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Discrete-time output feedback nonlinear control for combined low-and high-frequency disturbance compensation

Authors:

Abstract

We present a discrete-time output feedback nonlinear control algorithm for reference tracking under both broad-band disturbances at low frequencies and narrow-band disturbances at high frequencies. A discrete-time nonlinear damping backstepping controller with an extended state observer is proposed to track the desired output and to compensate for low-frequency broad-band disturbances. A narrow-band disturbance observer is constructed for rejecting narrow-band high-frequency disturbances. This combination method gives us the merit that both broad-band disturbances at low frequencies and narrow-band disturbances at high frequencies can be simultaneously compensated.
Discrete-time Output Feedback Nonlinear Control for Combined Low-
and High-Frequency Disturbance Compensation
Wonhee Kim, Xu Chen, Chung Choo Chung, Masayoshi Tomizuka
Abstract— We present a discrete-time output feedback non-
linear control algorithm for reference tracking under both
broad-band disturbances at low frequencies and narrow-band
disturbances at high frequencies. A discrete-time nonlinear
damping backstepping controller with an extended state ob-
server is proposed to track the desired output and to com-
pensate for low-frequency broad-band disturbances. A narrow-
band disturbance observer is constructed for rejecting narrow-
band high-frequency disturbances. This combination method
gives us the merit that both broad-band disturbances at low
frequencies and narrow-band 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 high-frequency 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 narrow-band disturbances, but less
effective for broad-band disturbance attenuation.
In this paper, we present an output feedback nonlinear
control to track the desired output with attenuation of both
broad-band disturbances at low frequencies and narrow-
band disturbance at high frequencies. The proposed method
consists of two main parts: (i) a narrow-band disturbance
†: Corresponding Author
W. Kim is with the Department of Electrical Engineering, Dong-A
University, Busan 604-714, Korea (e-mail: whkim79@dau.ac.kr)
X. Chen is with the Department of Mechanical Engineering,
University of Connecticut, Storrs, CT 06269-3139, USA. (E-mail:
xchen@engr.uconn.edu)
C. C. Chung is with the Division of Electrical and Biomedi-
cal Engineering, Hanyang University, Seoul, 133-791, Korea (e-mail:
cchung@hanyang.ac.kr) Tel: +82-2-2220-1724, Fax: +82-2-2291-5307
M. Tomizuka is with the Department of Mechanical Engineering,
University of California, Berkeley, CA 94720-1740, USA. (E-mail:
tomizuka@me.berkeley.edu)
observer (DOB) and (ii) a nonlinear damping backstepping
controller with ESO. High-frequency 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 narrow-band distur-
bance at high frequencies. Compared to other IMP based
algorithms, DOB is a convenient addition to the ESO-based
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 broad-band disturbances at low frequencies.
The nonlinear damping backstepping controller with ESO is
then proposed for tracking control and broad-band 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 continuous-time ESO
algorithms, is the new direct discrete-time design of the
nonlinear backstepping control, with proofs of stability and
convergence directly in the discrete-time 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 H-F REQ UE N CY DI S TU RBANC E REJE CT I ON
This section discusses a discrete-time internal disturbance
observer for cancellation of the high-frequency disturbances
using selective model inversion [5], [9].
Let the plant dynamics be modeled as
Yz1
U(z1)=Pz1(1)
where Uz1is the Z transform of the system input u(k),
Yz1is the Z transform of the system output y(k), The
structure of the disturbance observer is shown in Fig. 1. Here,
zmP
nz1is 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 Yz1,Dz1,Uz1, respectively. Block-diagram
analysis gives
Yz1=Gyd z1Dz1+Gyuz1Uz1
(
)
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
Gyuz1=Pz1
1zmQ(z1) + P(z1)P1
n(z1)Q(z1)
(2)
Gyd z1=Pz11zmQz1
1zmQ(z1) + P(z1)P1
n(z1)Q(z1)
(3)
If zmQz11, then
Gyuz1zmP
nz1(4)
Gyd z10 (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 zmP
nz1.
