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Discretetime Output Feedback Nonlinear Control for Combined Low
and HighFrequency Disturbance Compensation
Wonhee Kim, Xu Chen, Chung Choo Chung†, Masayoshi Tomizuka
Abstract— We present a discretetime output feedback non
linear control algorithm for reference tracking under both
broadband disturbances at low frequencies and narrowband
disturbances at high frequencies. A discretetime nonlinear
damping backstepping controller with an extended state ob
server is proposed to track the desired output and to com
pensate for lowfrequency broadband disturbances. A narrow
band disturbance observer is constructed for rejecting narrow
band highfrequency disturbances. This combination method
gives us the merit that both broadband disturbances at low
frequencies and narrowband disturbances at high frequencies
can be simultaneously compensated.
I. INTRO DUC TIO N
A robust control design for systems affected by distur
bances is a fundamental issue in control engineering. One
approach that has been extensively studied for disturbance
rejection is to use an extended state observer (ESO) [1],
[2]. By regarding the disturbance as an extended state,
the system states and disturbance can be estimated using
output feedback. It has been shown that the transfer function
between the disturbance and the disturbance estimation error
is in the form of a high pass ﬁlter [2], which provides
the capability to estimate disturbances whose frequencies
are below the observer bandwidth. A high observer gain
is required to estimate highfrequency disturbances in ESO.
Yet, this will tend to amplify measurement noises at high
frequencies. Thus, it is difﬁcult for ESO based method to
compensate for the high frequency disturbances. Generally,
many high frequency disturbances are in the form of induced
vibrations [3], [4]. To reject this disturbance, the internal
model principle (IMP) [6] based perspective has been investi
gated in feedback control algorithms [7], [8]. These methods
are effective to cancel narrowband disturbances, but less
effective for broadband disturbance attenuation.
In this paper, we present an output feedback nonlinear
control to track the desired output with attenuation of both
broadband disturbances at low frequencies and narrow
band disturbance at high frequencies. The proposed method
consists of two main parts: (i) a narrowband disturbance
†: Corresponding Author
W. Kim is with the Department of Electrical Engineering, DongA
University, Busan 604714, Korea (email: whkim79@dau.ac.kr)
X. Chen is with the Department of Mechanical Engineering,
University of Connecticut, Storrs, CT 062693139, USA. (Email:
xchen@engr.uconn.edu)
C. C. Chung is with the Division of Electrical and Biomedi
cal Engineering, Hanyang University, Seoul, 133791, Korea (email:
cchung@hanyang.ac.kr) Tel: +82222201724, Fax: +82222915307
M. Tomizuka is with the Department of Mechanical Engineering,
University of California, Berkeley, CA 947201740, USA. (Email:
tomizuka@me.berkeley.edu)
observer (DOB) and (ii) a nonlinear damping backstepping
controller with ESO. Highfrequency disturbances are usu
ally challenging to compensate in regular servo control. The
DOB constructed based on inﬁnite impulse response (IIR)
ﬁlters [5], [9] is applied to reject the narrowband distur
bance at high frequencies. Compared to other IMP based
algorithms, DOB is a convenient addition to the ESObased
backstepping control, as the former can selectively reject
disturbances without altering the nominal plant dynamics,
which is needed in (ii). With the functionality of DOB, the
plant with mixed disturbances can be regarded as a nominal
plant with just broadband disturbances at low frequencies.
The nonlinear damping backstepping controller with ESO is
then proposed for tracking control and broadband distur
bance compensation. To improve servo performance when
the tracking error is large, nonlinear damping is implemented
in the backstepping controller. Another contribution of the
paper compared to [2] and general continuoustime ESO
algorithms, is the new direct discretetime design of the
nonlinear backstepping control, with proofs of stability and
convergence directly in the discretetime domain.
The proposed method was validated via simulation for a
motion control problem using linear motors and air bearings.
