sampled data active disturbance rejection output feedback control for systems with mismatched uncertainties

91 1–9 Ó The Authors 2017 DOI: 10.1177/1687814016682690 journals.sagepub.com/home/ade Sampled-data active disturbance rejection output feedback control for systems with mismatched uncert

Advances in Mechanical Engineering

2017, Vol 9(1) 1–9

Ó The Author(s) 2017 DOI: 10.1177/1687814016682690 journals.sagepub.com/home/ade

Sampled-data active disturbance

rejection output feedback control

for systems with mismatched

uncertainties

Jun You1, Jiankun Sun2, Shuaipeng He3and Jun Yang2

Abstract

This article investigates the sampled-data disturbance rejection control problem for a class of non-integral-chain systems with mismatched uncertainties Aiming to reject the adverse effects caused by general mismatched uncertainties via digi-tal control strategy, a new generalized discrete-time extended state observer is first proposed to estimate the lumped disturbances in the sampling point A disturbance rejection control law is then constructed in a sampled-data form, which will lead to easier implementation in practices By carefully selecting the control gains and a sampling period suffi-ciently small to restrain the state growth under a zero-order-holder input, the bounded-input bounded-output stability

of the hybrid closed-loop system and the disturbance rejection ability are delicately proved even the controller is dor-mant within two neighbor sampling points Numerical simulation results demonstrate the feasibility and efficacy of the proposed method

Keywords

Sampled-data control, disturbance rejection, mismatched uncertainty, discrete-time extended state observer

Date received: 26 August 2016; accepted: 14 October 2016

Academic Editor: Yongping Pan

Introduction

In this article, we consider a class of input

single-output (SISO) system with mismatched uncertainties of

the form

_x(t) = Ax(t) + Buu(t) + Bdf x(t), v(t), tð Þ

ym(t) = Cmx(t)

y0(t) = C0x(t)

ð1Þ

where x(t)2Rn is the system state vector, u(t)2R is

the control input, v(t)2R is the external disturbance,

ym(t)2Rr is the measurable outputs, y0 2R is the

con-trolled output, and A2Rn 3 n, Bu2Rn 3 1, Bd 2Rn 3 1,

Cm2Rr 3 n, C02R1 3 n are system matrices f (x(t),

v(t), t) is an uncertain function representing the lumped

disturbance in a general way which possibly includes

external disturbances, unmodeled dynamics, parameter variations, and complex nonlinear dynamics.1–8

In modern control practices, it is a trend that sampled-data controllers with a zero-order holder (ZOH) are being digitally implemented into real-life plants with the rapid development of computational hardware technology It is common that conventional

1

School of Electrical Engineering, Southeast University, Nanjing, China

2

Key Laboratory of Measurement and Control of CSE, Ministry of Education, School of Automation, Southeast University, Nanjing, China

3

College of Automation Engineering, Shanghai University of Electric Power, Shanghai, China

Corresponding author:

Jun You, School of Electrical Engineering, Southeast University, Nanjing

210096, China.

Email: youjun@seu.edu.cn

Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License

(http://www.creativecommons.org/licenses/by/3.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage).

control law design is always devoted to continuous-time

design for mathematically modeled continuous-time

systems due to the convenience of direct stability

analy-sis, typically based on the continuously differentiable

Lyapunov function analysis In most practical

imple-mentation processes, continuous-time controllers can

be discretized directly which exists on the fact that the

closed-loop system performance can always be

guaran-teed while the sampling frequency is fast enough

However, in most of the times, this is done without a

theoretical support, and the formulated restrictions of

the sampling frequency and how it affects the controlled

system performance are actually unknown Moreover,

only digital sensors are available for some real-life

plants, for instance, Global Positioning System (GPS)

