Recei ed: 28 July 2022 Re ised: 27 Ma ch 2023 Accep ed: 21 June 2023 IET Con ol Theo y & Applica ions
DOI: 10.1049/c h2.12515
ORIGINAL RESEARCH
Modi ied ac i e dis u bance ejec ion con ol scheme o sys ems
wi h ime delay
Josu Jugo1Ande Elejaga1Pablo Eche a ia2
1Elec ici y and Elec onics Depa men , Uni e si y
o he Basque Coun y UPV/EHU, Campus Leioa,
Spain
2Helmhol z-Zen um Be lin, Be lin, Ge many
Co espondence
Josu Jugo, Elec ici y and Elec onics Depa men ,
Uni e si y o he Basque Coun y UPV/EHU,
Campus Leioa 48940, Spain.
Email: josu.jug[email p o ec ed]
Funding in o ma ion
Ekonomia en Ga apen e a Lehiako asun Saila,
Eusko Jau la i za, G an /Awa d Numbe : KK-
2022/00026; Hezkun za, Hizkun za Poli ika E a
Kul u a Saila, Eusko Jau la i za, G an /Awa d
Numbe s: IT1533-22, Ph Deg ee g an o A. Elejaga
Abs ac
Ac i e dis u bance ejec ion con ol (ADRC) has been gaining a en ion in ecen yea s
and has shown i s pe o mance in mul iple applica ions including non-linea ones, wi h-
ou he need o accu a e models. Despi e he good esul s o his echnique, ime delay
can de e io a e he pe o mance o ADRC, limi ing i s applica ion. He e, he e ec o
ime delay on he s abili y o a linea ADRC is analysed, using an al e na i e ma hema -
ical desc ip ion, and a new e ec i e design echnique, based on a modi ied ADRC scheme,
is p oposed o o e come he delay e ec while main aining he dis u bance ejec ion p op-
e ies o he ADRC. An expe imen al example is discussed conside ing a sys em wi h low
damped mechanical esonances, showing good esul s using he p oposed echnique.
1 INTRODUCTION
Ac i e dis u bance ejec ion con ol (ADRC) has gained ele-
ance in ecen yea s, due o i s g owing success ul applica ion
[1–4] and he ecen esea ch inc easing i s heo e ical back-
g ound [1, 5–9]. Some imes p esen ed as an e olu ion o he
classical p opo ional in eg al de i a i e con ol (PID) [1], he
ADRC echnique is based in ou undamen al elemen s: a
simple di e en ial equa ion as a ansien ajec o y gene a o ,
a noise- ole an acking di e en ia o , he non-linea con ol
laws and he use o he concep o o al dis u bance es ima ion
and ejec ion.
One o he keys o he ADRC con ol echnique is he
ex ended s a e obse e (ESO), which es ima es he ex e nal
dis u bances and conside s he in e nal dynamics as ano he
dis u bance. The plan is educed o a simple chained in eg a-
o o m by using a con ol law, in which model-based design
me hods can be applied. Fu he mo e, eedback o he ex ended
s a e, consis ing o he in e nal dynamics and he ex e nal dis-
u bance, allows he educ ion o he o al dis u bance. This way
he desi ed sys em dynamics is ob ained simul aneously wi h he
dis u bance educ ion. In addi ion, he ADRC con ol e o is
usually lowe han ha equi ed wi h o he con ol echniques.
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© 2023 The Au ho s. IET Con ol Theo y & Applica ions published by John Wiley & Sons L d on behal o The Ins i u ion o Enginee ing and Technology.
Howe e , as wi h o he con ol echniques, ADRC con ol
has p ac ical limi a ions. An impo an limi a ion o eedback
con ol a ises om he p esence o ime delay, which educes
he s abili y ange. The delay in oduces a phase ha is lin-
ea ly dependen wi h he equency, limi ing he gain a highe
equencies in o de o main ain he s abili y. This e ec is espe-
cially ele an in he p esence o poo ly damped esonances, as
high gains wi h high phase shi a e di icul o manage wi h
eedback con ol. The delay limi s he dis u bance educ ion
bandwid h ha can be ob ained wi h he con en ional ADRC
con ol scheme. An example o a sys em wi h his p oblem is he
con ol o mic ophonics in supe conduc ing RF ca i ies used
in pa icle accele a o s, whe e dis o ion educ ion is c i ical o
keep he ca i y esonance a ound he nominal alue [10]. Se e al
wo ks deal wi h he ime delay e ec by mean o ADRC-based
schemes and se e al me hods ha e been p oposed [11–16]. One
possibili y p oposed by Han is o igno e he ime-delay and
design he ADRC o dynamics wi hou ime-delay, bu his
echnique limi s he pe o mance ob ained. Ano he possibil-
i y is he use o a Padé app oxima ion, inc easing he sys em
o de , bu i is only alid o small ime-delays [1, 13, 16]. To
ake in accoun he ime-delay, in [11, 12] i was sugges ed o
delay he con ol signal by he same amoun be o e i en e s
1992 wileyonlinelib a y.com/ie -c h IET Con ol Theo y Appl. 2023;17:1992–2003.
