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A ailable online a www.sciencedi ec .com
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Pee e iew unde esponsibili y o In e na ional Fede a ion o Au oma ic Con ol.
10.1016/j.i acol.2020.12.1738
10.1016/j.i acol.2020.12.1738 2405-8963
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2020 The Au ho s. This is an open access a icle unde he CC BY-NC-ND license
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)
Model-F ee Sliding-Mode Con olle o
So Landing o Reluc ance Ac ua o s
Edua do Moya-Lashe as ∗Edga Rami ez-Labo eo ∗
Ca los Sagues ∗
∗Depa amen o de In o ma ica e Ingenie ia de Sis emas (DIIS) and
Ins i u o de In es igacion en Ingenie ia de A agon (I3A),
Uni e sidad de Za agoza, Za agoza 50018, Spain,
(e-mail: emoya@uniza .es, ami lab@uniza .es, csagues@uniza .es)
Abs ac : Some elec omagne ic ac ua o s su e om high eloci y impac s du ing non-
con olled swi ching ope a ions, which cause con ac bouncing, mechanical wea , and acous ic
noise. So -landing con ol s a egies aim a minimizing he impac eloci ies o hese de ices
o imp o e hei pe o mance. This pape p esen s a sliding-mode con olle o so landing
o single-coil eluc ance ac ua o s. I is a swi ching model- ee con olle , which esul s in a
e y simple implemen a ion. A gene alized dynamical hyb id model o an ac ua o is u ilized
o de i ing he obus ness condi ion, based on he Lyapuno heo y. Then, he condi ion
is e alua ed o a dynamical model, based on a comme cial de ice, and se e al e e ence
ajec o ies. Finally, he con olle pe o mance is alida ed h ough simula ions. The e ec
o he sampling a e on he esul ing impac eloci ies is also analyzed.
Keywo ds: Ac ua o s, Elec omagne ic de ices, Modeling, Nonlinea con ol, Robus con ol,
Sliding-mode con ol, T acking
1. INTRODUCTION
A eluc ance ac ua o is a ype o nonlinea elec ome-
chanical de ice which gene a es a eluc ance-based mag-
ne ic o ce o mo e i s a ma u e. Pa icula ly, single-coil
ac ua o s a e used in an ex ensi e a ie y o indus ial
applica ions because o hei as esponse, compac ness,
high ene gy e iciency, and low cos . Thus, he e is a
g ea esea ch in e es conce ning modeling, iden i ica ion,
es ima ion, and con ol o his class o ac ua o s.
Rega ding he con ol, one o he main mo i a ions and
challenges is o achie e so landing du ing swi ching op-
e a ions; hus educing con ac bouncing, impac noise and
mechanical wea . In he li e a u e, he e a e se e al con-
ol p oposals o eluc ance ac ua o s, e.g., based on he
backs epping echnique (Kah eci and Kolmano sky, 2010),
ene gy compensa ion (Yang e al., 2013), a linea iza ion
me hod (Ka alenic e al., 2016), o cycle- o-cycle adap a-
ion (Moya-Lashe as e al., 2019), among o he s.
One impo an d awback o many mass-ma ke single-
coil eluc ance ac ua o s is he manu ac u ing a iabili y
among de ices om he same ensemble. Mo eo e , he
iden i ica ion o e e y uni may impose a p ohibi i e cos .
One majo app oach o deal wi h model unce ain ies is
he sliding-mode con ol (SMC) heo y (Slo ine and Li,
1991). The e a e se e al wo ks ha ake his app oach.
Mos commonly, he con ol law is di ided in o wo e ms:
This wo k was pa ially suppo ed by he A ag´on Regional
Go e nmen , he Spanish Go e nmen , and he Eu opean Union,
unde p ojec RTC-2017-5965-6, p ojec PGC2018-098719-B-I00
(MCIU/ AEI/FEDER, UE), esea ch g oup DGA-T45 17R, schol-
a ship FPU14/04171, and p og am FSE A ag´on 2014-2020.
an equi alen and a swi ching con ol e m (Lee e al.,
2015; Zhao e al., 2016). Al e na i ely, Eyabi and Wash-
ing on (2006) p oposed a SMC wi h only a swi ching
e m, which is hen app oxima ed o a p opo ional one.
One impo an aspec ha is omi ed in hese wo ks is
he de ini ion o he acking ajec o y, which di ec ly
a ec s he obus ness condi ions o he SMC. Ano he
impo an issue is he in luence o he sampling a e. In
gene al, he sliding accu acy is p opo ional o he squa e
o he swi ching delay (Le an , 1993). S ill, i s e ec on
he esul ing impac eloci ies needs o be e alua ed.
This pape p esen s a obus SMC con olle o single-
coil eluc ance ac ua o s. I is pu ely a swi ching con-
olle , which esul s in a e y simple and compu a ionally
inexpensi e implemen a ion. Al hough he esul ing con-
olle is model- ee—i.e. i does no depend on any model
unc ions o pa ame e s—a dynamic model is equi ed
du ing he design p ocess o gua an ee i s obus ness. The
gene alized sys em, which p esen s bo h con inuous and
disc e e dynamic beha io , is modeled wi h a hyb id au-
oma on. A obus ness condi ion is de i ed, which depends
on he sys em dynamics and he posi ion ajec o y. I is
hen e alua ed o a speci ic dynamic model, based on a
comme cial solenoid al e, and se e al ajec o ies. The
i s con ibu ion o he pape is he p oposal o a swi ching
model- ee SMC, which wo ks o e e y disc e e mode o
he sys em. The second con ibu ion is he analysis o he
in luence o he sampling a e on he impac eloci ies.
2. SYSTEM DYNAMICS
A gene al single-coil eluc ance ac ua o is ep esen ed in
Fig. 1. The magne ic co e is di ided in o wo pa s: a ixed
Model-F ee Sliding-Mode Con olle o
So Landing o Reluc ance Ac ua o s
Edua do Moya-Lashe as ∗Edga Rami ez-Labo eo ∗
Ca los Sagues ∗
∗Depa amen o de In o ma ica e Ingenie ia de Sis emas (DIIS) and
Ins i u o de In es igacion en Ingenie ia de A agon (I3A),
Uni e sidad de Za agoza, Za agoza 50018, Spain,
(e-mail: emoya@uniza .es, ami lab@uniza .es, csagues@uniza .es)
Abs ac : Some elec omagne ic ac ua o s su e om high eloci y impac s du ing non-
con olled swi ching ope a ions, which cause con ac bouncing, mechanical wea , and acous ic
noise. So -landing con ol s a egies aim a minimizing he impac eloci ies o hese de ices
o imp o e hei pe o mance. This pape p esen s a sliding-mode con olle o so landing
o single-coil eluc ance ac ua o s. I is a swi ching model- ee con olle , which esul s in a
e y simple implemen a ion. A gene alized dynamical hyb id model o an ac ua o is u ilized
o de i ing he obus ness condi ion, based on he Lyapuno heo y. Then, he condi ion
is e alua ed o a dynamical model, based on a comme cial de ice, and se e al e e ence
ajec o ies. Finally, he con olle pe o mance is alida ed h ough simula ions. The e ec
o he sampling a e on he esul ing impac eloci ies is also analyzed.
Keywo ds: Ac ua o s, Elec omagne ic de ices, Modeling, Nonlinea con ol, Robus con ol,
Sliding-mode con ol, T acking
1. INTRODUCTION
A eluc ance ac ua o is a ype o nonlinea elec ome-
chanical de ice which gene a es a eluc ance-based mag-
ne ic o ce o mo e i s a ma u e. Pa icula ly, single-coil
ac ua o s a e used in an ex ensi e a ie y o indus ial
applica ions because o hei as esponse, compac ness,
high ene gy e iciency, and low cos . Thus, he e is a
g ea esea ch in e es conce ning modeling, iden i ica ion,
es ima ion, and con ol o his class o ac ua o s.