is a discrete-time sensor and a radar is more naturally

represented using a discrete-time model.9 Hence, it is

always a challenging but crucial issue to address the

problem of designing sampled-data controllers which

pursue smaller steady-state error, faster dynamical

response, and milder noise susceptibility for system

per-turbed by matching or mismatched uncertainties

In the literature, one of the most popular methods

among those existing approaches developed to design

sampled-data controllers for nonlinear plants is the

dis-cretization method.10–12 It employs a discrete-time

approximation model of the plant (typically use Euler

approximation) to design discrete-time controllers since

it is almost impossible to obtain an exact discrete-time

model of the continuous-time nonlinear plant Hence,

the results regarding this method are always achieved

within a local or semi-global control goal The

emula-tion method, as a global control method, designs a

time controller based on the

continuous-time plant, followed by a discretization process to yield

sampled-data controllers which will assure the

continuous-time nonlinear plant by choosing an

appro-priate sampling period One can refer to papers13–19

and the references therein

In practical engineering, the control performance of

modern industrial systems is inevitably affected by

vari-ous uncertainties including parameter variations,

unmodeled dynamics, and external disturbances.20–22

Active anti-disturbance technique is generally required

in the controller design to access high-precision control

performance.1,23–25 The extended state observer–based

control (ESOBC) was originally proposed by Han,26,27

and it is made practical by the tuning method which

simplifies its implementation and makes the design

transparent to engineers.1Su et al.28presented the

rela-tionship between time-domain and frequency-domain

disturbance observers and its applications for further

information However, most of existing disturbance

rejection methods, including ESOBC, are concerned

with continuous control approaches, which lack sound

justification since most control approaches are digitally implemented in a sampled-data manner To this end, the development of an active disturbance rejection method for system (1) with mismatched uncertainties will be of interest for both theoretical and industrial communities Moreover, the mismatched uncertainties, rather than the so-called matching conditions4,20 are concerned to discover a generalized sampled-data dis-turbance rejection control law for system (1) Mismatched uncertainties may not act via the same channel with the control input and are regarded as a more general case concerned in uncertainty attenuation problems As an example, the lumped disturbance tor-ques in flight control systems always affect the states directly rather than through the input channels.23,29 This article presents a generalized sampled-data con-trol law design based on a discrete-time extended state observer (ESO), which estimates the unmeasurable states and the lumped disturbance information in the sampling point Explicit formulas to select the control gains and the tunable sampling period based on a detailed stability analysis for the hybrid closed-loop sys-tem are presented Numerical simulations are shown to demonstrate the effectiveness of the proposed method The proposed method will be a helpful guideline for direct digital implementation

Main results

In this section, we present a step-by-step procedure

to design a discrete-time ESO-based sampled-data control law to solve the global stabilization problem for system (1)

With an added extended variable

xn + 1(t) = d = f x(t), v(t), tð Þ system (1) can be extended to the following form

_x(t) = Ax(t) + Buu(t) + Eh(t)

with the denotation of

x(t) = x T(t), xn + 1(t)T h(t) = df x(t), v(t), tð Þ

dt



01 3 n 0

2R(n + 1) 3 (n + 1)



Bu= Bu

0

2R(n + 1) 3 1

E =½0, , 0, 1T2R(n + 1) 3 1



Cm=½Cm, 0r 3 1 2Rr 3 (n + 1)

Assumption 1 (A, Bu) is controllable and (A, Cm) is

observable

Assumption 2 The lumped disturbances satisfy the

fol-lowing conditions:20

 d(t) and h(t) are bounded, that is, there exist

two positive constants d,  h such that

jd(t)j  d, jh(t)j  h, respectively

lim t!‘_d(t) = lim

t!‘h(t) = 0

In the article, the Assumption 2 is made on the

dis-turbances d(t) that its derivative is close to zero

However, this assumption can be relaxed using

unknown observer theory.30,31

Construction of discrete-time ESO

In what follows, an ESO for system (1) will be built

using the sampled-data information ym(tk)

(tk= kT , k = 0, 1, 2, ), where T is the sampling

period The continuous-time state ^x(t)¼D ½^xT, ^xn + 1T

defined in the time region t2 ½tk, tk + 1)

_^x(t) = A^x(t) + Buu(t) L ^yð m(t) ym(tk)Þ, t2 ½tk, tk + 1)