II. SEL ECT IVE MO DE L IN VE R SI ON FO R
HIG HF REQ UE N CY DI S TU RBANC E REJE CT I ON
This section discusses a discretetime internal disturbance
observer for cancellation of the highfrequency disturbances
using selective model inversion [5], [9].
Let the plant dynamics be modeled as
Yz−1
U(z−1)=Pz−1(1)
where Uz−1is the Z transform of the system input u(k),
Yz−1is the Z transform of the system output y(k), The
structure of the disturbance observer is shown in Fig. 1. Here,
z−mP
nz−1is the nominal plant model that is used in model
based feedback and feedforward designs; and mis the relative
degree of the nominal model.
Remark: Note the difference between u(k)and u∗(k).u∗(k)
is the control command from the backstepping algorithm that
will be later designed.
Deﬁne the Z transforms of y(k),d(k), and u∗(k),
as Yz−1,Dz−1,U∗z−1, respectively. Blockdiagram
analysis gives
Yz−1=Gyd z−1Dz−1+Gyu∗z−1U∗z−1
(
)
1
P z
−
(
)
1
1 /
n
P z
−
(
)
1
Q z
−
m
z
−
(
)
d k
(
)
y k
(
)
*
u k
+
+
+
+
−
−
(
)
u k
(
)
DOB
u k
Fig. 1. Structure of the DOB
where
Gyu∗z−1=Pz−1
1−z−mQ(z−1) + P(z−1)P−1
n(z−1)Q(z−1)
(2)
Gyd z−1=Pz−11−z−mQz−1
1−z−mQ(z−1) + P(z−1)P−1
n(z−1)Q(z−1)
(3)
If z−mQz−1≈1, then
Gyu∗z−1≈z−mP
nz−1(4)
Gyd z−1≈0 (5)
namely, the plant uncertainty and the disturbance d(k)are
rejected in the local feedback loop, such that the overall
dynamics between u∗(k)and y(k)approximately equals the
nominal model z−mP
nz−1.
If Q(z−1)≈0, then
Gyu∗z−1=Gyd z−1=Pz−1(6)
and DOB is disengaged from the loop.
When combined with the nonlinear damping backstepping
control, it is desired that
i) (4) is valid in a large frequency range, and
ii) (5) holds at frequencies where strong external dis
turbances occur outside the control bandwidth of the
backstepping design.
For precision systems, good model information is usually
available such that P≈z−mP
n(z−1)in a large region [satisfac
tion of i)]. Highfrequency disturbances however are usually
challenging to compensate in regular servo control [challenge
in ii)]. Assume the highfrequency disturbance is centered at
ω
d(in rad/sec), the DOB can be designed as follows: let
Qz−1=Qoz−1m
where
Qoz−1= (
α
−1)(
α
+1)z−1−2 cos (
ω
dTs)
1−2
α
cos(
ω
dTs)z−1+
α
2z−2
This way, the Q ﬁlter has the frequency response as shown in
Fig. 2. Such a Qﬁlter design achieves z−mQz−1
z=ej
ω
dTs=
1. From (5), Gyd z−1
z=ej
ω
dTsthus equals zero, namely,
we achieve perfect disturbance rejection at the particular
frequency
ω
d. At frequencies other than
ω
d,Qhas small
gains and the system recovers to (6), hence maintaining
80
60
40
20
0
Magnitude (dB)
10
1
10
0
10
1
10
2
10
3
180
90
0
90
180
270
360
Phase (deg)
Bode Diagram
Frequency (Hz )
Fig. 2. Frequency response of an example Q ﬁlter
the original dynamic properties. Analogous design can be
applied for the case of disturbance rejection at multiple
frequencies [5], [9].