Rega ding he con ol, one o he main mo i a ions and
challenges is o achie e so landing du ing swi ching op-
e a ions; hus educing con ac bouncing, impac noise and
mechanical wea . In he li e a u e, he e a e se e al con-
ol p oposals o eluc ance ac ua o s, e.g., based on he
backs epping echnique (Kah eci and Kolmano sky, 2010),
ene gy compensa ion (Yang e al., 2013), a linea iza ion
me hod (Ka alenic e al., 2016), o cycle- o-cycle adap a-
ion (Moya-Lashe as e al., 2019), among o he s.
One impo an d awback o many mass-ma ke single-
coil eluc ance ac ua o s is he manu ac u ing a iabili y
among de ices om he same ensemble. Mo eo e , he
iden i ica ion o e e y uni may impose a p ohibi i e cos .
One majo app oach o deal wi h model unce ain ies is
he sliding-mode con ol (SMC) heo y (Slo ine and Li,
1991). The e a e se e al wo ks ha ake his app oach.
Mos commonly, he con ol law is di ided in o wo e ms:
This wo k was pa ially suppo ed by he A ag´on Regional
Go e nmen , he Spanish Go e nmen , and he Eu opean Union,
unde p ojec RTC-2017-5965-6, p ojec PGC2018-098719-B-I00
(MCIU/ AEI/FEDER, UE), esea ch g oup DGA-T45 17R, schol-
a ship FPU14/04171, and p og am FSE A ag´on 2014-2020.
an equi alen and a swi ching con ol e m (Lee e al.,
2015; Zhao e al., 2016). Al e na i ely, Eyabi and Wash-
ing on (2006) p oposed a SMC wi h only a swi ching
e m, which is hen app oxima ed o a p opo ional one.
One impo an aspec ha is omi ed in hese wo ks is
he de ini ion o he acking ajec o y, which di ec ly
a ec s he obus ness condi ions o he SMC. Ano he
impo an issue is he in luence o he sampling a e. In
gene al, he sliding accu acy is p opo ional o he squa e
o he swi ching delay (Le an , 1993). S ill, i s e ec on
he esul ing impac eloci ies needs o be e alua ed.
This pape p esen s a obus SMC con olle o single-
coil eluc ance ac ua o s. I is pu ely a swi ching con-
olle , which esul s in a e y simple and compu a ionally
inexpensi e implemen a ion. Al hough he esul ing con-
olle is model- ee—i.e. i does no depend on any model
unc ions o pa ame e s—a dynamic model is equi ed
du ing he design p ocess o gua an ee i s obus ness. The
gene alized sys em, which p esen s bo h con inuous and
disc e e dynamic beha io , is modeled wi h a hyb id au-
oma on. A obus ness condi ion is de i ed, which depends
on he sys em dynamics and he posi ion ajec o y. I is
hen e alua ed o a speci ic dynamic model, based on a
comme cial solenoid al e, and se e al ajec o ies. The
i s con ibu ion o he pape is he p oposal o a swi ching
model- ee SMC, which wo ks o e e y disc e e mode o
he sys em. The second con ibu ion is he analysis o he
in luence o he sampling a e on he impac eloci ies.
2. SYSTEM DYNAMICS
A gene al single-coil eluc ance ac ua o is ep esen ed in
Fig. 1. The magne ic co e is di ided in o wo pa s: a ixed
Model-F ee Sliding-Mode Con olle o
So Landing o Reluc ance Ac ua o s
Edua do Moya-Lashe as ∗Edga Rami ez-Labo eo ∗
Ca los Sagues ∗
∗Depa amen o de In o ma ica e Ingenie ia de Sis emas (DIIS) and
Ins i u o de In es igacion en Ingenie ia de A agon (I3A),
Uni e sidad de Za agoza, Za agoza 50018, Spain,
(e-mail: emoya@uniza .es, ami lab@uniza .es, csagues@uniza .es)
Abs ac : Some elec omagne ic ac ua o s su e om high eloci y impac s du ing non-
con olled swi ching ope a ions, which cause con ac bouncing, mechanical wea , and acous ic
noise. So -landing con ol s a egies aim a minimizing he impac eloci ies o hese de ices
o imp o e hei pe o mance. This pape p esen s a sliding-mode con olle o so landing
o single-coil eluc ance ac ua o s. I is a swi ching model- ee con olle , which esul s in a
e y simple implemen a ion. A gene alized dynamical hyb id model o an ac ua o is u ilized
o de i ing he obus ness condi ion, based on he Lyapuno heo y. Then, he condi ion
is e alua ed o a dynamical model, based on a comme cial de ice, and se e al e e ence
ajec o ies. Finally, he con olle pe o mance is alida ed h ough simula ions. The e ec
o he sampling a e on he esul ing impac eloci ies is also analyzed.
Keywo ds: Ac ua o s, Elec omagne ic de ices, Modeling, Nonlinea con ol, Robus con ol,
Sliding-mode con ol, T acking
1. INTRODUCTION
A eluc ance ac ua o is a ype o nonlinea elec ome-
chanical de ice which gene a es a eluc ance-based mag-
ne ic o ce o mo e i s a ma u e. Pa icula ly, single-coil
ac ua o s a e used in an ex ensi e a ie y o indus ial
applica ions because o hei as esponse, compac ness,
high ene gy e iciency, and low cos . Thus, he e is a
g ea esea ch in e es conce ning modeling, iden i ica ion,
es ima ion, and con ol o his class o ac ua o s.
Rega ding he con ol, one o he main mo i a ions and
challenges is o achie e so landing du ing swi ching op-
e a ions; hus educing con ac bouncing, impac noise and
mechanical wea . In he li e a u e, he e a e se e al con-
ol p oposals o eluc ance ac ua o s, e.g., based on he
backs epping echnique (Kah eci and Kolmano sky, 2010),
ene gy compensa ion (Yang e al., 2013), a linea iza ion
me hod (Ka alenic e al., 2016), o cycle- o-cycle adap a-
ion (Moya-Lashe as e al., 2019), among o he s.
One impo an d awback o many mass-ma ke single-
coil eluc ance ac ua o s is he manu ac u ing a iabili y
among de ices om he same ensemble. Mo eo e , he
iden i ica ion o e e y uni may impose a p ohibi i e cos .
One majo app oach o deal wi h model unce ain ies is
he sliding-mode con ol (SMC) heo y (Slo ine and Li,
1991). The e a e se e al wo ks ha ake his app oach.
Mos commonly, he con ol law is di ided in o wo e ms:
This wo k was pa ially suppo ed by he A ag´on Regional
Go e nmen , he Spanish Go e nmen , and he Eu opean Union,
unde p ojec RTC-2017-5965-6, p ojec PGC2018-098719-B-I00
(MCIU/ AEI/FEDER, UE), esea ch g oup DGA-T45 17R, schol-
a ship FPU14/04171, and p og am FSE A ag´on 2014-2020.
an equi alen and a swi ching con ol e m (Lee e al.,
2015; Zhao e al., 2016). Al e na i ely, Eyabi and Wash-
ing on (2006) p oposed a SMC wi h only a swi ching
e m, which is hen app oxima ed o a p opo ional one.
One impo an aspec ha is omi ed in hese wo ks is
he de ini ion o he acking ajec o y, which di ec ly
a ec s he obus ness condi ions o he SMC. Ano he
impo an issue is he in luence o he sampling a e. In
gene al, he sliding accu acy is p opo ional o he squa e
o he swi ching delay (Le an , 1993). S ill, i s e ec on
he esul ing impac eloci ies needs o be e alua ed.