JUGO ET AL.1993
he ESO. In [14], a gene alized p opo ional in eg al (PI) con-
ol based on Smi h’s p edic o , and he ADRC philosophy,
is p oposed o a class o delayed-inpu non-linea mechani-
cal sys ems. In [15], acking con ol o unce ain ime-delayed
sys ems is p oposed, which is implemen ed in a p edic o
scheme o ime-delay compensa ion. In [16], a wo-deg ee-o -
eedom (2DOF) con ol s uc u e is p oposed o uns able
ime-delayed sys ems. In gene al, hese me hods imp o e
s abili y in he p esence o ime-delay bu educe he dis u -
bance ejec ion e ec . Mo e ecen ly, p obabilis ic obus ness-
based ADRC design has also been applied o sys ems wi h
delay [17].
Dis u bance educ ion is a key con ol objec i e in sys ems
wi h low damped esonances o many applica ions, such as
con ol o mic ophonics in supe conduc ing ca i ies o pa -
icle accele a o s, [2–4]. ADRC is a good candida e o hese
applica ions, bu he e ec o ela i ely small ime-delays can be
c ucial due o he s abili y p oblems in oduced by his delay,
which limi s i s applica ion and e ec i e dis u bance ejec ion.
This ela i ely small delay can be easily in oduced by ac ua o s.
This wo k s udies he e ec o he delay in he s abili y o
a linea ADRC (LADRC) con ol sys em. The esul o his
s udy is a no el s a egy based on LADRC o inc ease he s a-
bili y in he p esence o ime delay, ob aining an imp o emen
in dis u bance ejec ion. The main con ibu ions a e:
∙A ma hema ical desc ip ion o LADRC ha acili a es he
analysis o s abili y and he e ec o he ime delay.
∙The analysis o he ime-delay e ec using LADRC, based on
he ma hema ical desc ip ion p esen ed.
∙A new modi ied LADRC con ol scheme (MLADRC), alid
o inc easing he delay s abili y ma gin.
Fi s , in Sec ion 2, he s abili y analysis o a delayed LADRC
sys em is pe o med, a e ew i ing he basic scheme ollow-
ing a simila way o he one in [12]. The sys em desc ip ion
ob ained allows he de ini ion o he MLADRC algo i hm,
p esen ed in Sec ion 3, which acili a es he s abili y analy-
sis and adds a new con ol elemen , inc easing he designe ’s
eedom. The design o his con ol elemen o s abiliz-
ing he closed-loop sys em in he p esence o ime-delay is
discussed, using loop shaping as design echnique. The esul -
ing scheme allows o imp o e he s abili y in he p esence
o ime-delay, main aining as much as possible he so-called
ma ching condi ion and, he e o e, imp o ing he dis u bance
educ ion. An expe imen al applica ion example in ol ing
mechanical esonances is used o discuss in de ail he ull p o-
cedu e, which shows good esul s and imp o ed dis u bance
ejec ion.
2STABILITY ANALYSIS OF LINEAR
ADRC SYSTEMS WITH TIME-DELAY
In his sec ion, he analysis o an LADRC is ca ied ou , ollow-
ing he desc ip ion p oposed in [18] and including he e ec o
an inpu ime-delay.
Conside a linea sys em desc ibed in he Laplace domain by
he exp ession:
Y(s)=e−𝜏sP(s)(U(s)+𝜉
1(s))+𝜉
2(s)(1)
whe e P(s) is a ans e unc ion ep esen ing he sys em dynam-
ics, 𝜉1,2(s) a e he ex e nal dis u bances o he sys em and τis
he inpu ime-delay.