Stability of the disturbanceobserver loop: From (2) and
(3), the poles of the local disturbance observer loop contain
•poles of Qz−1, and
•roots of
1−z−mQz−1+Pz−1P−1
nz−1Qz−1=0 (7)
From (7), a sufﬁcient condition for its roots to be stable is
that
Qe−j
ω
<1
P(e−j
ω
)P−1
n(e−j
ω
)−e−jm
ω
=1
∆(e−j
ω
)
III. DES IG N O F OUTP UT FEE DBACK NON LIN EAR
DAM PI NG BACK S TE P PI N G CON T ROL L ER FO R
LOWFR EQU ENC Y DIS T UR BA NCE RE JE C TI O N
This section discusses the proposed tracking control with
compensation of strong external lowfrequency disturbances,
using an ESO and a discretetime nonlinear damping back
stepping controller for the nominal plant z−mP
n(z−1).
Let the nominal system model z−mP
n(z−1)be given by
z−mP
n(z−1) = bn−1zn−1+···+b1z+b0
zn+an−1zn−1+···+a1z+a0
.(8)
Under the assumption that the numerator and the denomina
tor of (8) are coprime, a minimal statespace realizations of
(8) can be represented as
x(k+1) =Ax(k) + Bu∗(k) + Bd(k)
y(k) =Cx(k)(9)
where x(k) = x1(k)x2(k)··· xn(k)T∈Rn×1is the
state vector, u∗(·)is the control input, d(·)is the resid
ual disturbance after narrowband disturbance compensation.
Additionally,
A=
010··· 0 0
001··· 0 0
.
.
..
.
..
.
..........
000··· 0 1
−a0−a1−a2··· −an−2−an−1
∈Rn×n
B=01×(n−1)1T∈Rn×1
C=b0b1b2··· bn−2bn−1∈R1×n.
Here, the transfer function from yto x1is
X1(z−1)
Y(z−1)=1
bn−1zn−1+bn−2zn−2+···+b1z+b0
,(10)
Given a designed yd,x1d–the desired state of x1(required
for the controller design)–can be made from the following
equation
X1d(z−1) = 1
bn−1zn−1+bn−2zn−2+···+b1z+b0
Yd(z−1).
(11)
Here we assume that d(k)is slowly timevarying such that
d(k+1) = d(k) +
δ
(k).(12)
where 
δ
(k) ≤
δ
max.
A. Design of ESO
With the disturbance d(k)deﬁned as an extended state
xn+1(k) = d(k)(13)
the augmented statespace plant model (10) becomes
xex(k+1) =Aexxex (k) + Bexu∗(k) + Bd
δ
(k)
y(k) =Cexxex (k)(14)
where xex(k) = xT(k)xn+1(k)T,
Aex =
0 1 0 ··· 0 0
0 0 1 ··· 0 0
.
.
..
.
..
.
..........
0 0 0 ··· 1 0
−a0−a1−a2··· −an−11
0 0 0 ··· 0 1
∈R(n+1)×(n+1)
Bex =01×(n−1)1 0T∈R(n+1)×1
Bd=01×n1T∈R(n+1)×1
Cex =b0b1b2··· bn−10∈R1×(n+1).
We assume that the pair (Aex ,Cex)is observable. This is
true if the plant does not have a zero at z=1 to cancel
the disturbance mode in (12) under the assumption that the
numerator and the denominator of (8) are coprime. The ESO
is designed as
ˆxex(k+1) =Aex ˆxex(k) + Bexu∗(k) + L(y(k)−ˆy(k))
ˆy(k) =Cex ˆxex(k)(15)
where ˆxex =ˆx1··· ˆxn+1Tis the estimated xex and L=
l1l2··· lnln+1T∈R(n+1)×1is the observer gain
matrix. Deﬁne the estimation error
˜xex =
˜x1
˜x2
.
.
.
˜xn
˜xn+1
=
x1−ˆx1
x2−ˆx2
.
.