This pape p esen s a obus SMC con olle o single-
coil eluc ance ac ua o s. I is pu ely a swi ching con-
olle , which esul s in a e y simple and compu a ionally
inexpensi e implemen a ion. Al hough he esul ing con-
olle is model- ee—i.e. i does no depend on any model
unc ions o pa ame e s—a dynamic model is equi ed
du ing he design p ocess o gua an ee i s obus ness. The
gene alized sys em, which p esen s bo h con inuous and
disc e e dynamic beha io , is modeled wi h a hyb id au-
oma on. A obus ness condi ion is de i ed, which depends
on he sys em dynamics and he posi ion ajec o y. I is
hen e alua ed o a speci ic dynamic model, based on a
comme cial solenoid al e, and se e al ajec o ies. The
i s con ibu ion o he pape is he p oposal o a swi ching
model- ee SMC, which wo ks o e e y disc e e mode o
he sys em. The second con ibu ion is he analysis o he
in luence o he sampling a e on he impac eloci ies.
2. SYSTEM DYNAMICS
A gene al single-coil eluc ance ac ua o is ep esen ed in
Fig. 1. The magne ic co e is di ided in o wo pa s: a ixed
Model-F ee Sliding-Mode Con olle o
So Landing o Reluc ance Ac ua o s
Edua do Moya-Lashe as ∗Edga Rami ez-Labo eo ∗
Ca los Sagues ∗
∗
Depa amen o de In o ma ica e Ingenie ia de Sis emas (DIIS) and
Ins i u o de In es igacion en Ingenie ia de A agon (I3A),
Uni e sidad de Za agoza, Za agoza 50018, Spain,
(e-mail: [email p o ec ed], [email p o ec ed], [email p o ec ed])
Abs ac : Some elec omagne ic ac ua o s su e om high eloci y impac s du ing non-
con olled swi ching ope a ions, which cause con ac bouncing, mechanical wea , and acous ic
noise. So -landing con ol s a egies aim a minimizing he impac eloci ies o hese de ices
o imp o e hei pe o mance. This pape p esen s a sliding-mode con olle o so landing
o single-coil eluc ance ac ua o s. I is a swi ching model- ee con olle , which esul s in a
e y simple implemen a ion. A gene alized dynamical hyb id model o an ac ua o is u ilized
o de i ing he obus ness condi ion, based on he Lyapuno heo y. Then, he condi ion
is e alua ed o a dynamical model, based on a comme cial de ice, and se e al e e ence
ajec o ies. Finally, he con olle pe o mance is alida ed h ough simula ions. The e ec
o he sampling a e on he esul ing impac eloci ies is also analyzed.
Keywo ds: Ac ua o s, Elec omagne ic de ices, Modeling, Nonlinea con ol, Robus con ol,
Sliding-mode con ol, T acking
1. INTRODUCTION
A eluc ance ac ua o is a ype o nonlinea elec ome-
chanical de ice which gene a es a eluc ance-based mag-
ne ic o ce o mo e i s a ma u e. Pa icula ly, single-coil
ac ua o s a e used in an ex ensi e a ie y o indus ial
applica ions because o hei as esponse, compac ness,
high ene gy e iciency, and low cos . Thus, he e is a
g ea esea ch in e es conce ning modeling, iden i ica ion,
es ima ion, and con ol o his class o ac ua o s.
Rega ding he con ol, one o he main mo i a ions and
challenges is o achie e so landing du ing swi ching op-
e a ions; hus educing con ac bouncing, impac noise and
mechanical wea . In he li e a u e, he e a e se e al con-
ol p oposals o eluc ance ac ua o s, e.g., based on he
backs epping echnique (Kah eci and Kolmano sky, 2010),
ene gy compensa ion (Yang e al., 2013), a linea iza ion
me hod (Ka alenic e al., 2016), o cycle- o-cycle adap a-
ion (Moya-Lashe as e al., 2019), among o he s.
One impo an d awback o many mass-ma ke single-
coil eluc ance ac ua o s is he manu ac u ing a iabili y
among de ices om he same ensemble. Mo eo e , he
iden i ica ion o e e y uni may impose a p ohibi i e cos .
One majo app oach o deal wi h model unce ain ies is
he sliding-mode con ol (SMC) heo y (Slo ine and Li,
1991). The e a e se e al wo ks ha ake his app oach.
Mos commonly, he con ol law is di ided in o wo e ms:
This wo k was pa ially suppo ed by he A ag´on Regional
Go e nmen , he Spanish Go e nmen , and he Eu opean Union,
unde p ojec RTC-2017-5965-6, p ojec PGC2018-098719-B-I00
(MCIU/ AEI/FEDER, UE), esea ch g oup DGA-T45 17R, schol-
a ship FPU14/04171, and p og am FSE A ag´on 2014-2020.
an equi alen and a swi ching con ol e m (Lee e al.,
2015; Zhao e al., 2016). Al e na i ely, Eyabi and Wash-
ing on (2006) p oposed a SMC wi h only a swi ching
e m, which is hen app oxima ed o a p opo ional one.
One impo an aspec ha is omi ed in hese wo ks is
he de ini ion o he acking ajec o y, which di ec ly
a ec s he obus ness condi ions o he SMC. Ano he
impo an issue is he in luence o he sampling a e. In
gene al, he sliding accu acy is p opo ional o he squa e
o he swi ching delay (Le an , 1993). S ill, i s e ec on
he esul ing impac eloci ies needs o be e alua ed.
This pape p esen s a obus SMC con olle o single-
coil eluc ance ac ua o s. I is pu ely a swi ching con-
olle , which esul s in a e y simple and compu a ionally
inexpensi e implemen a ion. Al hough he esul ing con-
olle is model- ee—i.e. i does no depend on any model
unc ions o pa ame e s—a dynamic model is equi ed
du ing he design p ocess o gua an ee i s obus ness. The
gene alized sys em, which p esen s bo h con inuous and
disc e e dynamic beha io , is modeled wi h a hyb id au-
oma on. A obus ness condi ion is de i ed, which depends
on he sys em dynamics and he posi ion ajec o y. I is
hen e alua ed o a speci ic dynamic model, based on a
comme cial solenoid al e, and se e al ajec o ies. The
i s con ibu ion o he pape is he p oposal o a swi ching
model- ee SMC, which wo ks o e e y disc e e mode o
he sys em. The second con ibu ion is he analysis o he
in luence o he sampling a e on he impac eloci ies.
2. SYSTEM DYNAMICS
A gene al single-coil eluc ance ac ua o is ep esen ed in
Fig. 1. The magne ic co e is di ided in o wo pa s: a ixed
Model-F ee Sliding-Mode Con olle o
So Landing o Reluc ance Ac ua o s
Edua do Moya-Lashe as ∗Edga Rami ez-Labo eo ∗
Ca los Sagues ∗
∗Depa amen o de In o ma ica e Ingenie ia de Sis emas (DIIS) and
Ins i u o de In es igacion en Ingenie ia de A agon (I3A),
Uni e sidad de Za agoza, Za agoza 50018, Spain,
(e-mail: emoya@uniza .es, ami lab@uniza .es, csagues@uniza .es)
Abs ac : Some elec omagne ic ac ua o s su e om high eloci y impac s du ing non-
con olled swi ching ope a ions, which cause con ac bouncing, mechanical wea , and acous ic
noise. So -landing con ol s a egies aim a minimizing he impac eloci ies o hese de ices
o imp o e hei pe o mance. This pape p esen s a sliding-mode con olle o so landing
o single-coil eluc ance ac ua o s. I is a swi ching model- ee con olle , which esul s in a
e y simple implemen a ion. A gene alized dynamical hyb id model o an ac ua o is u ilized
o de i ing he obus ness condi ion, based on he Lyapuno heo y. Then, he condi ion
is e alua ed o a dynamical model, based on a comme cial de ice, and se e al e e ence
ajec o ies. Finally, he con olle pe o mance is alida ed h ough simula ions. The e ec
o he sampling a e on he esul ing impac eloci ies is also analyzed.
Keywo ds: Ac ua o s, Elec omagne ic de ices, Modeling, Nonlinea con ol, Robus con ol,
Sliding-mode con ol, T acking
1. INTRODUCTION
A eluc ance ac ua o is a ype o nonlinea elec ome-
chanical de ice which gene a es a eluc ance-based mag-
ne ic o ce o mo e i s a ma u e. Pa icula ly, single-coil
ac ua o s a e used in an ex ensi e a ie y o indus ial
applica ions because o hei as esponse, compac ness,
high ene gy e iciency, and low cos . Thus, he e is a
g ea esea ch in e es conce ning modeling, iden i ica ion,
es ima ion, and con ol o his class o ac ua o s.