ð3Þ where ^ym(t) = Cm^x(t) and L2R(n + 1) 3 r is the observer

gain matrix to be assigned later

The observer (3) can be rewritten as follows

_^x(t) = A^x(t) + Buu(tk) L ^yð m(t) ym(tk)Þ

= ðA L CmÞ^x(t) + Buu(tk) + Lym(tk), t2 ½tk, tk + 1)

ð4Þ Integrating the continuous-time observer (4) from tk

to tk + 1, it can be concluded that ^x(tk + 1) can also be

generated by the following discrete-time ESO

^

x(tk + 1) = eðAL  Cm ÞT^x(tk) +

ðT 0

eðAL Cm Þsds ðBuu(tk) + Lym(tk)Þ

¼D F^x(tk) + Gu(tk) + Nym(tk)

ð5Þ with the denotation of

F = eðAL Cm ÞT

G =

ðT

0

eðAL Cm Þs

dsBu

N =

ðT

eðAL  Cm Þs

dsL

The state and disturbances errors are defined as e(t) =½eT

x(t), ed(t)T where ex(t) = ^x(t) x(t), ed(t) =

^ d(t) d(t), respectively With equations (2) and (3), the error dynamics are

_e(t) = ðA L CmÞe(t)  Eh(t)  L Cmðx(t) x(tk)Þ,

t2 ½tk, tk + 1) ð6Þ

Sampled-data disturbance rejection law design

The sampled-data disturbance rejection law can be designed as

u(t) = u(tk) = K^x(tk) = Kx^x(tk) + Kdd(t^ k), t2 ½tk, tk + 1)

ð7Þ where K =½Kx, Kd Kx is the control gain to be designed Kd is the disturbance compensation gain Motivated by the results,20Kdis assigned by the follow-ing equation

Kd= C0ðA + BuKxÞ1Bu

C0ðA + BuKxÞ1Bd

Remark 1 With equation (7) in mind, the designed discrete-time extended observer can also be presented as

^ x(tk + 1) = M^x(tk) + Nym(tk) ð8Þ where M = F GK 2R(n + 1) 3 (n + 1) and N are two matrices dependent of the sampling period T

Hybrid closed-loop system stability analysis

Combing equations (1), (6), and (7) together, one can obtain the hybrid closed-loop system as

_x(t) _e(t)

= A + BuKx BuK

0 A L Cm

e(t)

+ 0 BuKd+ Bd

d(t)

+ BuK 0

0 L Cm

^ x(t) ^x(tk)

 x(t) x(tk)

If we choose the observer gain L and the feedback gain Kx such that A + BuKx and A L Cm are Hurwitz matrices, it is easy to verify that the following matrix

A + BuKx BuK

0 A L Cm

¼D L

is also a Hurwitz one Hence, there exists a positive defi-nite matrix P = PT 2R(2n + 1) 3 (2n + 1)such that

LTP + PL =aI where a.0 is a constant will be determined later

Construct a positive definite and proper Lyapunov

function V (Z(t)) = ZT(t)PZ(t) where

Z(t) =½xT(t), eT(t)T The derivative of V (Z(t)) along

system (9) is given as follows

_

V Z(t)ð Þ = a k Z(t)k2+ 2Z(t)T

P 0 BuKd+ Bd

d(t)

+ 2Z(t)TP BuK 0

0 L Cm

x(t) ^x(tk)

x(t) x(tk)

,

t2 ½tk, tk + 1)

ð10Þ

Now, we will estimate the items in the right hand

side of equation (10) First, with jh(t)j  h,

k P k = lmax(P), k E k = 1 in mind, we have

2 Z(t)TP 0 BuKd+ Bd

d(t)



 2c1k P k  h + d

k Z(t) k

= c1 h + d

k Z(t) k

ð11Þ

where c1.0 and c1= 2c1lmax(P) are constants

Since K, L are set, one can conclude that

BuK 0

0 L Cm

where c2.0 is a constant dependent on the value of

K, L

Similar to equation (11), there exists a constant

c2.0, such that the following estimation holds

2 Z(t)TP B u K 0

0 L  C m

x(t)  ^  x(t k )

 x(t)  x(t k )