.
xn−ˆxn
xn+1−ˆxn+1
∈R(n+1)×1.(16)
Then the estimation error dynamics are
˜xex(k+1) = ¯
A˜xex(k) + Bd
δ
(k)(17)
where ¯
A= (Aex −LCex). For stability analysis, we deﬁne the
Lyapunov candidate function Voas
Vo(k) = ˜xT
ex(k)P
o˜xex(k)(18)
where P
ois positive deﬁnite. Since P
ois positive deﬁnite,
ζ
TP
o
ζ
≥0 for any
ζ
. If we choose
ζ
=√
ν
¯
A˜xex(k)−
Bd
δ
(k)/√
ν
, then
√
ν
¯
A˜xex −Bd
δ
√
ν
T
P
o√
ν
¯
A˜xex −Bd
δ
√
ν
≥0.(19)
Thus, we obtain
˜xT
ex(k)¯
ATP
oBd
δ
(k) +
δ
(k)BT
dP
o¯
A˜xex(k)
≤
ν
˜xT
ex(k)¯
AP
o¯
A˜xex(k) +
δ
(k)BT
dP
oBd
δ
(k)
ν
.(20)
Then, ∆Vo(k) = Vo(k+1)−Vo(k)is
∆Vo(k) = ˜xT
ex(k+1)P
o˜xex(k+1)−˜xT
ex(k)P
o˜xex(k)
=¯
A˜xex(k) + Bd
δ
(k)TP
o¯
A˜xex(k) + Bd
δ
(k)
−˜xT
ex(k)P
o˜xex(k)
=˜xT
ex[¯
ATP
o¯
A−P
o]˜xex +˜xT
ex(k)¯
ATP
oBd
δ
(k)
+
δ
(k)BT
dP
o¯
A˜xex(k) + BT
dP
oBd
δ
2(k).
(21)
From (20) and (21),
∆Vo(k)≤˜xT
ex[(1+
ν
)¯
ATP
o¯
A−P
o]˜xex
+1+1
ν
BT
dP
oBd
δ
2(k).(22)
If eigenvalues of ¯
A= (Aex −LCex)are all inside the unit cir
cle, the positive deﬁnite matrix solution P
oto the Lyapunov
matrix equation
Qo=−[¯
ATP
o¯
A−P
o](23)
exists such that Qois positive deﬁnite. Note that

λ
(√1+
ν
¯
A)=√1+
ν

λ
(¯
A)where
λ
(A)denotes the
eigenvalues of A. Replacing ¯
Aby √1+
ν
¯
Ain (23), the matrix
equation
Qo=−[(1+
ν
)¯
ATP
o¯
A−P
o](24)
has a unique positive deﬁnite solution P
oif 
λ
(¯
A)<
1
√1+
ν
[12]. Consequently,
∆Vo(k)≤˜xT
exQo˜xex +1+1
ν
BT
dP
oBd
δ
2
max(k)
≤−
λ
min(Qo)k˜xexk2
2+1+1
ν
kP
ok2
δ
2
max(k).
(25)
From (25), we conclude that if 
λ
(¯
A)<1
√1+
ν
,xex converges
to the bounded ball Br=˜xexk˜xex k2≤√kP
ok2
δ
max
√
νλ
min(Qo).
Analogous to the continuoustime case in [2], the transfer
function H(z)from the disturbance dto the estimation error
˜
dis in the form of the high pass ﬁlter as
H(z):=˜
D(s)
D(s)
=zr(zn+l1zn−1+···+ln)
zn+r+l1zn+r−1+···+ln+r−1z+ln+r
.
(26)
Thus the extended observer can effectively estimate distur
bances whose frequencies are below the cutoff frequency of
H(z).
B. Design of Nonlinear Damping Backstepping
For output tracking, we will design the controller
via backstepping. We deﬁne the tracking error e(k) =
e1(k)e2(k)··· en(k)Tas
e1(k) =x1d(k)−x1(k)
.
.
.
en−1(k) =xn−1d(k)−xn−1(k)
en(k) =xnd(k)−xn(k)
(27)
where xid,i∈[2,n]will be designed in a moment. Based
on (17), the tracking error system is
e1(k+1) =x1d(k+1)−x2(k)
.
.