In he o iginal ADRC desc ip ion, he sys em model is no
a equi emen , his being one o he powe ul cha ac e is ics o
his con ol app oach. The only in o ma ion needed is he ela-
i e o de o P(s), ha is, p:=n−m, and i s high- equency gain
b=bm∕an, whe e nand ma e he o de o he denomina o and
nume a o o he ans e unc ion P(s), espec i ely, and anand
bma e he coe icien s o he highes deg ees.
Igno ing o he momen he delay o simpli y he de elop-
men , he con olled plan is desc ibed by he ollowing model:
y(p)( )=bu ( )+ ( )(2)
whe e ( ) is a combina ion o he unknown plan dynamics and
he ex e nal plan dis u bance, which is called he gene alized
dis u bance and assumed o be unknown in he ADRC design
amewo k.
Following his design scheme, he cen al idea is o es i-
ma e he unknown gene alized dis u bance ( ( )). To do so, a
Luenbe ge - ype ex ended linea obse e (ESO) is de ined. Le
z1=y,z2=⋅
y,…,zp=y(p−1),zp+1= (3)
Assume ha is di e en iable and le ⋅
=h. Then, (2)can
be w i en as {z=Aez+Beu+Eeh
y=Cez
whe e z=[z1,z2…,zp,zp+1]Tand
Ae=⎡⎢⎢⎢⎢⎢⎢⎢⎣
010…0
001…0
⋮⋮⋮⋱⋮
000…1
000…0
⎤⎥⎥⎥⎥⎥⎥⎥⎦(p+1)×(p+1)
(4)
Be=[00…b0]T
(p+1)×1
Ee=[00…01
]T
(p+1)×1
Ce=[100…0]T
(p+1)×1
A ull-o de Luenbe ge s a e obse e can be designed as
ollows: {
z=Aez+Beu+Lo(y−y)
y=Cez
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1994 JUGO ET AL.
FIGURE 1 S uc u e o he classical LADRC, including he plan an inpu delay.
whe e Lois he obse e gain ec o Lo=[𝛽1𝛽2…𝛽
p𝛽p+1]T.
The gain ec o Lois a designe ’s ool, which de ines he
bandwid h o he obse e 𝜔oand delimi s he equency ange
o he dis u bances o be conside ed and mus be selec ed
o ensu e ha Ae−LoCis asymp o ically s able. The alues
z1( ),…,zp( ) es ima e y( ), y( ), and i s de i a i es, and zp+1( )
es ima es he gene alized dis u bance ( ). This es ima ed dis-
u bance is he key con ol elemen in ADRC con ol, since i
can be used o o ce he sys em o ollow he desi ed dynamics,
while he ex e nal dis u bance is minimized, when he s abili y
is main ained.
In his wo k, he e e ence acking is no conside ed nec-
essa y, since he main objec i e is he dis u bance educ ion.
Fo his eason, he use o a acking di e en ia o (TD) is no
conside ed he eina e , simpli ying he discussion. Unde hese
condi ions, he con ol can be de ined:
u( )= −K′
o[z1( ),…,zp( )]T−zp+1( )
b
whe e K′
ois he con olle gain o he p h-o de in eg al plan
and he e e ence signal. This second gain ec o is ano he
designe ’s ool, which de ines he con olle bandwid h 𝜔c,[18].
De ining Ko=[K′
o,1], he LADRC can be desc ibed:
⎧
⎪
⎪
⎨
⎪
⎪
⎩
z( )=Aez( )+Beu( )+Lo(y( )−Cez( ))
=(Ae−LoCe)z( )+Beu( )+Loy( )
u( )= −Ko[z1( ),…,zp+1( )]T
b
(5)
In his classical desc ip ion, he LADRC is a “gene al” con-
ol s uc u e which is independen o he o iginal plan model,
excep o he ela i e o de po he model and he high-
equency gain b. Mo eo e , a LADRC can be uned wi h wo
pa ame e s (𝜔cand 𝜔o), and, hus, is easy o unde s and by p ac-
ical con ol enginee s [17]. The scheme o his sys em can be
obse ed in Figu e 1, including he plan an inpu ime delay.
The designe does no need o know he de ailed s uc u e and
he pa ame e s o he model, so i is qui e simila o PID con-
ol which has a ixed con ol s uc u e ha is independen o
he plan models.
2.1 Al e na i e sys em desc ip ion
To acili a e he analysis, he sys em shown in Figu e 1can be
es uc u ed as in Figu e 2 ollowing a simila way o he one
p oposed in [11], and a gene alized ESO Heso can be de ined
[19].