If Q(z1)0, then
Gyuz1=Gyd z1=Pz1(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 PzmP
n(z1)in a large region [satisfac-
tion of i)]. High-frequency disturbances however are usually
challenging to compensate in regular servo control [challenge
in ii)]. Assume the high-frequency disturbance is centered at
ω
d(in rad/sec), the DOB can be designed as follows: let
Qz1=Qoz1m
where
Qoz1= (
α
1)(
α
+1)z12 cos (
ω
dTs)
12
α
cos(
ω
dTs)z1+
α
2z2
This way, the Q filter has the frequency response as shown in
Fig. 2. Such a Q-filter design achieves zmQz1
z=ej
ω
dTs=
1. From (5), Gyd z1
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 disturbance-observer loop: From (2) and
(3), the poles of the local disturbance observer loop contain
poles of Qz1, and
roots of
1zmQz1+Pz1P1
nz1Qz1=0 (7)
From (7), a sufficient condition for its roots to be stable is
that
Qej
ω
<1
P(ej
ω
)P1
n(ej
ω
)ejm
ω
=1
|(ej
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
LOW-FR 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 low-frequency disturbances,
using an ESO and a discrete-time nonlinear damping back-
stepping controller for the nominal plant zmP
n(z1).
Let the nominal system model zmP
n(z1)be given by
zmP
n(z1) = bn1zn1+···+b1z+b0
zn+an1zn1+···+a1z+a0
.(8)
Under the assumption that the numerator and the denomina-
tor of (8) are coprime, a minimal state-space 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)TRn×1is the
state vector, u(·)is the control input, d(·)is the resid-
ual disturbance after narrow-band disturbance compensation.
Additionally,
A=
010··· 0 0
001··· 0 0
.
.
..
.
..
.
..........
000··· 0 1
a0a1a2··· −an2an1
Rn×n
B=01×(n1)1TRn×1
C=b0b1b2··· bn2bn1R1×n.
Here, the transfer function from yto x1is
X1(z1)
Y(z1)=1
bn1zn1+bn2zn2+···+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(z1) = 1
bn1zn1+bn2zn2+···+b1z+b0
Yd(z1).
(11)
Here we assume that d(k)is slowly time-varying 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 state-space 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
a0a1a2··· −an11
0 0 0 ··· 0 1
R(n+1)×(n+1)
Bex =01×(n1)1 0TR(n+1)×1
Bd=01×n1TR(n+1)×1
Cex =b0b1b2··· bn10R1×(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+1TR(n+1)×1is the observer gain
matrix. Define the estimation error
˜xex =
˜x1
˜x2
.
.
.
˜xn
˜xn+1
=
x1ˆx1
x2ˆx2
.
.
.
xnˆxn
xn+1ˆxn+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¯
AP
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¯
AP
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¯
AP
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¯
AP
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=˜xex|k˜xex k2kP
ok2
δ
max
νλ
min(Qo).
Analogous to the continuous-time 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+l1zn1+···+ln)
zn+r+l1zn+r1+···+ln+r1z+ln+r
.
(26)
Thus the extended observer can effectively estimate distur-
bances whose frequencies are below the cut-off 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)
.
.
.
en1(k) =xn1d(k)xn1(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)
.
.
.
en1(k+1) =xn1d(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)
.
.
.
en1(k+1) =xn1d(k+1)xnd(k) + en(k)
en(k+1) =xnd(k+1) +
n
i=1
ai1xi(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) = cn1en1(k) + xnd(k+1)
u(k) = cnen(k) + xnd(k+1) +
n
i=1
ai1xi(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=x1dˆx1. In u(k), the term cnen(k) + xnd(k+1) +
n
i=1ai1xi(k)ˆxn+1(k)is for stabilization of the system.
The term cn[1exp(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 uget 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)
.
.