.
en−1(k+1) =xn−1d(k+1)−xn(k)
en(k+1) =xnd(k+1)−xn(k+1)
(28)
which can be written as
e1(k+1) =x1d(k+1)−x2d(k) + e2(k)
.
.
.
en−1(k+1) =xn−1d(k+1)−xnd(k) + en(k)
en(k+1) =xnd(k+1) +
n
∑
i=1
ai−1xi(k)−xn+1(k)−u∗(k).
(29)
The desired state variables and control input in the nonlinear
damping backsteppig controller are designed as
x2d(k) = −c1e1(k) + x1d(k+1)
.
.
.
xnd(k) = −cn−1en−1(k) + xnd(k+1)
u∗(k) = −cnen(k) + xnd(k+1) +
n
∑
i=1
ai−1xi(k)
−ˆxn+1(k) + cd(ˆe1,ˆxn+1)en(k)
(30)
where ci<1,i∈[1,n],cd(ˆe1,ˆxn+1) = cn[1−
exp(−cd1ˆe2
1(k)−cd2ˆx2
n+1(k)−
ν
)],cd1>0, cd2>0,
ν
>0,
and ˆe1=x1d−ˆx1. In u∗(k), the term −cnen(k) + xnd(k+1) +
∑n
i=1ai−1xi(k)−ˆxn+1(k)is for stabilization of the system.
The term cn[1−exp(−cd1ˆe2
1(k)−cd2ˆx2
n+1(k)−
ν
)] is the
nonlinear damping term. Generally, the estimated output
tracking error ˆe1(k)and the estimated disturbance ˆxn+1(k)
increase when the disturbance estimation error ˜xn+1(k)
increases. The role of the nonlinear damping term makes
the control gain −cn+cd(ˆe1,ˆxn+1)of u∗get close to zero
for a large disturbance estimation error ˜xn+1(k), which will
later provide a tighter bound of the errors in (35).
With the controller (30), the tracking error system (29)
becomes
e1(k+1) =c1e1(k) + e2(k)
.
.
.
en−1(k+1) =cn−1en−1(k) + en(k)
en(k+1) =cns(k)en(k)−˜xn+1(k)
(31)
where cns(k) = cn−cd(ˆe1(k),ˆxn+1(k)). Now we show the
boundedness of the tracking errors in (31) and provide
equations about how the design parameters control the error
bound.
From en(k+1)in (31), we have
en(k) =
k−1
∏
j=0
cns(j)en(0) +
k−1
∑
j=0
ck−j
ns(j)˜xn+1(j).(32)
Since cns(k)<1 for all kand ˆxn+1(k)is bounded, en(k)is
also bounded. From en−1(k+1)in (31), we also have
en−1(k) = ck
n−1e1(0) +
k−1
∑
j=0
ck−j
n−1en(j).(33)
Since cn−1<1 and en(k)is bounded, en−1(k)is also
bounded. Thus, we have
e1(k) =ck
1e1(0) +
k−1
∑
j=0
ck−j
1e2(j)
.
.
.
en−1(k) =ck
n−1e1(0) +
k−1
∑
j=0
ck−j
n−1en(j)
en(k) =
k−1
∏
j=0
cns(j)en(0) +
k−1
∑
j=0
ck−j
ns(j)˜xn+1(j).
(34)
Ex.Ob.
Non. Back.
(
)
d k
(
)
d
y k
(
)
y k
(
)
*
ˆ
u k
(
)
(
)
(
)
1 1 1
ˆˆ
, , ,
n
x k x k x k
+
⋯
(
)
1
m
n
z P z
− −
+
+
Fig. 3. Structure of the control system
Using the technique in [10], it can be derived that
lim
k→∞e1(k) ≤sup
k
1
1−c1e2(k)
.
.
.
lim
k→∞en−1(k) ≤sup
k
1
1−cn−1en(k)
lim
k→∞en(k) ≤sup
k
1
1−cn−cd(ˆe1(k),ˆxn+1(k))˜xn+1(k).