F om his scheme and (5), his gene alized ESO (GESO) can
be desc ibed by he ollowing s a e-space ep esen a ion:
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
z( )=Aeˆz( )+Bo[ ( )
y( )]
yo( )=Koˆz( )
Ao=Ae−LoCe−Ko
bBe
Bo=[Be
bLo]
(6)
whe e Kode ines he desi ed dynamics, which is now included
in he GESO desc ip ion. No e ha (6) is equi alen o (5).
Following a simila desc ip ion o he s a e-space ealiza ion
(6) using he Laplace T ans o m p esen ed in [19], he GESO
can be desc ibed as ollows:
ˆ
Z(s)=Aoˆ
Z(s)+Be
bR(s)+LoY(s)
Yo(s)=Koˆ
Z(7)
Dele ing he in e media e a iable
Z(s), he GESO ou pu
Yo(s)is:
Yo(s)=Ko(sI(p+1)×(p+1)−Ao)−1Be
bR
+Ko(sI(p+1)×(p+1)−Ao)−1
LoY(s)
(8)
which can be ew i en by a wo-deg ee-o - eedom con en-
ional eedback s uc u e as shown in Figu e 3:
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JUGO ET AL.1995
FIGURE 2 Al e na i e desc ip ion o he classical LADRC s uc u e, which eases he sys em analysis.
FIGURE 3 Equi alen sys em desc ip ion o he classical LADRC using
ans e unc ions.
Yo(s)=Heso (s)Y(s)+H
eso (s)R(s)
Heso =Ko(sI(p+1)×(p+1)−Ao)−1
Lo
H
eso =Ko(sI(p+1)×(p+1)−Ao)−1Be
b
(9)
No e ha he denomina o o Heso(s)andH
eso(s) ans e
unc ions is he same.
This desc ip ion acili a es he s abili y analysis o he
LADRC sys em. Using he ans e unc ions in (9), he closed
loop cha ac e is ic equa ion de i ed om Figu e 3can be
w i en:
1+e−𝜏sKHeso (s)P(s)=0 (10)
The oo s o (10) a e he poles o he closed loop ans e
unc ion and gi es he s abili y o he sys em which can be s ud-
ied as unc ion o he di ec -loop gain K, o a pa icula ime
delay τ. No e ha K=1
bis he nominal alue ha sa is ies he
ma ching condi ion, since his desc ip ion is alid o any ime
delay, including 𝜏=0.
When τis no 0, he analy ical solu ion o equa ion (10)is
no easy, since he numbe o oo s is in ini y. Howe e , e-
quency domain me hods, such as he Nyquis c i e ion, a e alid
in his case [11, 20]. By ob aining he Nyquis diag am (o Bode
diag am o open loop s able sys ems) om he exp ession
e−𝜏sKHeso(s)P(s), he s abili y o he sys em can be analysed by
he Nyquis c i e ion and he delay s abili y ma gin can be easily
ob ained.
As a s aigh o wa d esul , i he o iginal sys em is open-loop
s able, o any delay 𝜏a su icien ly low Kmakes he closed-
loop sys em s able [20]. Howe e , i his K alue is lowe han
1/b he ma ching condi ion is no ul illed, and he dis u bance
educ ion is diminished.
The equi alen desc ip ion o Figu e 3isonly alid o anal-
ysis and no as an al e na i e implemen a ion scheme since
nume ical issues may a ise.
Now, conside he nex exp ession:
Heso (s)=Hc
eso (s)+H
eso (s)(11)
being
Hc
eso (s)=[K′
o,0](sI(p+1)×(p+1)−Ao)−1
Lo
H
eso (s)=[0,…,0,1](sI(p+1)×(p+1)−Ao)−1
Lo
(12)
whe e H
eso de ines he eedback loop dependen on ( ), ha
is, he ex ended s a e which es ima es he sys em dynamic and
he ex e nal dis u bances, and Hc
eso depends on he es o he
s a e ec o . No e ha he denomina o o he ans e unc-
ions Hc
eso(s)andH
eso(s) a e equal and ha e a pole in he o igin,
since he ma ching condi ion is sa is ied by de ini ion o ma ix
Ao. The use o hese ans e unc ions mus be done ca e-
ully o a oid nume ical p oblems, since bo h ha e he same
denomina o .
Using he ela ion (11), he eedback loop in Figu e 3can be
sepa a ed in o wo loops, which can be analysed independen ly.
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1996 JUGO ET AL.
FIGURE 4 Open loop Bode diag ams wi h 𝜔c=5 ad/s and τ=0.005 s, o di e en alues o he obse e bandwid h 𝜔o.Le :e−𝜏sH
esoP,Righ :e−𝜏sHc
esoP.