" #













 2c 2 k Z(t) k  k P k  e(t) e(tk) + x(t) x(tk)

 x(t)  x(t k )

 4c 2 lmaxfPg k Z(t) k k Z(t)  Z(t ð k ) k + k d(t)  d(t k ) k Þ

 c 2 k Z(t) kk Z(t)  Z(t k ) k + c 2 d  k Z(t) k , t 2 ½t k , tk + 1)

ð13Þ Substituting equations (11) and (13) into equation

(10) yields

_

V Z(t)ð Þ   a k Z(t)k2+ c 1h + c1d + c2

k Z(t) k + c2 k Z(t) kk Z(t)  Z(tk)k , t2 ½tk, tk + 1)

ð14Þ Next, we introduce a lemma which plays a key role

in the main theorem

Lemma 1 There exist proper constants c3.0, c4.0, such that the following inequality holds

k Z(t)  Z(tk)k  c4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

V (Z(tk))

p

+ h + d

ec3 (tkT) 1

,

t2 ½tk, tk + 1) ð15Þ

Proof 1 Denote the whole closed-loop system (9) to be _Z(t) = C(u(tk), Z(t)) With the fact that h(t) and d(t) are bounded in mind, the following inequality holds for

s2 ½tk, t)

k C u(tð k), Z(s)Þ k  c3ðk Z(s)  Z(tk)k + k Z(tk)k + h + dÞ ð16Þ where c3.0 is a constant By equation (16), one can obtain that

k Z(t)  Z(tk)k 

ðt

t k

k C(u(tk), Z(s))k ds

 c3 k Z(tk)k + h + d

(t kT ) +

ðt

t k

c3k Z(s)  Z(tk)k ds, t2 ½tk, tk + 1)

ð17Þ

Based on equation (17), applying the Gronwall inequality, we have

k Z(t)  Z(tk)k  c3 k Z(tk)k + h + d

(t kT ) + c23 k Z(tk)k + c2

3h + c2

3

t k

(s kT )ec 3 (ts)

ds

 c4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

V (Z(tk))

p

+ h + d

ec3 (tkT) 1

, t2 ½tk, tk + 1)

ð18Þ where c4.0 is a constant

With Lemma 1 in mind, equation (14) becomes

_

V Z(t)ð Þ   a k Z(t)k2+ c 1h + c1d + c2

k Z(t) k + c2c4k Z(t) k ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

V Z(tð k)Þ

p

+ h + d

ec3 (tkT ) 1

  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia

lminfPg

!

V Z(t)ð Þ + c6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

V Z(t)ð Þ

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

V Z(tð k)Þ

p

ec3 (tkT ) 1

+ c 5+ c6 ec3 (tkT ) 1

(h + d) ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

V Z(t)ð Þ

p

,

t2 ½tk, tk + 1)

ð19Þ with two positive constants c5, c6.0

Theorem 1 Under Assumptions 1 and 2, with the

fol-lowing selection formulas for the parameter a and the

allowable sampling period T

a

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

lminfPg

p  c5+ 1; T \ 1

c3

ln 1 + 1

c6

ð20Þ

and the observer gain L and the feedback control gain

Kx are selected such that A + BuKx and A L Cm are

Hurwitz matrices; system (1) under the proposed

sampled-data control law (5)–(7) can be rendered

bounded stable for any bounded d(t), h(t)

Proof Define j(t) = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

V (Z(t))

p

= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

V (Z(tk))

p

It can be concluded that equation (19) equals to the following

inequality

_j(t)  1

2

a ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

lminfPg

! j(t) +c6

2 e

c 3 (tkT ) 1

+1

2 c5+ c6 e

c 3 (tkT ) 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

V Z(tð k)Þ

t2 ½tk, tk + 1)

ð21Þ With equation (20) in mind, the following inequality

can be concluded

_j(t)  1

2j(t) +

c6

2 e

c 3 T 1

+1

2 c5+ c6 e

c 3 T 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

V Z(tð k)Þ

p , t2 ½tk, tk + 1)

ð22Þ Solving the above inequality with j(tk) = 1, we have

j(tk + 1) c6 ec3 T 1

+ c 5+ c6 ec3 T 1 h + d

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

V Z(tð k)Þ p

!