(35)
Equation (35) shows the inputtostate stability (ISS) prop
erty of the tracking error system (31). As ˆe1and ˆ
d=ˆxn+1get
larger, 1
1−cn−cd(ˆe1,ˆxn+1)gets closer to 1, yielding a smaller
gain for the effect of ˜xn+1(k).
C. Stability Analysis of the Closedloop System
In practice, usually only the output yis available. The other
states in (28) is replaced by its estimate ˆxifor implementation
of (30). Thus the control law (30) becomes
ˆx2d(k) = −c1ˆe1(k) + x1d(k+1)
.
.
.
ˆxnd(k) = −cn−1ˆen−1(k) + ˆxnd(k+1)
ˆu∗(k) = −cnˆen(k) + ˆxnd(k+1) +
n
∑
i=1
ai−1ˆxi(k)−ˆxn+1(k)
+cn[1−exp(−cd1ˆe2
1(k)−cd2ˆx2
n+1(k)−
ν
)] ˆen(k)
(36)
where ˆei=ˆxi−ˆxid,i∈[1,n]. Fig. 3 shows the structure of
the control system that consists of the nominal plant, the
ESO, and the nonlinear damping backstepping controller.
Now we study the stability of the closedloop system. The
nth subsystem of the tracking error dynamics (31) becomes
en(k+1) =[cn−cd(ˆe1(k),ˆxn+1(k))]en(k)
−˜xn+1(k) + u∗
2(k)−u∗
1(k).(37)
As shown in (31) and (37), since the observer affects only
the nth subsystem of the tracking error dynamics (31), it is
sufﬁcient to investigate the behavior of the nth subsystem
owing to the cascade ISS property. In u∗(k)and ˆu∗(k), the
different desired state variables, xd
iand ˆxidi∈[3,n], are used
respectively. On the other hand, x1,x1d, and x2dare used in
both (30) and (36). Thus, a positive constant
γ
exists such
that
˜xn+1+ˆu∗−u∗ ≤
γ
kˆxex −xexk2.(38)
(
)
1
P z
−
(
)
1
1 /
n
P z
−
(
)
1
Q z
−
m
z
−
Ex.Ob.
Non. Back.
(
)
d k
(
)
d
y k
(
)
y k
(
)
*
ˆ
u k
(
)
(
)
1 1
ˆˆ
, ,
n
x k x k
+
⋯
(
)
1
m
n
z P z
− −
+
+
+
+
−
−
(
)
u k
Fig. 4. Structure of the overall control system
From (37)
en(k+1)≤(cn−cd(ˆe1(k),ˆxn+1(k)))en(k)
+˜xn+1(k) + ˆu∗(k)−u∗(k)(39)
and based on (35), ensatisﬁes
lim
k→∞en(k) ≤sup
k
˜xn+1(k) + u∗
2(k)−u∗
1(k)
1−cn−cd(ˆe1(k),ˆxn+1(k))
≤sup
k
γ
k˜xex(k)k2
1−cn−cd(ˆe1(k),ˆxn+1(k)).
(40)
Finally, from (35) and (40), we have
lim
k→∞e1(k) ≤sup
k
1
1−c1e2(k)
.
.
.
lim
k→∞en−1(k) ≤sup
k
1
1−cn−1en(k)
lim
k→∞en(k) ≤sup
k
γ
k˜xex(k)k2
1−cn−cd(ˆe1(k),ˆxn+1(k))
(41)
Equation (41) thus shows the ISS property of the tracking
error system (31) with (37).
IV. OVER ALL CO NTROL SY ST E M
Fig. 4 shows the structure of the overall control system.
With DOB compensating the highfrequency disturbances,
the plant with mixed disturbances can be regarded as the
nominal plant with just broadband disturbances at low
frequencies.
For implementation, the output of P(z−1)with the narrow
band DOB is used instead of that of z−mP
n(z−1)in ESO (15).
