response due
to the input nonsmoothness.
Assume that the system coefficients are constant in (4). The
transient response after t0is the solution to the ODE:
y(n)(t) + an−1y(n−1)(t) + ·· · +a0y(t) = 0,(18)
with the initial condition ny(i)(t+
0)(= y(i)(t−
0) + ey,i)on−1
i=0.
Proposition 5. Let the same assumptions in Theorem 2 hold.
Let G(s) = B(s)/A(s)be stable with A (s) = sn+an−1sn−1+
··· +a0and B(s) = bnsn+bn−1sn−1+· · · +b1s+b0. The
transient response due to {eu,i}n
i=0—the input nonsmoothness
at t0—has the 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,n−2
eu,n−1
A(s).
(19)
The full transient response after t0is
Y(s) = Y(s,ei) + e−t0s
A(s)×
sn−1+an−1sn−2+· · · +a2s+a1
sn−2+an−1sn−3+· · · +a2
.
.
.
s+an−1
1
T
yt−
0
˙yt−
0
.
.
.
y(n−2)t−
0
y(n−1)t−
0
.(20)
Proof. Consider t+
0as the initial time. The Laplace-
domain quantities 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+
0give:
L[y(t),˙y(t),...,y(n)(t)]T= [1,s,...,sn]TY(s)
−
0... . . . 0
1 0 .
.
.
s1....
.
.
.
.
.......0
sn−1... s1
yt+
0
˙yt+
0
.
.
.
y(n−1)t+
0
.(21)
Writing the Laplace transform of (18) as
[a0,a1,...,an−1,1]L[y(t),˙y(t),...,y(n−1)(t),y(n)(t)]T=0
and using (21), we can solve for Y(s). The solution is
Y(s) =
e−t0s
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)
A(s).
(22)
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)—obtained by replacing y(t+
0)in (22) with y(t−
0)and
ei, respectively—are the Laplace transforms of the natural
transient and the transient due to nonsmoothness in the input.
Writing out Ys,t−
0explicitly gives (20). To obtain the
specific form of Y(s,ei)in (19), using (5) and (22) gives
Y(s,ei) =
e−t0s
sn−1
sn−2
.
.
.
1
T
1 0 .. . 0
an−1
.......
.
.
.
.
.......0
a1... an−11
ey,0
ey,1
.
.
.
ey,n−1
A(s)
=
e−t0s
sn−1
sn−2
.
.
.
1
T
bn0... 0
bn−1
.......
.
.
.
.
.......0
b1... bn−1bn
eu,0
eu,1
.
.
.
eu,n−1
A(s),(23)
which, after simplifications, is equivalent to (19).
Notice that Proposition 5 quantifies the post-switching
response using only properties of the pre-switching system
and signal. The prophetic result can provide guidance on the
switching instance as well as conditions for small/smooth post-
switching transient response. In (19) and (20), the weighting
of the input discontinuities is characterized by the norms of
the scaling transfer functions. Take (19) for instance. The
transient Y(s,ei)is a linear combination of the delayed impulse
responses of
GB,n(s):=bn
A(s),GB,n−1(s):=bns+bn−1
A(s),.... (24)
The overall transient speed depends on the poles and ze-
ros in the individual modes. The 2 norm of the impulse
response of GB,iequals the H2norm of the stable trans-
fer function GB,i(s). Thus the 2 norm of the overall time-
domain response yei(t) = L−1{Y(s,ei)}is upper bounded
by ||yei(t)||2≤ ||GB,n(s)||2|eu,n−1|+||GB,n−1(s)||2|eu,n−2|+
... ||GB,1(s)||2|eu,0|. Computing the H2norms of the GB,i(s)’s
provides a quantitative understanding of the significant terms
in the discontinuities of the input and its derivatives; and
hence can guide designers about the selection of the switching
instance.