Rega ding he con ol, one o he main mo i a ions and
challenges is o achie e so landing du ing swi ching op-
e a ions; hus educing con ac bouncing, impac noise and
mechanical wea . In he li e a u e, he e a e se e al con-
ol p oposals o eluc ance ac ua o s, e.g., based on he
backs epping echnique (Kah eci and Kolmano sky, 2010),
ene gy compensa ion (Yang e al., 2013), a linea iza ion
me hod (Ka alenic e al., 2016), o cycle- o-cycle adap a-
ion (Moya-Lashe as e al., 2019), among o he s.
One impo an d awback o many mass-ma ke single-
coil eluc ance ac ua o s is he manu ac u ing a iabili y
among de ices om he same ensemble. Mo eo e , he
iden i ica ion o e e y uni may impose a p ohibi i e cos .
One majo app oach o deal wi h model unce ain ies is
he sliding-mode con ol (SMC) heo y (Slo ine and Li,
1991). The e a e se e al wo ks ha ake his app oach.
Mos commonly, he con ol law is di ided in o wo e ms:
This wo k was pa ially suppo ed by he A ag´on Regional
Go e nmen , he Spanish Go e nmen , and he Eu opean Union,
unde p ojec RTC-2017-5965-6, p ojec PGC2018-098719-B-I00
(MCIU/ AEI/FEDER, UE), esea ch g oup DGA-T45 17R, schol-
a ship FPU14/04171, and p og am FSE A ag´on 2014-2020.
an equi alen and a swi ching con ol e m (Lee e al.,
2015; Zhao e al., 2016). Al e na i ely, Eyabi and Wash-
ing on (2006) p oposed a SMC wi h only a swi ching
e m, which is hen app oxima ed o a p opo ional one.
One impo an aspec ha is omi ed in hese wo ks is
he de ini ion o he acking ajec o y, which di ec ly
a ec s he obus ness condi ions o he SMC. Ano he
impo an issue is he in luence o he sampling a e. In
gene al, he sliding accu acy is p opo ional o he squa e
o he swi ching delay (Le an , 1993). S ill, i s e ec on
he esul ing impac eloci ies needs o be e alua ed.
This pape p esen s a obus SMC con olle o single-
coil eluc ance ac ua o s. I is pu ely a swi ching con-
olle , which esul s in a e y simple and compu a ionally
inexpensi e implemen a ion. Al hough he esul ing con-
olle is model- ee—i.e. i does no depend on any model
unc ions o pa ame e s—a dynamic model is equi ed
du ing he design p ocess o gua an ee i s obus ness. The
gene alized sys em, which p esen s bo h con inuous and
disc e e dynamic beha io , is modeled wi h a hyb id au-
oma on. A obus ness condi ion is de i ed, which depends
on he sys em dynamics and he posi ion ajec o y. I is
hen e alua ed o a speci ic dynamic model, based on a
comme cial solenoid al e, and se e al ajec o ies. The
i s con ibu ion o he pape is he p oposal o a swi ching
model- ee SMC, which wo ks o e e y disc e e mode o
he sys em. The second con ibu ion is he analysis o he
in luence o he sampling a e on he impac eloci ies.
2. SYSTEM DYNAMICS
A gene al single-coil eluc ance ac ua o is ep esen ed in
Fig. 1. The magne ic co e is di ided in o wo pa s: a ixed
Model-F ee Sliding-Mode Con olle o
So Landing o Reluc ance Ac ua o s
Edua do Moya-Lashe as ∗Edga Rami ez-Labo eo ∗
Ca los Sagues ∗
∗Depa amen o de In o ma ica e Ingenie ia de Sis emas (DIIS) and
Ins i u o de In es igacion en Ingenie ia de A agon (I3A),
Uni e sidad de Za agoza, Za agoza 50018, Spain,
(e-mail: emoya@uniza .es, ami lab@uniza .es, csagues@uniza .es)
Abs ac : Some elec omagne ic ac ua o s su e om high eloci y impac s du ing non-
con olled swi ching ope a ions, which cause con ac bouncing, mechanical wea , and acous ic
noise. So -landing con ol s a egies aim a minimizing he impac eloci ies o hese de ices
o imp o e hei pe o mance. This pape p esen s a sliding-mode con olle o so landing
o single-coil eluc ance ac ua o s. I is a swi ching model- ee con olle , which esul s in a
e y simple implemen a ion. A gene alized dynamical hyb id model o an ac ua o is u ilized
o de i ing he obus ness condi ion, based on he Lyapuno heo y. Then, he condi ion
is e alua ed o a dynamical model, based on a comme cial de ice, and se e al e e ence
ajec o ies. Finally, he con olle pe o mance is alida ed h ough simula ions. The e ec
o he sampling a e on he esul ing impac eloci ies is also analyzed.
Keywo ds: Ac ua o s, Elec omagne ic de ices, Modeling, Nonlinea con ol, Robus con ol,
Sliding-mode con ol, T acking
1. INTRODUCTION
A eluc ance ac ua o is a ype o nonlinea elec ome-
chanical de ice which gene a es a eluc ance-based mag-
ne ic o ce o mo e i s a ma u e. Pa icula ly, single-coil
ac ua o s a e used in an ex ensi e a ie y o indus ial
applica ions because o hei as esponse, compac ness,
high ene gy e iciency, and low cos . Thus, he e is a
g ea esea ch in e es conce ning modeling, iden i ica ion,
es ima ion, and con ol o his class o ac ua o s.
Rega ding he con ol, one o he main mo i a ions and
challenges is o achie e so landing du ing swi ching op-
e a ions; hus educing con ac bouncing, impac noise and
mechanical wea . In he li e a u e, he e a e se e al con-
ol p oposals o eluc ance ac ua o s, e.g., based on he
backs epping echnique (Kah eci and Kolmano sky, 2010),
ene gy compensa ion (Yang e al., 2013), a linea iza ion
me hod (Ka alenic e al., 2016), o cycle- o-cycle adap a-
ion (Moya-Lashe as e al., 2019), among o he s.
One impo an d awback o many mass-ma ke single-
coil eluc ance ac ua o s is he manu ac u ing a iabili y
among de ices om he same ensemble. Mo eo e , he
iden i ica ion o e e y uni may impose a p ohibi i e cos .
One majo app oach o deal wi h model unce ain ies is
he sliding-mode con ol (SMC) heo y (Slo ine and Li,
1991). The e a e se e al wo ks ha ake his app oach.
Mos commonly, he con ol law is di ided in o wo e ms:
This wo k was pa ially suppo ed by he A ag´on Regional
Go e nmen , he Spanish Go e nmen , and he Eu opean Union,
unde p ojec RTC-2017-5965-6, p ojec PGC2018-098719-B-I00
(MCIU/ AEI/FEDER, UE), esea ch g oup DGA-T45 17R, schol-
a ship FPU14/04171, and p og am FSE A ag´on 2014-2020.
an equi alen and a swi ching con ol e m (Lee e al.,
2015; Zhao e al., 2016). Al e na i ely, Eyabi and Wash-
ing on (2006) p oposed a SMC wi h only a swi ching
e m, which is hen app oxima ed o a p opo ional one.
One impo an aspec ha is omi ed in hese wo ks is
he de ini ion o he acking ajec o y, which di ec ly
a ec s he obus ness condi ions o he SMC. Ano he
impo an issue is he in luence o he sampling a e. In
gene al, he sliding accu acy is p opo ional o he squa e
o he swi ching delay (Le an , 1993). S ill, i s e ec on
he esul ing impac eloci ies needs o be e alua ed.