The output of z−mP
n(z−1)was deﬁned as y=Cx =Cexxex.
The actually output, the output of Pwith the DOB, is deﬁned
as yac. Due to the difference between yand yac, the estimation
error dynamics (17) becomes
˜xex(k+1) = (Aex −LCex)˜xex(k) + Bd∆d(k) + L(yac(k)−y(k)).
(42)
Since the DOB guarantees the stability of the disturbance
observer loop, yac −yis bounded. Thus the results from (18)
to (21) give us the boundedness of the estimation error. Con
sequently, the ISS property of the tracking error system (31)
guarantees the boundedness of the output tracking error.
0 1 2 3 4 5 6 7
−0.02
0
0.02
0.04
0.06
0.08
0.1
Time [second]
Position [m]
Fig. 5. The desired output yd
V. SI MU L ATION EVAL UATI ON
This section provides applications of the algorithm to a
motion control system for photolithography that has been
described in [11]. It consists of linear motors and air bear
ings. The precision system has a quite accurate model. The
transfer function of the plant is
P(z−1) = bz +0.8b
z3−2z2+z=z−2b+0.8bz−1
1−2z−1+z−2(43)
where b=3.4766 ×10−7. Then m=2, and
P−1
nz−1=z2−2z+1
bz2+0.8bz.
A minimum statespace realizations of the plant is
x(k+1) =
010
001
0−1 2
x(k) +
0
0
1
u(k)
y(k) =0.8b b 0x(k)
(44)
where x(k) = x1(k)x2(k)x3(k)T. The model of the
plant including the disturbance is
x(k+1) =
010
001
0−1 2
x(k) +
0
0
1
u(k) +
0
0
1
d(k)
y(k) =0.8b b 0x(k)
(45)
Using (10), x1dis obtained by
X1d(z−1) = 1
bz +0.8bYd(z−1).(46)
In these simulations, the desired output ydis shown in Fig. 5.
For comparison, we tested three cases:
[Case 1:] w/o DOB; under disturbances at only low
frequencies
[Case 2:] w/o DOB; under disturbances at both low and
high frequencies
[Case 3:] w/ DOB; under disturbances at both low and
high frequencies.
In the three cases, the nonlinear damping backstepping
controller (30) with the ESO (15) was used. 2 sin(k)and
sin(2k)were used as the disturbances at low frequencies;
sin(160
π
k)was used as the disturbance at high frequency.
Fig 6 shows the simulation results of case 1. Since the
disturbance d=x4was accurately estimated, the nonlinear
damping backstepping controller (30) achieved good tracking
0 1 2 3 4 5 6 7
−0.02
0
0.02
0.04
0.06
0.08
0.1
Time [second]
Position [m]
yd
y
(a) Output tracking performance of y
0 1 2 3 4 5 6 7
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
x 10−6
Time [second]
Tracking error [m]
(b) Output tracking error yd−y
0 1 2 3 4 5 6 7
−4
−3
−2
−1
0
1
2
3
4
Time [second]
x4
x4
Estimated x4
(c) Estimation performance of d=x4
Fig. 6. Simulation results of case 1
of yd. Estimation performance of case 1 is shown in Fig. 7. It
is observed that the estimated state variables tracked the state
variables well. The simulation results of case 2 are shown
in Fig 8. Unlike case 1, the system was subjected to high
frequency disturbances whose frequency was higher than the
bandwidth of ESO. Thus highfrequency ripples appeared
in the output tracking error. Fig 9 shows the simulation
results of case 3. With the narrowband DOB, the high
frequency disturbance was signiﬁcantly attenuated and the
output tracking error of case 3 was reduced to the same level
as that in case 1.
VI. CO N CL USI ONS
A discretetime output feedback nonlinear control was
proposed to track the desired output with both broadband
disturbances at low frequencies and narrowband disturbance
at high frequencies. In the proposed algorithm, an ESO
estimates the full state and broadband disturbance at low
frequencies; and design of a narrowband DOB was provided
to reject the narrowband disturbance at high frequencies.