To examine the transient speed in add-on control, consider
first the natural response Y s,t−
0, which is (22) with t+
0
replaced by t−
0, namely,
Ys,t−
0=e−t0sG0(s)yt−
0+G1(s)˙yt−
0+... ,(25)
where
G0(s) = sn−1+an−1sn−2+· · · +a1
A(s)=1
s−a0
A(s)
1
s(26)
G1(s) =
.
.
.
sn−2+an−1sn−3+· · · +a2
A(s)=1
sG0(s)−a1
A(s)
1
s
(27)
Gi(s) = 1
sGi−1(s)−ai
A(s)
1
s,∀i∈{1,2,...,n−1}.(28)
In the time domain, from Final Value Theorem, all elements
L−1{Gi(s)}∀iin (25) have zero steady-state values. Hence
the transient indeed eventually converges to zero. From (26),
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. L−1{G1(s)}from (27) is the differ-
ence between L−11
sG0(s)—the integral of L−1{G0(s)}(a
signal with zero steady-state value)—and the step response
of a1/A(s). Due to the integral effect, the transient speed
of L−1[G1(s)] is slower than L−1[G0(s)]. Analogous results
hold for the general case (28).
For the response due to input discontinuities, with eu,i=
0−u(i)(t−
0), similar construction gives that Y(s,ei)in (23) is
Y(s,ei) = −e−t0sGB,0(s)u(t−
0) + GB,1˙u(t−
0) + ... (29)
GB,0(s) = bnsn−1+· · · +b2s+b1
A(s)=G(s)1
s−b0
A(s)
1
s(30)
GB,i(s) = 1
sGB,i−1(s)−bi
A(s)
1
s,∀i∈{1,2,...,n−1}.
Thus, L−1{GB,0(s)}is the transient difference between the
step responses of G(s)and b0/A(s). In (30) and (26), the
scaling a0and b0change only the relative magnitude of the
response. For a fast overall transient response, (29) needs to
match the transient speed of (25) as Y(s) = Ys,t−
0+Y(s,ei).
If the step response of G(s)is slow, namely, L−1{G(s)/s}in
(30) is slow compared to L−1{1/s}in (26), 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 add-on compensation
is directly applied on the output y(which is, of course, not
feasible in practice).
Recall in Fig. 2, that G(s)—the dynamics between the add-
on control command and the plant output—is P/(1+PC)in
UC and PC/(1+PC)in UR/UE, respectively. Among the
closed-loop transfer functions in a general feedback block
diagram in Fig. 2, PC/(1+PC)has the dynamic response
that is closest to G(s) = 1, and hence will provide the fastest
transient response from the perspective of add-on injection.
On the other hand, from the pole-zero point of view, UR
also has faster transient. As G(s) = P/(1+PC)in UC and
G(s) = PC/(1+PC)in UR, 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 open-loop unstable zeros slow down
the transient and will yield various fundamental limitations
in the steady-state performance of a feedback system (see,
e.g., [25]–[27]). For the poles of C, marginally stable poles or
poles close to the imaginary axis are often needed for high-
gain feedback at low frequencies (consider, e.g., the case of
PID control).
If again the add-on command is not ideal or there exists
actuation delay, similar to (17), we can decompose the actual
add-on command into two parts: uadd−on =uid eal +ε. In this
case, (18) becomes y(n)(t) + an−1y(n−1)(t) + ·· · +a1˙y(t) +
a0y(t) = bnε(n)(t) + bn−1ε(n−1)(t) + · ·· +b1˙
ε(t) + b0ε(t). The
transient response can now be decomposed into three parts:
Y(s) = Ys,t−
0+Y(s,ei) + Y(s,ε), where the term Y(s,ε)is
the frequency domain response of the residual disturbance ε(t).
Assume that the relative magnitude of the residual disturbance
ε(t)over uideal is small and on the same scale for both UC and
UR. The conclusion that UR provides faster transient response
than UC is then still valid.
VI. SIMULATION AND EXPE RI ME NT S
In this section, the theory is verified on the wafer scanner
system. The plant has a nominal model P(s) = 1/(0.2556s2+
0.279s)and is stabilized (in negative feedback) by a PID
controller C(s) = 10000(1+2/s+0.012s). The noncausal dif-
ferentiation action in the PID controller is approximated by
s/(εs+1)with ε=1/9000.