This pape p esen s a obus SMC con olle o single-
coil eluc ance ac ua o s. I is pu ely a swi ching con-
olle , which esul s in a e y simple and compu a ionally
inexpensi e implemen a ion. Al hough he esul ing con-
olle is model- ee—i.e. i does no depend on any model
unc ions o pa ame e s—a dynamic model is equi ed
du ing he design p ocess o gua an ee i s obus ness. The
gene alized sys em, which p esen s bo h con inuous and
disc e e dynamic beha io , is modeled wi h a hyb id au-
oma on. A obus ness condi ion is de i ed, which depends
on he sys em dynamics and he posi ion ajec o y. I is
hen e alua ed o a speci ic dynamic model, based on a
comme cial solenoid al e, and se e al ajec o ies. The
i s con ibu ion o he pape is he p oposal o a swi ching
model- ee SMC, which wo ks o e e y disc e e mode o
he sys em. The second con ibu ion is he analysis o he
in luence o he sampling a e on he impac eloci ies.
2. SYSTEM DYNAMICS
A gene al single-coil eluc ance ac ua o is ep esen ed in
Fig. 1. The magne ic co e is di ided in o wo pa s: a ixed
Edua do Moya-Lashe as e al. / IFAC Pape sOnLine 53-2 (2020) 6256–6261 6257
Copy igh ©
2020 The Au ho s. This is an open access a icle unde he CC BY-NC-ND license
(
h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0
)
Model-F ee Sliding-Mode Con olle o
So Landing o Reluc ance Ac ua o s
Edua do Moya-Lashe as ∗Edga Rami ez-Labo eo ∗
Ca los Sagues ∗
∗Depa amen o de In o ma ica e Ingenie ia de Sis emas (DIIS) and
Ins i u o de In es igacion en Ingenie ia de A agon (I3A),
Uni e sidad de Za agoza, Za agoza 50018, Spain,
(e-mail: emoya@uniza .es, ami lab@uniza .es, csagues@uniza .es)
Abs ac : Some elec omagne ic ac ua o s su e om high eloci y impac s du ing non-
con olled swi ching ope a ions, which cause con ac bouncing, mechanical wea , and acous ic
noise. So -landing con ol s a egies aim a minimizing he impac eloci ies o hese de ices
o imp o e hei pe o mance. This pape p esen s a sliding-mode con olle o so landing
o single-coil eluc ance ac ua o s. I is a swi ching model- ee con olle , which esul s in a
e y simple implemen a ion. A gene alized dynamical hyb id model o an ac ua o is u ilized
o de i ing he obus ness condi ion, based on he Lyapuno heo y. Then, he condi ion
is e alua ed o a dynamical model, based on a comme cial de ice, and se e al e e ence
ajec o ies. Finally, he con olle pe o mance is alida ed h ough simula ions. The e ec
o he sampling a e on he esul ing impac eloci ies is also analyzed.
Keywo ds: Ac ua o s, Elec omagne ic de ices, Modeling, Nonlinea con ol, Robus con ol,
Sliding-mode con ol, T acking
1. INTRODUCTION
A eluc ance ac ua o is a ype o nonlinea elec ome-
chanical de ice which gene a es a eluc ance-based mag-
ne ic o ce o mo e i s a ma u e. Pa icula ly, single-coil
ac ua o s a e used in an ex ensi e a ie y o indus ial
applica ions because o hei as esponse, compac ness,
high ene gy e iciency, and low cos . Thus, he e is a
g ea esea ch in e es conce ning modeling, iden i ica ion,
es ima ion, and con ol o his class o ac ua o s.
Rega ding he con ol, one o he main mo i a ions and
challenges is o achie e so landing du ing swi ching op-
e a ions; hus educing con ac bouncing, impac noise and
mechanical wea . In he li e a u e, he e a e se e al con-
ol p oposals o eluc ance ac ua o s, e.g., based on he
backs epping echnique (Kah eci and Kolmano sky, 2010),
ene gy compensa ion (Yang e al., 2013), a linea iza ion
me hod (Ka alenic e al., 2016), o cycle- o-cycle adap a-
ion (Moya-Lashe as e al., 2019), among o he s.
One impo an d awback o many mass-ma ke single-
coil eluc ance ac ua o s is he manu ac u ing a iabili y
among de ices om he same ensemble. Mo eo e , he
iden i ica ion o e e y uni may impose a p ohibi i e cos .
One majo app oach o deal wi h model unce ain ies is
he sliding-mode con ol (SMC) heo y (Slo ine and Li,
1991). The e a e se e al wo ks ha ake his app oach.
Mos commonly, he con ol law is di ided in o wo e ms:
This wo k was pa ially suppo ed by he A ag´on Regional
Go e nmen , he Spanish Go e nmen , and he Eu opean Union,
unde p ojec RTC-2017-5965-6, p ojec PGC2018-098719-B-I00
(MCIU/ AEI/FEDER, UE), esea ch g oup DGA-T45 17R, schol-
a ship FPU14/04171, and p og am FSE A ag´on 2014-2020.
an equi alen and a swi ching con ol e m (Lee e al.,
2015; Zhao e al., 2016). Al e na i ely, Eyabi and Wash-
ing on (2006) p oposed a SMC wi h only a swi ching
e m, which is hen app oxima ed o a p opo ional one.
One impo an aspec ha is omi ed in hese wo ks is
he de ini ion o he acking ajec o y, which di ec ly
a ec s he obus ness condi ions o he SMC. Ano he
impo an issue is he in luence o he sampling a e. In
gene al, he sliding accu acy is p opo ional o he squa e
o he swi ching delay (Le an , 1993). S ill, i s e ec on
he esul ing impac eloci ies needs o be e alua ed.
This pape p esen s a obus SMC con olle o single-
coil eluc ance ac ua o s. I is pu ely a swi ching con-
olle , which esul s in a e y simple and compu a ionally
inexpensi e implemen a ion. Al hough he esul ing con-
olle is model- ee—i.e. i does no depend on any model
unc ions o pa ame e s—a dynamic model is equi ed
du ing he design p ocess o gua an ee i s obus ness. The
gene alized sys em, which p esen s bo h con inuous and
disc e e dynamic beha io , is modeled wi h a hyb id au-
oma on. A obus ness condi ion is de i ed, which depends
on he sys em dynamics and he posi ion ajec o y. I is
hen e alua ed o a speci ic dynamic model, based on a
comme cial solenoid al e, and se e al ajec o ies. The
i s con ibu ion o he pape is he p oposal o a swi ching
model- ee SMC, which wo ks o e e y disc e e mode o
he sys em. The second con ibu ion is he analysis o he
in luence o he sampling a e on he impac eloci ies.
2. SYSTEM DYNAMICS
A gene al single-coil eluc ance ac ua o is ep esen ed in
Fig. 1. The magne ic co e is di ided in o wo pa s: a ixed
Model-F ee Sliding-Mode Con olle o
So Landing o Reluc ance Ac ua o s
Edua do Moya-Lashe as ∗Edga Rami ez-Labo eo ∗
Ca los Sagues ∗
∗Depa amen o de In o ma ica e Ingenie ia de Sis emas (DIIS) and
Ins i u o de In es igacion en Ingenie ia de A agon (I3A),
Uni e sidad de Za agoza, Za agoza 50018, Spain,
(e-mail: emoya@uniza .es, ami lab@uniza .es, csagues@uniza .es)
Abs ac : Some elec omagne ic ac ua o s su e om high eloci y impac s du ing non-
con olled swi ching ope a ions, which cause con ac bouncing, mechanical wea , and acous ic
noise. So -landing con ol s a egies aim a minimizing he impac eloci ies o hese de ices
o imp o e hei pe o mance. This pape p esen s a sliding-mode con olle o so landing
o single-coil eluc ance ac ua o s. I is a swi ching model- ee con olle , which esul s in a
e y simple implemen a ion. A gene alized dynamical hyb id model o an ac ua o is u ilized
o de i ing he obus ness condi ion, based on he Lyapuno heo y. Then, he condi ion
is e alua ed o a dynamical model, based on a comme cial de ice, and se e al e e ence
ajec o ies. Finally, he con olle pe o mance is alida ed h ough simula ions. The e ec
o he sampling a e on he esul ing impac eloci ies is also analyzed.