1 eT2

 + eT2 ¼D r(T ) ð23Þ which leads to

V Z(tð k + 1)Þ  r(T )V Z(tð k)Þ ð24Þ

According to the condition (20), we have

16¼ c6(ec 3 T 1) Hence, the difference equation

DV (Z) = V (Z(tk + 1)) V (Z(tk)) 0 if

V Z(tð k)Þ  c5+ c6 e

c 3 T 1

1 c6ðec 3 T 1Þ

ð25Þ

which implies that Z(tk) decreases for any Z(tk) satisfies

condition (25); hence, the state Z(tk) is bounded for any

bounded d(t) and h(t) if those parameters are properly

selected This completes the proof of Theorem 1

Remark 2 As one can mention, the discrete-time obser-ver (8) will generate the same sampled information

^x(kT ) at the sampling point for the same initial condi-tion with the continuous-time observer (3) Hence, in the design, we could use the continuous-time form (3)

to take fully advantages of the continuous-time design method.16,18As a matter of fact, we do not need to con-sider the signal information for t2 (kT, (k + 1)T ) for the observer (3), what we need is only the sampled information ^x(kT ) which can be generated from the sampled output information y(kT ), k = 0, 1, As one can observe from the proof of Theorem 1 that even the controller is dormant within two neighbor sampling points and the inter-sample information is not ana-lyzed, the fact that the difference Lyapunov function

DV (Z) is semi-negative definite can assure the bounded-input bounded-output stability of the hybrid closed-loop system

Disturbance rejection analysis

In what follows, we will give a detailed disturbance rejection analysis to show the proposed ESO-based sampled-data disturbance rejection control law will effectively reject the lumped disturbances

Theorem 2 Under Assumptions 1 and 2, the lumped disturbances from system (1) can be effectively rejected from the output channel in steady-state under the pro-posed sampled-data control law (7) provided that the observer gain L and the feedback control gain Kx are selected such that A + BuKx, A L Cm are Hurwitz matrices and C0(A + BuKx)1Buis invertible

Proof Substituting the sampled-data controller (7) into system (1) yields

_x(t) = Ax(t) + BuKxex(tk) + BuKxx(tk) + Bd(d(t) d(tk)) Bded(tk), t2 ½tk, tk + 1)

ð26Þ

By Theorem 1, the following relations hold for the hybrid closed-loop system (1)–(7)–(8)

lim t!‘_x(t) = lim

k!‘_x(tk) = 0, lim

k!‘e(tk) = 0 ð27Þ Hence, one can conclude from equations (26) and (27) that

lim t!‘y0(t) = lim k!‘y0(tk) = C0ðA + BuKxÞ1_x(tk) C0ðA + BuKxÞ1

3 Bð uKxex(tk) + Bdðd(tk) d(tk)Þ  Bded(tk)Þ = 0 ð28Þ

which concludes that the lumped disturbances can be

effectively rejected from the output channel in

steady-state

Remark 3 In this article, we assume some states of the

systems are not measurable; hence, the proposed

sampled-data control strategy has to be implemented

by output feedback of the form of equation (7) If we

consider the case that the states are available or can be

easily measured, the sampled-data state-feedback

con-trol law can be simply designed as

u(t) = u(tk) = Kxx(tk) + Kdd(t^ k), t2 ½tk, tk + 1)

Kd= C 0(A + BuKx)1Bu1

C0ðA + BuKxÞ1Bd

ð29Þ where ^d(tk) can be generated from the discrete-time

ESO (8)

Remark 4 Figure 1 depicts the implementation

struc-ture of practical plant with the proposed ESO-based

sampled-data disturbance rejection controller (7) and

(8) In practical engineering, the dominated dynamics

of a system has been stabilized by feedback control,

and the nonlinear character of those uncertainties

con-tained in the system are relatively weak, which means

the presence of the uncertainties will not cause much

damage to the closed-loop system’s stability Hence,

such uncertainties can be regarded as part of the

lumped disturbances and can be reasonably handled by

sampled-data control law of the form (7) This fact will

support the general availability of the proposed control

method

Example and simulation results

Next, we use an example and numerical simulations to show how an ESOBC law is designed and the effective-ness of the proposed method