Simulation results showed that the proposed method can
compensate for both broadband disturbance at low frequen
cies and narrowband disturbance at high frequencies.
REF ERE NCE S
[1] M. Zeitz, “The extended Luenberger observer for nonlinear systems,”
Syst. Control Lett., vol. 9, no. 2, pp. 149156, 1987.
0 1 2 3 4 5 6 7
0
5
10
15
20x 104
Time [second]
x1
x1
Estimated x1
(a) Estimation performance of x1
0 1 2 3 4 5 6 7
0
5
10
15
20x 104
Time [second]
x2
x2
Estimated x2
(b) Estimation performance of x1
0 1 2 3 4 5 6 7
0
5
10
15
20x 104
Time [second]
x3
x3
Estimated x3
(c) Estimation performance of x1
Fig. 7. Estimation performance of case 1
[2] W. Kim and C. C. Chung, “Robust high order augmented observer based
control for nonlinear systems,” in Proc. IEEE Conf. Dec. Control, 2012,
pp. 919924.
[3] C. Kinney, R. de Callafon, E. Dunens, R. Bargerhuff, and C. Bash,
“Optimal periodic disturbance reduction for active noise cancelation,”
J. Sound Vib., vol. 305, no. 12, pp. 2233, 2007.
[4] X. Chen and M. Tomizuka, “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, 2012.
[5] X. Chen and M. Tomizuka, “Overview and new results in disturbance
observer based adaptive vibration rejection with application to advanced
manufacturing,” International J. Adap. Control & Signal Processing, to
appear.
[6] B. A. Francis and W. M. Wonham, “The internal model principle of
control theory,” Automatica, vol. 12, no. 5, pp. 457465, 1976.
[7] F. BenAmara, P. T. Kabamba,and a. G. Ulsoy, “Adaptive sinusoidal
disturbance rejection in linear discretetime systems part I:theory,”
ASME J. Dyn. Syst. Meas. Control, vol 121, no. pp. 648654., 1999.
[8] W. Kim, H. Kim, C. Chung, and M. Tomizuka, “Adaptive output
regulation for the rejection of a periodic disturbance with an unknown
frequency,” IEEE Trans. Control Syst. Technol., vol. 19, no. 5, pp. 1296
1304, 2011.
[9] X. Chen and M. Tomizuka, “Selective model inversion and adaptive
disturbance observer for timevarying vibration rejection on an active
suspension benchmark,”European J. Control, vol. 19, no. 4, pp. 300
312, 2013.
[10] Z.P. Jiang and Y. Wang, “Inputtostate stability for discretetime
nonlinear systems,” Automatica, vol. 37, no. 6, pp. 857869, 2001.
[11] 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. 664675, 2014.
[12] A. Weinmann, Uncertain Models and Robust Control, SpringerVerlag,
1991.
0 1 2 3 4 5 6 7
−0.02
0
0.02
0.04
0.06
0.08
0.1
Time [second]
Position [m]
yd
y
(a) Output tracking performance of y
0 1 2 3 4 5 6 7
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
x 10−6
Time [second]
Tracking error [m]
(b) Output tracking error yd−y
0 1 2 3 4 5 6 7
−4
−3
−2
−1
0
1
2
3
4
Time [second]
x4
x4
Estimated x4
(c) Estimation performance of d=x4
Fig. 8. Simulation results of case 2
0 1 2 3 4 5 6 7
−0.02
0
0.02
0.04
0.06
0.08
0.1
Time [second]
Position [m]
yd
y
(a) Output tracking performance of y
0 1 2 3 4 5 6 7
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
x 10−6
Time [second]
Tracking error [m]
(b) Output tracking error yd−y
0 1 2 3 4 5 6 7
−4
−3
−2
−1
0
1
2
3
4
Time [second]
x4
x4
Estimated x4
(c) Estimation performance of d=x4
Fig. 9. Simulation results of case 3