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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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