A two-dimensional example

Consider a two-dimensional uncertain nonlinear system with mismatching condition of the form

_x1(t) = x2(t) + f x(t), v(t), tð Þ _x2(t) =2x1(t) x2(t) + u(t) y(t) = x1(t)

ð30Þ

which can also be written as the state-space form

_x(t) = Ax(t) + Buu(t) + Bdd(t)

with the denotation of A = 0 1

, Bu= 0

1

  ,

Bd= 0 1

  , Cm= C0= C =½1, 0, and d(t) = f (x1(t),

x2(t), v(t), t)

Clearly, system (30) can be seen as an example of system (1) and can satisfy Assumptions 1 and 2 Let d(t) = x3(t), one can obtain an extended system as

_x(t) = Ax(t) + Bu(t) + Eh(t)

where A =

2 4

3

5, Bu=

0 1 0

2 4

3

5, C = ½1, 0, 0, and E =½0, 0, 1T One can construct a discrete-time Figure 1 Implementation structure of extended state observer–based sampled-data controller.

ESO-based sampled-data output feedback controller of

the following form

^

x tðk + 1Þ = M^x(tk) + Ny(tk) ð33Þ

u(t) = K^x(tk), t2 ½tk, tk + 1) ð34Þ

where ^x =½^x1, ^x2, ^dT, K =½k1, k2, k3, M = e( AL  C)T+

ÐT

0 e(AL C)sdsBuK, and N =ÐT

0 eL( AL  C)sdsL

Numerical simulations

In what follows, we will use numerical simulations for

system (30) to demonstrate the effectiveness of the

pro-posed method

The disturbance is considered as d(t) = ex 1 (t)+ v,

where v represents the external disturbance assigned as

v= 2 acting on the system at the time instant t = 5s,

the observer gain vector L is selected as

L =½14,  66, 125T, and the controller gain

K =½4,  4,  5 based on the guideline (7) Hence,

in this case, we can verify that A + BuKx and A L C

are both Hurwitz By Theorem 1, we can choose a

proper sampling period T = 0:05s The discrete-time

observer matrices can be calculated as

M =

0:5286 1:0833 1:0913

26:3854 2:8041 3:3161

5:6081 4:6100 6:0125

0

@

1 A,

N =

0:5278

437:8961 500:2623

0

@

1 A

Figures 2–4 show the response curves of the system

and estimated states One can observe from Figures 2

and 3 that the system states converge to the equilibrium

quickly in the presence of both internal uncertainties

and external disturbances It is illustrated by Figure 2

that the lumped disturbance can be successfully rejected

in the output channel And from Figure 4, it is shown

that the discrete-time ESO works effectively and leads

to high-precision observation of the disturbance d(t)

The time history of the sampled-data output feedback

control law (34) is shown in Figure 5

Conclusion

A novel discrete-time ESO-based sampled-data active

disturbance rejection output feedback control law has

been proposed in this article to achieve high-precision

control performances for a class of systems with

mis-matched uncertainties With a delicate design

proce-dure, the careful selection of the involved parameters

ensures the global stability of the hybrid closed-loop

system and disturbance rejection ability Direct digital

design strategy will lead to easier implementation Numerical simulations have shown the effectiveness of the proposed method

Figure 2 Trajectories of x1(t) and ^ x1(t k ) of the closed-loop system (30)–(33)–(34).

Figure 3 Trajectories of x2(t) and ^ x2(t k ) of the closed-loop system (30)–(33)–(34).

Figure 4 Trajectories of the disturbance d(t) and estimated disturbance ^ d(tk) from the ESO (33).

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with

respect to the research, authorship, and/or publication of this

article.

Funding

The author(s) received no financial support for the research,

authorship, and/or publication of this article.

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