The phase inc eases linea ly wi h τ.
To illus a e he e ec o each loop, an example wi h wo
mechanical esonances is conside ed:
P(s)=1.4
(s+1)+g0+g1
g0=𝜔2
0
(s2+2𝛿0𝜔0s+𝜔2
0)
g1=𝜔2
1
(s2+2𝛿1𝜔1s+𝜔2
1)
(13)
wi h 𝜔0=10 ad/s, 𝛿0=0.1, 𝜔1=40 ad/s, 𝛿1=0.05 and a
cons an delay 𝜏=0.005 s. Those alues ha e been chosen o
de ining a sys em wi h low ela i e s abili y.
Fo his sys em, se e al LADRC con olle s ( i s o de )
ha e been de ined om (6), using di e en alues o he ESO
obse e bandwid h 𝜔o( wo poles in −𝜔o) o ob aining Lo,
and o he con olle bandwid h 𝜔c(one pole in −𝜔c) o
compu ing Ko. Fi s , he con olle bandwid h 𝜔c=5 ad/s is
main ained cons an and di e en alues o he ESO obse e
bandwid h 𝜔oa e used, ob aining he open-loop ans e unc-
ions e−𝜏sKH
eso(s)P(s)ande−𝜏sKH c
eso(s)P(s) using (12), o he
analysis o i s in luence on he sys em s abili y. Figu e 4shows
he Bode diag ams o bo h ans e unc ions wi h di e en
alues o 𝜔o.
Simila ly, se e al LADRC con olle s (again i s o de ) has
been de ined keeping cons an 𝜔o=1000 ad/s o di e -
en alues o he LADRC con ol bandwid h 𝜔c. Figu e 5
shows he Bode diag ams o he open-loop ans e unc ions
e−𝜏sKH
eso(s)P(s)ande−𝜏sKH c
eso(s)P(s), o such sys ems.
Obse ing Figu es 4and 5, he e ec o educing o inc eas-
ing he obse e bandwid h is closely ela ed o he change o
he gain alue in he dis u bance es ima ion loop (H
eso(s)P(s))),
ha is, K.
O he ele an conclusion, obse ed in all he igu es, is
he e iden e ec o he delay 𝜏=0.005 s on he s abili y
o he LADRC sys em. This e ec is especially impo an o
he loop wi h eedback o he es ima ed dis u bance. The high
gain esponsible o he dis u bance educ ion makes he sys-
em uns able wi h ela i ely low ime delay, since he delay
inc eases he phase lag p opo ionally wi h he equency. In
his case, ollowing he Nyquis c i e ion o s able open-loop
sys ems, he sys em becomes uns able i he sys em ampli ude
is highe han 0 db a equencies wi h a phase lag highe
han −180 deg ees. The igu es show ha he men ioned phase
limi is always gained due o he delay e ec and, hen, all
cases ha e a maximum sys em gain ha gua an ees he s abili y.
This maximum allowable gain limi s he capaci y o dis o ion
educ ion.
This example shows ha , compa ing bo h eedback loops,
he dis u bance es ima ion loop (H
eso(s)P(s)) is he dominan
one, ha is, i s open-loop gain is much la ge and can be
conside ed he main con ol e ec o he LADRC scheme.
F om he p e ious analysis, i is clea ha he exp ession (10)
acili a es he s abili y analysis o he LADRC con olle , espe-
cially in he p esence o ime delay. Taking in o accoun hese
esul s and wi h he aim o inc easing he design lexibili y, a
new scheme based on he LADRC con olle is p oposed in he
nex sec ion.
3MODIFIED LADRC CONTROLLER
The o iginal design p ocedu e o an ADRC con olle is based
in wo s eps. Fi s , he obse e gains Loa e selec ed conside ing
he canonical sys em (4) and he desi ed es ima ion bandwid h
𝜔oand, second, a con olle gain K′
ois selec ed o de ine he
desi ed con ol bandwid h 𝜔c. The con olle and es ima ion
bandwid h a e selec ed o no in e e e wi h each o he .
In his sec ion, a modi ied LADRC (MLADRC) con olle ,
which ollows o he design p ocedu e, is p esen ed. This
MLADRC is based on a no el GESO scheme, which has some
simila i ies wi h he p oposed one in [20], and i has been
de i ed om he p e ious discussion and obse ing he ans e
unc ions (9).