Evaluating the transient properties such as the impulse
and the step responses reveals that PC/(1+PC)provides
much faster transient response with respect to rapid changing
input signals. For actual disturbance rejection, abrupt step
disturbances are first injected to the plant similar to that in Fig.
3, at around 0.12 sec. This time, the add-on compensation is
turned on much faster than the case in Fig. 3. Fig. 7 shows the
effect of add-on compensation 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, add-on compensation is turned on
at t0=0.5 sec and t0=0.25 sec, using the UC and UR
configurations in Section III-B.
0 0.5 1 1.5 2 2.5 3 3.5
0
1
2
position (m)
#10-5 w/o add-on compensation
0 0.5 1 1.5 2 2.5 3 3.5
-1
0
1
2
position (m)
#10-5 add-on compensation turned on at 0.5sec
UC
UR
0 0.5 1 1.5 2 2.5 3 3.5
time (sec)
0
1
2
position (m)
#10-5 add-on compensation turned on at 0.25sec
UC
UR
Fig. 7: Effects of add-on compensation in simulation.
Both add-on compensation schemes affect the system in the
direction of reducing the error. Comparing the two bottom sub-
plots in Fig. 7, it can be seen that the earlier the compensation
is turned on, the more significant the rejection effect is. For
UC compensation, however, there is a large undershoot due
to the system dynamics, yielding a much slower and worse
transient compared to UR. Experiments on the physical system
also strongly verify the results as shown in Fig. 8, where it is
observed that the simulation and experimental results almost
overlap with each other, and UR provides almost zero transient
response in the actual system. Notice that compared to the
results in Fig. 3, although the transient of UC is better in Fig.
8 (thanks to an earlier application of the add-on enhancement),
the UC transient is still significantly long compared to that of
UR. The adverse effect of UC transient always exists as long
as there are delays in turning on the add-on control.
0 0.5 1 1.5 2 2.5 3 3.5
0
1
2
x 10−5 w/o add−on compensation
position (m)
0 0.5 1 1.5 2 2.5 3 3.5
−1
0
1
2
x 10−5
position (m)
w/ add−on UC compensation
simulation
experiment
simulation
experiment
0 0.5 1 1.5 2 2.5 3 3.5
−1
0
1
2
x 10−5
time (sec)
position (m)
w/ add−on UR compensation
simulation
experiment
Fig. 8: Simulation and experimental comparison of transients
in UC and UR compensation.
Furthermore, applying the obtained results (23), we can get
the transient due to input discontinuity in UR:
L−1e−t0s[(120s+10000)eu,0+120eu,1]
0.2556s3+120.3s2+10000s+20000 ,(31)
and in UC:
L−1e−t0s˜eu,0
0.2556s3+120.3s2+10000s+20000 ,(32)
where eu,0,eu,1, and ˜eu,0are the input discontinuities at time
0.5 sec. Fig. 9 provides the computed transient response due to
input discontinuity. The red dashed lines provide a zoomed-in
view of the experimental data in Fig. 8. The blue solid lines are
directly computed using only the nominal system model and
input derivatives at time 0.5. It is clear that except for some
effects from noises in actual hardware (there is a permanent 18
Hz force ripple in the motor), the calculated responses closely
matches those in the actual system.
Of course, as discussed in the theoretical derivations,
the transient problem exists not just for the case of
step disturbances but for any add-on design with input-
discontinuities. Fig. 10 reveals the add-on transient in com-
pensating frequency-dependent disturbances (a 500 Hz vibra-
tion). A similar superior performance of UR add-on design
is observed. In this example, it is no longer possible to
straightforwardly tell the direction of the adverse transient in
UC, as the high-frequency input and its derivatives change very
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
−1
0
1
2
3x 10−5
Position (m)
UC: computed response (starts at 0.5s)
UC: actual response (experimental result)
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.5
1
1.5
2
2.5
3x 10−5
Time (sec)
Position (m)
UR: computed response (starts at 0.5s)
UR: actual response (experimental result)
Fig. 9: Verification of transient computation algorithm in UC
and UR: the red dashed lines are from actual experiments; the
blue solid lines are computed from the developed transient
computation equations (the equations compute only the add-
on transient response, which starts at 0.5 sec).
rapidly with respect to time; and the obtained conclusions in
the paper are increasingly important for avoiding large servo
errors during controller implementation.