Keywo ds: Ac ua o s, Elec omagne ic de ices, Modeling, Nonlinea con ol, Robus con ol,
Sliding-mode con ol, T acking
1. INTRODUCTION
A eluc ance ac ua o is a ype o nonlinea elec ome-
chanical de ice which gene a es a eluc ance-based mag-
ne ic o ce o mo e i s a ma u e. Pa icula ly, single-coil
ac ua o s a e used in an ex ensi e a ie y o indus ial
applica ions because o hei as esponse, compac ness,
high ene gy e iciency, and low cos . Thus, he e is a
g ea esea ch in e es conce ning modeling, iden i ica ion,
es ima ion, and con ol o his class o ac ua o s.
Rega ding he con ol, one o he main mo i a ions and
challenges is o achie e so landing du ing swi ching op-
e a ions; hus educing con ac bouncing, impac noise and
mechanical wea . In he li e a u e, he e a e se e al con-
ol p oposals o eluc ance ac ua o s, e.g., based on he
backs epping echnique (Kah eci and Kolmano sky, 2010),
ene gy compensa ion (Yang e al., 2013), a linea iza ion
me hod (Ka alenic e al., 2016), o cycle- o-cycle adap a-
ion (Moya-Lashe as e al., 2019), among o he s.
One impo an d awback o many mass-ma ke single-
coil eluc ance ac ua o s is he manu ac u ing a iabili y
among de ices om he same ensemble. Mo eo e , he
iden i ica ion o e e y uni may impose a p ohibi i e cos .
One majo app oach o deal wi h model unce ain ies is
he sliding-mode con ol (SMC) heo y (Slo ine and Li,
1991). The e a e se e al wo ks ha ake his app oach.
Mos commonly, he con ol law is di ided in o wo e ms:
This wo k was pa ially suppo ed by he A ag´on Regional
Go e nmen , he Spanish Go e nmen , and he Eu opean Union,
unde p ojec RTC-2017-5965-6, p ojec PGC2018-098719-B-I00
(MCIU/ AEI/FEDER, UE), esea ch g oup DGA-T45 17R, schol-
a ship FPU14/04171, and p og am FSE A ag´on 2014-2020.
an equi alen and a swi ching con ol e m (Lee e al.,
2015; Zhao e al., 2016). Al e na i ely, Eyabi and Wash-
ing on (2006) p oposed a SMC wi h only a swi ching
e m, which is hen app oxima ed o a p opo ional one.
One impo an aspec ha is omi ed in hese wo ks is
he de ini ion o he acking ajec o y, which di ec ly
a ec s he obus ness condi ions o he SMC. Ano he
impo an issue is he in luence o he sampling a e. In
gene al, he sliding accu acy is p opo ional o he squa e
o he swi ching delay (Le an , 1993). S ill, i s e ec on
he esul ing impac eloci ies needs o be e alua ed.
This pape p esen s a obus SMC con olle o single-
coil eluc ance ac ua o s. I is pu ely a swi ching con-
olle , which esul s in a e y simple and compu a ionally
inexpensi e implemen a ion. Al hough he esul ing con-
olle is model- ee—i.e. i does no depend on any model
unc ions o pa ame e s—a dynamic model is equi ed
du ing he design p ocess o gua an ee i s obus ness. The
gene alized sys em, which p esen s bo h con inuous and
disc e e dynamic beha io , is modeled wi h a hyb id au-
oma on. A obus ness condi ion is de i ed, which depends
on he sys em dynamics and he posi ion ajec o y. I is
hen e alua ed o a speci ic dynamic model, based on a
comme cial solenoid al e, and se e al ajec o ies. The
i s con ibu ion o he pape is he p oposal o a swi ching
model- ee SMC, which wo ks o e e y disc e e mode o
he sys em. The second con ibu ion is he analysis o he
in luence o he sampling a e on he impac eloci ies.
2. SYSTEM DYNAMICS
A gene al single-coil eluc ance ac ua o is ep esen ed in
Fig. 1. The magne ic co e is di ided in o wo pa s: a ixed
Model-F ee Sliding-Mode Con olle o
So Landing o Reluc ance Ac ua o s
Edua do Moya-Lashe as ∗Edga Rami ez-Labo eo ∗
Ca los Sagues ∗
∗Depa amen o de In o ma ica e Ingenie ia de Sis emas (DIIS) and
Ins i u o de In es igacion en Ingenie ia de A agon (I3A),
Uni e sidad de Za agoza, Za agoza 50018, Spain,
(e-mail: emoya@uniza .es, ami lab@uniza .es, csagues@uniza .es)
Abs ac : Some elec omagne ic ac ua o s su e om high eloci y impac s du ing non-
con olled swi ching ope a ions, which cause con ac bouncing, mechanical wea , and acous ic
noise. So -landing con ol s a egies aim a minimizing he impac eloci ies o hese de ices
o imp o e hei pe o mance. This pape p esen s a sliding-mode con olle o so landing
o single-coil eluc ance ac ua o s. I is a swi ching model- ee con olle , which esul s in a
e y simple implemen a ion. A gene alized dynamical hyb id model o an ac ua o is u ilized
o de i ing he obus ness condi ion, based on he Lyapuno heo y. Then, he condi ion
is e alua ed o a dynamical model, based on a comme cial de ice, and se e al e e ence
ajec o ies. Finally, he con olle pe o mance is alida ed h ough simula ions. The e ec
o he sampling a e on he esul ing impac eloci ies is also analyzed.
Keywo ds: Ac ua o s, Elec omagne ic de ices, Modeling, Nonlinea con ol, Robus con ol,
Sliding-mode con ol, T acking
1. INTRODUCTION
A eluc ance ac ua o is a ype o nonlinea elec ome-
chanical de ice which gene a es a eluc ance-based mag-
ne ic o ce o mo e i s a ma u e. Pa icula ly, single-coil
ac ua o s a e used in an ex ensi e a ie y o indus ial
applica ions because o hei as esponse, compac ness,
high ene gy e iciency, and low cos . Thus, he e is a
g ea esea ch in e es conce ning modeling, iden i ica ion,
es ima ion, and con ol o his class o ac ua o s.
Rega ding he con ol, one o he main mo i a ions and
challenges is o achie e so landing du ing swi ching op-
e a ions; hus educing con ac bouncing, impac noise and
mechanical wea . In he li e a u e, he e a e se e al con-
ol p oposals o eluc ance ac ua o s, e.g., based on he
backs epping echnique (Kah eci and Kolmano sky, 2010),
ene gy compensa ion (Yang e al., 2013), a linea iza ion
me hod (Ka alenic e al., 2016), o cycle- o-cycle adap a-
ion (Moya-Lashe as e al., 2019), among o he s.
One impo an d awback o many mass-ma ke single-
coil eluc ance ac ua o s is he manu ac u ing a iabili y
among de ices om he same ensemble. Mo eo e , he
iden i ica ion o e e y uni may impose a p ohibi i e cos .
One majo app oach o deal wi h model unce ain ies is
he sliding-mode con ol (SMC) heo y (Slo ine and Li,
1991). The e a e se e al wo ks ha ake his app oach.
Mos commonly, he con ol law is di ided in o wo e ms:
This wo k was pa ially suppo ed by he A ag´on Regional
Go e nmen , he Spanish Go e nmen , and he Eu opean Union,
unde p ojec RTC-2017-5965-6, p ojec PGC2018-098719-B-I00
(MCIU/ AEI/FEDER, UE), esea ch g oup DGA-T45 17R, schol-
a ship FPU14/04171, and p og am FSE A ag´on 2014-2020.
an equi alen and a swi ching con ol e m (Lee e al.,
2015; Zhao e al., 2016). Al e na i ely, Eyabi and Wash-
ing on (2006) p oposed a SMC wi h only a swi ching
e m, which is hen app oxima ed o a p opo ional one.