.
en1(k+1) =cn1en1(k) + en(k)
en(k+1) =cns(k)en(k)˜xn+1(k)
(31)
where cns(k) = cncd(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) =
k1
j=0
cns(j)en(0) +
k1
j=0
ckj
ns(j)˜xn+1(j).(32)
Since |cns(k)|<1 for all kand ˆxn+1(k)is bounded, en(k)is
also bounded. From en1(k+1)in (31), we also have
en1(k) = ck
n1e1(0) +
k1
j=0
ckj
n1en(j).(33)
Since |cn1|<1 and en(k)is bounded, en1(k)is also
bounded. Thus, we have
e1(k) =ck
1e1(0) +
k1
j=0
ckj
1e2(j)
.
.
.
en1(k) =ck
n1e1(0) +
k1
j=0
ckj
n1en(j)
en(k) =
k1
j=0
cns(j)en(0) +
k1
j=0
ckj
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|c1||e2(k)|
.
.
.
lim
k|en1(k)| ≤sup
k
1
1|cn1||en(k)|
lim
k|en(k)| ≤sup
k
1
1|cncd(ˆe1(k),ˆxn+1(k))||˜xn+1(k)|.
(35)
Equation (35) shows the input-to-state stability (ISS) prop-
erty of the tracking error system (31). As ˆe1and ˆ
d=ˆxn+1get
larger, 1
1−|cncd(ˆe1,ˆxn+1)|gets closer to 1, yielding a smaller
gain for the effect of |˜xn+1(k)|.
C. Stability Analysis of the Closed-loop 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) = cn1ˆen1(k) + ˆxnd(k+1)
ˆu(k) = cnˆen(k) + ˆxnd(k+1) +
n
i=1
ai1ˆxi(k)ˆxn+1(k)
+cn[1exp(cd1ˆe2
1(k)cd2ˆx2
n+1(k)
ν
)] ˆen(k)
(36)
where ˆei=ˆxiˆxid,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 closed-loop system. The
nth subsystem of the tracking error dynamics (31) becomes
en(k+1) =[cncd(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+ˆuu| ≤
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)|≤|(cncd(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|cncd(ˆe1(k),ˆxn+1(k))|
sup
k
γ
k˜xex(k)k2
1|cncd(ˆe1(k),ˆxn+1(k))|.
(40)
Finally, from (35) and (40), we have
lim
k|e1(k)| ≤sup
k
1
1|c1||e2(k)|
.
.
.
lim
k|en1(k)| ≤sup
k
1
1|cn1||en(k)|
lim
k|en(k)| ≤sup
k
γ
k˜xex(k)k2
1|cncd(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 high-frequency disturbances,
the plant with mixed disturbances can be regarded as the
nominal plant with just broad-band disturbances at low
frequencies.
For implementation, the output of P(z1)with the narrow-
band DOB is used instead of that of zmP
n(z1)in ESO (15).
The output of zmP
n(z1)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) + Bdd(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(z1) = bz +0.8b
z32z2+z=z2b+0.8bz1
12z1+z2(43)
where b=3.4766 ×107. Then m=2, and
P1
nz1=z22z+1
bz2+0.8bz.
A minimum state-space realizations of the plant is
x(k+1) =
010
001
01 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
01 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(z1) = 1
bz +0.8bYd(z1).(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 ydy
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 high-frequency ripples appeared
in the output tracking error. Fig 9 shows the simulation
results of case 3. With the narrow-band 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 discrete-time output feedback nonlinear control was
proposed to track the desired output with both broad-band
disturbances at low frequencies and narrow-band disturbance
at high frequencies. In the proposed algorithm, an ESO
estimates the full state and broad-band disturbance at low
frequencies; and design of a narrow-band DOB was provided
to reject the narrow-band disturbance at high frequencies.
Simulation results showed that the proposed method can
compensate for both broad-band disturbance at low frequen-
cies and narrow-band disturbance at high frequencies.
REF ERE NCE S
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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
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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 ydy
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 ydy
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
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