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55
−7
−6
−5
−4
−3
−2
−1
0
1
2x 10−6
time(s)
position(m)
w/o compensation
UC
UR
Fig. 10: Experimental comparison of add-on vibration com-
pensations: 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).
As a second example, we apply the developed tools to ana-
lyze a switched control scheme. Let d=0 in Fig. 2. Consider
the case of tracking a reference ras shown in the top plot of
Fig. 11a, which consists of a 10 Hz periodic signal and a 100
Hz signal that starts at around 0.6 sec. ris designed to contain
no discontinuities itself. To track the more aggressive 100 Hz
reference signal, the feedback controller Cswitches to a more
aggressive mode C2=40000 ×(1+3/s+0.02s/(18000s+1))
at around 0.75 sec, resulting in the improved tracking in Fig.
11a. However, a detailed look at the control output indicates
a significant increase of |u(t)|as shown in Fig. 11b. As the
saturation limits of the control input are -10 V and 10 V, such
high-amplitude control inputs are extremely dangerous for
application in practice, despite that the tracking error appears
to be well controlled in simulation. Applying Theorem 2 to
analyze the overlooked danger, one can find that due to the
jump in the input to C2, a significant discontinuity occurs in
the output of C2:u(t+
0)−u(t−
0) = −991.2 V; ˙u(t+
0)−˙u(t−
0) =
1.76255×107V/s. The calculated -991.2 V jump in the control
command can be seen to match well with the actual signal in
Fig. 11b. Furthermore, applying Proposition 5 gives the star-
marked solid line in Fig. 11c, which shows that the transient
induced from the discontinuity in C2indeed is the main
contributor of the abruptness in the overall control command.
0 0.2 0.4 0.6 0.8 1
−2
0
2x 10−4
position (m)
0 0.2 0.4 0.6 0.8 1
−2
0
2x 10−4
position error (m)
time (s)
(a) Reference and tracking error.
0 0.2 0.4 0.6 0.8 1
−1000
−800
−600
−400
−200
0
time (s)
magnitude (V)
Detail during switch: 0.6s~0.8s
zoom in: −20V~20V
(b) Corresponding control input.
0.7555 0.756 0.7565 0.757
−1000
−800
−600
−400
−200
0
time (s)
magnitude (V)
full control command
transient due to discontinuity
(c) Decomposition of control command: the transient due to discontinuity
dominates in the post-switching transient control command.
Fig. 11: Closed-loop signals with direct controller switching.
With the prediction in Fig. 11c, one can turn on the input
to C2first and slightly delay the engagement of the output
of C2, to avoid injecting the high-amplitude signals in the
closed loop. For instance, a 20-step delay in turning on the
output of C2gives the servo results in Fig. 12, where in the
top plot the control command is seen to be maintained well
under the saturation limits (actually no visual discontinuity or
overshoot is observable from the new control command); and
in the bottom plot the error remains to be controlled with a
slight 0.05sec-longer transient compared to Fig. 11a.2
0 0.2 0.4 0.6 0.8 1
−10
0
10
control input (V)
0 0.2 0.4 0.6 0.8 1
−2
0
2x 10−4
position error (m)
time (s)
Fig. 12: Closed-loop signals with smoothened switching.
VII. CONCLUSIONS
This paper addresses the general input-to-output discontinu-
ity problem and applies the results to the transient improve-
ment of add-on 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 problem is not only important for
add-on compensation schemes, but also for other applications
such as the switching between multiple controllers.
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