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JUGO ET AL.1997
FIGURE 5 Open loop Bode diag ams wi h 𝜔0=1000 ad/s and τ=0.005 s, o di e en alues o he con ol bandwid h 𝜔c.Le :e−𝜏sH
esoP,Righ :
e−𝜏sHc
esoP. The phase inc eases linea ly wi h τ.
Fi s , he new GESO s uc u e is p esen ed, which can be
desc ibed by he ollowing s a e space ep esen a ion:
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
z( )=Aoˆz( )+Bo[ ( )
y( )]
yo( )=Koˆz( )
Ao=Ae−LoCe−Ko
bBe
Bo=[Be
bLo]
(14)
being Ko=[K′
o,1], z=[z1,z2⋯, zp,zp+1]Tand
Ae=⎡⎢⎢⎢⎢⎢⎢⎢⎣
010…0
001…0
⋮⋮⋮⋱⋮
000…1
000…0
⎤⎥⎥⎥⎥⎥⎥⎥⎦(p+1)×(p+1)
(15)
Be=[00⋯b0]T
(p+1)×1
Ce=[100⋯0]T
(p+1)×1
The design pa ame e s a e selec ed as ollows:
∙The con olle gains K′
oa e calcula ed o ge he desi ed
closed-loop dynamics by sol ing he pole placemen p oblem
de ined by 𝜔c. The ec o o desi ed poles o he con olle is
[𝜔c,𝜔
c,⋯,𝜔
c]1xp . Using his gain ec o , an auxilia y ESO
ma ix is de ined, Ae−[K′
o,0]
bB.
∙Using he auxilia y ma ix, he obse e is designed o a cho-
sen obse e bandwid h, 𝜔o. The ec o o desi ed poles o
he obse e is [𝜔o,𝜔
o,…,𝜔
o]1x(p+1) . Wi h a alid Lo ec-
o , he esul ing GESO dynamics will be s able since he
FIGURE 6 Modi ied LADRC con ol s uc u e, based on he eedback o
he gene alized dis u bance.
in e ac ion be ween he con olle and obse e bandwid h
is included in he obse e design.
∙De ine he de ini i e GESO ma ix Ao=Ae−LoCe−Ko
bBe.
This ma ix assu es he ul ilmen o he ma ching condi ion
in he GESO.
The GESO, as shown in Figu e 6, uses as inpu s he e e ence
( ) and he ou pu o he plan o be con olled. One o he
main no el ies i his GESO is he di ec in oduc ion o he
e ec o he dis u bance es ima ion eedback in ma ix Ao,as
can be obse ed in he de ini ion o Aoin Equa ion (15). This
change isola es he ma ching condi ion wi h espec o he di ec
loop gain (o iginally 1/b). So, by changing he di ec loop gain,
he GESO main ains he ma ching condi ion.
In addi ion, his scheme allows he use o wo possible
al e na i es by de ining Co=Koo Co=[0,…,1]:
∙In he i s case, he ull eedback loop (Heso) is used, simila ly
o he ypical LADRC scheme.
∙In he second case, he eedback is ob ained only by means
o es ima ing he o al dis u bance ( )(H
eso). This is a new
p oposal, bu aking in o accoun he na u e o he o al dis-
u bance and he analysis o he example p esen ed in he
p e ious sec ion, he expec ed dynamics ob ained om he
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1998 JUGO ET AL.
FIGURE 7 Compa ison o he dis u bance educ ion using o iginal
LADRC scheme (blue line) and he p oposed MLADRC scheme (o ange line).
The g een line shows he dis u bance e ec a open loop. The open loop gain
has been no malized o one, o a be e compa ison.
new scheme is no e y di e en om he o iginal scheme.
The ob ained ad an age is a simple con olle s uc u e,
which can be mo e easily analysed.
In bo h cases, one o he main ea u es o he p oposed
GESO is he easy s abili y analysis as unc ion o he di ec loop
gain. This is discussed in he nex subsec ion.
Using his no el GESO, a Modi ied LADRC (MLADRC)
s uc u e is p oposed, shown in Figu e 6. The p oposed scheme
includes an addi ional con ol elemen , subs i u ing he gain
K=1
b. Then,
e( )= ( )−yo( )
u( )=e ( )(16)
whe e e ( ) is he ou pu o he ans e unc ion C(s), ha is,
E (s)=C(s)E(s). This elemen adds lexibili y o he designe ,
which is alid o inc easing he delay s abili y ma gin o he
esul ing sys em. On he o he hand, he s abili y analysis
MLADRC (and C(s) design) can be easily pe o med in he e-
quency domain, since i only depends on a scala signal and is
applicable wi h he p esence o ime delay.