One impo an aspec ha is omi ed in hese wo ks is
he de ini ion o he acking ajec o y, which di ec ly
a ec s he obus ness condi ions o he SMC. Ano he
impo an issue is he in luence o he sampling a e. In
gene al, he sliding accu acy is p opo ional o he squa e
o he swi ching delay (Le an , 1993). S ill, i s e ec on
he esul ing impac eloci ies needs o be e alua ed.
This pape p esen s a obus SMC con olle o single-
coil eluc ance ac ua o s. I is pu ely a swi ching con-
olle , which esul s in a e y simple and compu a ionally
inexpensi e implemen a ion. Al hough he esul ing con-
olle is model- ee—i.e. i does no depend on any model
unc ions o pa ame e s—a dynamic model is equi ed
du ing he design p ocess o gua an ee i s obus ness. The
gene alized sys em, which p esen s bo h con inuous and
disc e e dynamic beha io , is modeled wi h a hyb id au-
oma on. A obus ness condi ion is de i ed, which depends
on he sys em dynamics and he posi ion ajec o y. I is
hen e alua ed o a speci ic dynamic model, based on a
comme cial solenoid al e, and se e al ajec o ies. The
i s con ibu ion o he pape is he p oposal o a swi ching
model- ee SMC, which wo ks o e e y disc e e mode o
he sys em. The second con ibu ion is he analysis o he
in luence o he sampling a e on he impac eloci ies.
2. SYSTEM DYNAMICS
A gene al single-coil eluc ance ac ua o is ep esen ed in
Fig. 1. The magne ic co e is di ided in o wo pa s: a ixed
Model-F ee Sliding-Mode Con olle o
So Landing o Reluc ance Ac ua o s
Edua do Moya-Lashe as ∗Edga Rami ez-Labo eo ∗
Ca los Sagues ∗
∗Depa amen o de In o ma ica e Ingenie ia de Sis emas (DIIS) and
Ins i u o de In es igacion en Ingenie ia de A agon (I3A),
Uni e sidad de Za agoza, Za agoza 50018, Spain,
(e-mail: emoya@uniza .es, ami lab@uniza .es, csagues@uniza .es)
Abs ac : Some elec omagne ic ac ua o s su e om high eloci y impac s du ing non-
con olled swi ching ope a ions, which cause con ac bouncing, mechanical wea , and acous ic
noise. So -landing con ol s a egies aim a minimizing he impac eloci ies o hese de ices
o imp o e hei pe o mance. This pape p esen s a sliding-mode con olle o so landing
o single-coil eluc ance ac ua o s. I is a swi ching model- ee con olle , which esul s in a
e y simple implemen a ion. A gene alized dynamical hyb id model o an ac ua o is u ilized
o de i ing he obus ness condi ion, based on he Lyapuno heo y. Then, he condi ion
is e alua ed o a dynamical model, based on a comme cial de ice, and se e al e e ence
ajec o ies. Finally, he con olle pe o mance is alida ed h ough simula ions. The e ec
o he sampling a e on he esul ing impac eloci ies is also analyzed.
Keywo ds: Ac ua o s, Elec omagne ic de ices, Modeling, Nonlinea con ol, Robus con ol,
Sliding-mode con ol, T acking
1. INTRODUCTION
A eluc ance ac ua o is a ype o nonlinea elec ome-
chanical de ice which gene a es a eluc ance-based mag-
ne ic o ce o mo e i s a ma u e. Pa icula ly, single-coil
ac ua o s a e used in an ex ensi e a ie y o indus ial
applica ions because o hei as esponse, compac ness,
high ene gy e iciency, and low cos . Thus, he e is a
g ea esea ch in e es conce ning modeling, iden i ica ion,
es ima ion, and con ol o his class o ac ua o s.
Rega ding he con ol, one o he main mo i a ions and
challenges is o achie e so landing du ing swi ching op-
e a ions; hus educing con ac bouncing, impac noise and
mechanical wea . In he li e a u e, he e a e se e al con-
ol p oposals o eluc ance ac ua o s, e.g., based on he
backs epping echnique (Kah eci and Kolmano sky, 2010),
ene gy compensa ion (Yang e al., 2013), a linea iza ion
me hod (Ka alenic e al., 2016), o cycle- o-cycle adap a-
ion (Moya-Lashe as e al., 2019), among o he s.
One impo an d awback o many mass-ma ke single-
coil eluc ance ac ua o s is he manu ac u ing a iabili y
among de ices om he same ensemble. Mo eo e , he
iden i ica ion o e e y uni may impose a p ohibi i e cos .
One majo app oach o deal wi h model unce ain ies is
he sliding-mode con ol (SMC) heo y (Slo ine and Li,
1991). The e a e se e al wo ks ha ake his app oach.
Mos commonly, he con ol law is di ided in o wo e ms:
This wo k was pa ially suppo ed by he A ag´on Regional
Go e nmen , he Spanish Go e nmen , and he Eu opean Union,
unde p ojec RTC-2017-5965-6, p ojec PGC2018-098719-B-I00
(MCIU/ AEI/FEDER, UE), esea ch g oup DGA-T45 17R, schol-
a ship FPU14/04171, and p og am FSE A ag´on 2014-2020.
an equi alen and a swi ching con ol e m (Lee e al.,
2015; Zhao e al., 2016). Al e na i ely, Eyabi and Wash-
ing on (2006) p oposed a SMC wi h only a swi ching
e m, which is hen app oxima ed o a p opo ional one.
One impo an aspec ha is omi ed in hese wo ks is
he de ini ion o he acking ajec o y, which di ec ly
a ec s he obus ness condi ions o he SMC. Ano he
impo an issue is he in luence o he sampling a e. In
gene al, he sliding accu acy is p opo ional o he squa e
o he swi ching delay (Le an , 1993). S ill, i s e ec on
he esul ing impac eloci ies needs o be e alua ed.
This pape p esen s a obus SMC con olle o single-
coil eluc ance ac ua o s. I is pu ely a swi ching con-
olle , which esul s in a e y simple and compu a ionally
inexpensi e implemen a ion. Al hough he esul ing con-
olle is model- ee—i.e. i does no depend on any model
unc ions o pa ame e s—a dynamic model is equi ed
du ing he design p ocess o gua an ee i s obus ness. The
gene alized sys em, which p esen s bo h con inuous and
disc e e dynamic beha io , is modeled wi h a hyb id au-
oma on. A obus ness condi ion is de i ed, which depends
on he sys em dynamics and he posi ion ajec o y. I is
hen e alua ed o a speci ic dynamic model, based on a
comme cial solenoid al e, and se e al ajec o ies. The
i s con ibu ion o he pape is he p oposal o a swi ching
model- ee SMC, which wo ks o e e y disc e e mode o
he sys em. The second con ibu ion is he analysis o he
in luence o he sampling a e on he impac eloci ies.
2. SYSTEM DYNAMICS
A gene al single-coil eluc ance ac ua o is ep esen ed in
Fig. 1. The magne ic co e is di ided in o wo pa s: a ixed
Model-F ee Sliding-Mode Con olle o
So Landing o Reluc ance Ac ua o s
Edua do Moya-Lashe as ∗Edga Rami ez-Labo eo ∗
Ca los Sagues ∗
∗Depa amen o de In o ma ica e Ingenie ia de Sis emas (DIIS) and
Ins i u o de In es igacion en Ingenie ia de A agon (I3A),
Uni e sidad de Za agoza, Za agoza 50018, Spain,
(e-mail: emoya@uniza .es, ami lab@uniza .es, csagues@uniza .es)
Abs ac : Some elec omagne ic ac ua o s su e om high eloci y impac s du ing non-
con olled swi ching ope a ions, which cause con ac bouncing, mechanical wea , and acous ic
noise. So -landing con ol s a egies aim a minimizing he impac eloci ies o hese de ices
o imp o e hei pe o mance. This pape p esen s a sliding-mode con olle o so landing
o single-coil eluc ance ac ua o s. I is a swi ching model- ee con olle , which esul s in a
e y simple implemen a ion. A gene alized dynamical hyb id model o an ac ua o is u ilized
o de i ing he obus ness condi ion, based on he Lyapuno heo y. Then, he condi ion
is e alua ed o a dynamical model, based on a comme cial de ice, and se e al e e ence
ajec o ies. Finally, he con olle pe o mance is alida ed h ough simula ions. The e ec
o he sampling a e on he esul ing impac eloci ies is also analyzed.