To illus a e he good beha iou o he new MLADRC using
Co=[0,…,1], i has been applied o he sys em (13)wi h𝜏=
0.0001 s and using C(s)=1
b,b=1.4. The MLADRC and
he o iginal LADRC ha e same design pa ame e s: i s o de ,
𝜔o=1000 ad/s and 𝜔c=5 ad/s. Figu e 7shows he esul s
compa ing bo h LADRC and MLADRC wi h he open-loop
esponse using a s ep signal as e e ence and whi e noise as inpu
dis u bance. The open-loop gain has been no malized o one
o be e compa ison. I can be obse ed ha he dis u bance
ejec ion is e y good and e y simila using bo h schemes. In
his example, he delay 𝜏has been selec ed low o main ain he
sys em s able wi hou he necessi y o a mo e complex C(s), he
sys em being s able by using he o iginal LADRC.
FIGURE 8 Delay s abili y ma gin, in seconds, depending o he alue o
he gain K o he sys em (20).
3.1 S abili y analysis o he modi ied
LADRC
Based on he ma hema ical desc ip ion o he MLADRC (14),
(15), he GESO ou pu can be exp essed:
Yo(s)=Heso (s)Y(s)+H
eso (s)R(s)
Heso =Ko(sI(p+1)×(p+1)−Ao)−1
Lo
H
eso =Ko(sI(p+1)×(p+1)−Ao)−1Be
b
(17)
Now, closing he loop wi h a plan P(s) in he p esence o
ime-delay, he cha ac e is ic equa ion o he MLADRC sys em
is:
1+e−𝜏sC(s)Heso (s)P(s)=0 (18)
wi h Heso(s)=Co(sI −Ao)−1
oLo.
Using Equa ion (18), he s abili y analysis as a unc ion o
he loop gain can be easily pe o med using he Nyquis c i e-
ion. This analysis is especially s aigh o wa d when he plan is
minimum phase, [21], since he sys em is s able o a gain ha
sa is ies K<Kmin; ha is, i he open loop plan is minimum
phase, he sys em is closed-loop s able o a su icien ly low gain
alue.
I is impo an o ema k ha , in his case, he gain o C(s)is
di ec ly ela ed o he educ ion o he dis u bances and o gain
alues lowe han he alue o he ma ching condi ion 1/b, he
dis u bance educ ion is limi ed.
As an example, he analysis o he sys em (13) can be used
o illus a e he p ocedu e using a i s -o de MLADRC wi h
𝜔o=1000 ad/s and 𝜔c=5 ad/s. Figu e 8shows he delay
s abili y ma gin o his sys em choosing C(s)=K, as a unc-
ion o he gain K, ob ained om (18). This con igu a ion is
equi alen o a classical LADRC con olle . As is obse ed, by
educing he alue o he eedback gain, he s abili y ma gin
imp o es, bu a he cos o losing dis u bance educ ion. This
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JUGO ET AL.1999
FIGURE 9 Example showing he dis u bance esponse o (18) wi h a ime delay 𝜏=0.001 s, using C(s)=K.AlowK alue (0.002) is used o main aining he
s abili y, ob aining a limi ed dis u bance educ ion.
ac can be obse ed in Figu e 9. In his case, being he ime
delay 𝜏=0.001 s, he gain K=0.002 is chosen om Figu e 8
o main aining he closed loop s abili y, and a poo dis u bance
educ ion is ob ained.
3.2 Loop shaping design
As he p e ious example illus a es, gain educ ion is a limi ed
design s a egy, which may be use ul o some sys ems and wi h
small ime-delays. Howe e , he MLADRC p esen ed opens he
doo o imp o ing he s abili y by designing C(s), allowing an
inc emen o he loop gain and, as a esul , he dis u bance
educ ion. This is one o he main no el ies in oduced by
MLADRC. Fo ins ance, he di ec gain Kcan be subs i u ed by
a phase compensa o , which can be used o shaping he sys em
equency esponse.
Loop shaping is a common design me hodology which, in
ac , can be done ollowing di e en s a egies o ob aining
a desi ed equency esponse shape [22]. The con olle s can
be PIDs, lead o lag phase compensa o s, no ch il e s, among
o he s.
In he case o ime delayed sys ems, loop shaping can be
applied o compensa e he equency esponse in he high gain,
high phase lag a eas. Two possible ac ions can be in oduced:
gain educ ion o in oduc ion o a lead phase in he c i ical e-
quency ange. In he case o esonances, lead compensa o s can
be used o imp o e he phase and no ch il e s o educe he sys-
em gain in such sensi i e equency anges. A high equencies,
a gain- educing il e can be necessa y, as he lag in oduced by
he delay may be excessi e o be able o compensa e.
The example (13) p esen ed in he p e ious sec ions can
be used again o illus a e he p ocedu e, using he same i s -
o de MLADRC wi h 𝜔o=1000 ad/s 𝜔c=5 ad/s and he
ime delay 𝜏=0.001 s. F om Figu e 8, he alue o he gain
should be K<2×10−3 o main ain he sys em s abili y. How-
e e , he dis o ion educ ion is e y poo wi h such a low gain
(see Figu e 9).
To inc ease he gain o a alue ha gi es good dis u bance
ejec ion, o ins ance, K=0.3, a lead compensa o is in o-
duced as a di ec loop con olle C(s). F om he Bode diag am
o he open-loop sys em wi h K=0.3 (Figu e 10), he compen-
sa o pa ame e s a e designed o s abilize he sys em, ob aining
he ollowing ans e unc ion:
C(s)=1+bTbs
1+Tbs
wi h b=5.8284 and Tb=9.204710−4. This con olle inc eases
he phase a ound he c i ical equency o 55 ad/s. Figu e 10
shows he Bode diag am o he o iginal open-loop sys em, wi h
and wi hou delay and he e ec o he designed compensa o ,
s abilizing he sys em. The phase o he MLADRC, he g een
line, inc eases om 40 ad/s o 1000 ad/s app oxima ely,
hanks o he con olle designed C(s), compa ed wi h he
MLADRC con olle using only a cons an K. This inc emen
compensa es he e ec o he delay, inc easing he s abili y
ma gin.
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2000 JUGO ET AL.
FIGURE 10 Open-loop equency esponse compa ison, showing he loop shaping e ec o he compensa o , inc easing he phase a ound he c i ical
equency: ed line, sys em wi hou delay; blue line, sys em wi h delay; g een line, sys em wi h delay and compensa o .
FIGURE 11 Dis u bance educ ion using he modi ied LADRC wi h
K=0.3 and a lead compensa o o ime delay 𝜏=0.001 s.
Figu e 11 shows he dis u bance ejec ion ob ained using
he gain alue K=0.3, hanks o he inc eased s abili y ma gin
ob ained by he compensa o . No e ha he inpu noise is
equi alen o he one used in he Figu e 9, being he ob ained
dis u bance wo o de s o magni ude lowe . The compensa ed
sys em is s able o 𝜏<0.0038 s. To compa e pe o mance,
using he scheme p oposed in [12] and he gain K=0.3, he
sys em becomes uns able o 𝜏>0.00022 s, ha is, he s abili y
ma gin is one o de o magni ude lowe .
4EXPERIMENTAL VALIDATION
In his sec ion, he a o emen ioned app oach is implemen ed in
a eal mechanical sys em wi h ele an esonan modes o anal-
yse he easibili y o he design p ocess and he pe o mance o
he esul ing con olle .
4.1 Sys em desc ip ion
The mechanical sys em used o analyse he s abiliza ion p ocess
mus mee ce ain cha ac e is ics ha a e impo an o p ope
es ing o he con ol algo i hm and i s abili y o ejec dis u -
bances. This cha ac e is ic is a high sensi i i y o mechanical
ib a ions. This is an added di icul y o con ol he sys em and
makes a pe ec scena io o es ing he algo i hm, because his
ype o sys ems is e y sensi i e o ime-delay, which can become
uns able. In his way, he mechanical sys em selec ed o his
es was a passi e lexible s uc u e moun ed on a single-axis
seismic able o he s udy o ac i e mass dampe s, comme cial-
ized by Quanse [23], and shown in Figu e 12. The s uc u e
has a capaci i e accele ome e on i s op in o de o measu e
he ib a ion o he sys em and is con olled by he linea mo e-
men o he shaking able. A he same ime, he shaking able
is con olled by a high- o que mo o connec ed o i ia a
ack-and-pinion sys em. The mo o also has a high- esolu ion
op ical encode wi h which he posi ion o he shaking able
is measu ed. The goal o his implemen a ion is o con ol
he posi ion o he shaking able and he ib a ion (accele a-
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