Keywo ds: Ac ua o s, Elec omagne ic de ices, Modeling, Nonlinea con ol, Robus con ol,
Sliding-mode con ol, T acking
1. INTRODUCTION
A eluc ance ac ua o is a ype o nonlinea elec ome-
chanical de ice which gene a es a eluc ance-based mag-
ne ic o ce o mo e i s a ma u e. Pa icula ly, single-coil
ac ua o s a e used in an ex ensi e a ie y o indus ial
applica ions because o hei as esponse, compac ness,
high ene gy e iciency, and low cos . Thus, he e is a
g ea esea ch in e es conce ning modeling, iden i ica ion,
es ima ion, and con ol o his class o ac ua o s.
Rega ding he con ol, one o he main mo i a ions and
challenges is o achie e so landing du ing swi ching op-
e a ions; hus educing con ac bouncing, impac noise and
mechanical wea . In he li e a u e, he e a e se e al con-
ol p oposals o eluc ance ac ua o s, e.g., based on he
backs epping echnique (Kah eci and Kolmano sky, 2010),
ene gy compensa ion (Yang e al., 2013), a linea iza ion
me hod (Ka alenic e al., 2016), o cycle- o-cycle adap a-
ion (Moya-Lashe as e al., 2019), among o he s.
One impo an d awback o many mass-ma ke single-
coil eluc ance ac ua o s is he manu ac u ing a iabili y
among de ices om he same ensemble. Mo eo e , he
iden i ica ion o e e y uni may impose a p ohibi i e cos .
One majo app oach o deal wi h model unce ain ies is
he sliding-mode con ol (SMC) heo y (Slo ine and Li,
1991). The e a e se e al wo ks ha ake his app oach.
Mos commonly, he con ol law is di ided in o wo e ms:
This wo k was pa ially suppo ed by he A ag´on Regional
Go e nmen , he Spanish Go e nmen , and he Eu opean Union,
unde p ojec RTC-2017-5965-6, p ojec PGC2018-098719-B-I00
(MCIU/ AEI/FEDER, UE), esea ch g oup DGA-T45 17R, schol-
a ship FPU14/04171, and p og am FSE A ag´on 2014-2020.
an equi alen and a swi ching con ol e m (Lee e al.,
2015; Zhao e al., 2016). Al e na i ely, Eyabi and Wash-
ing on (2006) p oposed a SMC wi h only a swi ching
e m, which is hen app oxima ed o a p opo ional one.
One impo an aspec ha is omi ed in hese wo ks is
he de ini ion o he acking ajec o y, which di ec ly
a ec s he obus ness condi ions o he SMC. Ano he
impo an issue is he in luence o he sampling a e. In
gene al, he sliding accu acy is p opo ional o he squa e
o he swi ching delay (Le an , 1993). S ill, i s e ec on
he esul ing impac eloci ies needs o be e alua ed.
This pape p esen s a obus SMC con olle o single-
coil eluc ance ac ua o s. I is pu ely a swi ching con-
olle , which esul s in a e y simple and compu a ionally
inexpensi e implemen a ion. Al hough he esul ing con-
olle is model- ee—i.e. i does no depend on any model
unc ions o pa ame e s—a dynamic model is equi ed
du ing he design p ocess o gua an ee i s obus ness. The
gene alized sys em, which p esen s bo h con inuous and
disc e e dynamic beha io , is modeled wi h a hyb id au-
oma on. A obus ness condi ion is de i ed, which depends
on he sys em dynamics and he posi ion ajec o y. I is
hen e alua ed o a speci ic dynamic model, based on a
comme cial solenoid al e, and se e al ajec o ies. The
i s con ibu ion o he pape is he p oposal o a swi ching
model- ee SMC, which wo ks o e e y disc e e mode o
he sys em. The second con ibu ion is he analysis o he
in luence o he sampling a e on he impac eloci ies.
2. SYSTEM DYNAMICS
A gene al single-coil eluc ance ac ua o is ep esen ed in
Fig. 1. The magne ic co e is di ided in o wo pa s: a ixed
Model-F ee Sliding-Mode Con olle o
So Landing o Reluc ance Ac ua o s
Edua do Moya-Lashe as ∗Edga Rami ez-Labo eo ∗
Ca los Sagues ∗
∗Depa amen o de In o ma ica e Ingenie ia de Sis emas (DIIS) and
Ins i u o de In es igacion en Ingenie ia de A agon (I3A),
Uni e sidad de Za agoza, Za agoza 50018, Spain,
(e-mail: emoya@uniza .es, ami lab@uniza .es, csagues@uniza .es)
Abs ac : Some elec omagne ic ac ua o s su e om high eloci y impac s du ing non-
con olled swi ching ope a ions, which cause con ac bouncing, mechanical wea , and acous ic
noise. So -landing con ol s a egies aim a minimizing he impac eloci ies o hese de ices
o imp o e hei pe o mance. This pape p esen s a sliding-mode con olle o so landing
o single-coil eluc ance ac ua o s. I is a swi ching model- ee con olle , which esul s in a
e y simple implemen a ion. A gene alized dynamical hyb id model o an ac ua o is u ilized
o de i ing he obus ness condi ion, based on he Lyapuno heo y. Then, he condi ion
is e alua ed o a dynamical model, based on a comme cial de ice, and se e al e e ence
ajec o ies. Finally, he con olle pe o mance is alida ed h ough simula ions. The e ec
o he sampling a e on he esul ing impac eloci ies is also analyzed.
Keywo ds: Ac ua o s, Elec omagne ic de ices, Modeling, Nonlinea con ol, Robus con ol,
Sliding-mode con ol, T acking
1. INTRODUCTION
A eluc ance ac ua o is a ype o nonlinea elec ome-
chanical de ice which gene a es a eluc ance-based mag-
ne ic o ce o mo e i s a ma u e. Pa icula ly, single-coil
ac ua o s a e used in an ex ensi e a ie y o indus ial
applica ions because o hei as esponse, compac ness,
high ene gy e iciency, and low cos . Thus, he e is a
g ea esea ch in e es conce ning modeling, iden i ica ion,
es ima ion, and con ol o his class o ac ua o s.
Rega ding he con ol, one o he main mo i a ions and
challenges is o achie e so landing du ing swi ching op-
e a ions; hus educing con ac bouncing, impac noise and
mechanical wea . In he li e a u e, he e a e se e al con-
ol p oposals o eluc ance ac ua o s, e.g., based on he
backs epping echnique (Kah eci and Kolmano sky, 2010),
ene gy compensa ion (Yang e al., 2013), a linea iza ion
me hod (Ka alenic e al., 2016), o cycle- o-cycle adap a-
ion (Moya-Lashe as e al., 2019), among o he s.
One impo an d awback o many mass-ma ke single-
coil eluc ance ac ua o s is he manu ac u ing a iabili y
among de ices om he same ensemble. Mo eo e , he
iden i ica ion o e e y uni may impose a p ohibi i e cos .
One majo app oach o deal wi h model unce ain ies is
he sliding-mode con ol (SMC) heo y (Slo ine and Li,
1991). The e a e se e al wo ks ha ake his app oach.
Mos commonly, he con ol law is di ided in o wo e ms:
This wo k was pa ially suppo ed by he A ag´on Regional
Go e nmen , he Spanish Go e nmen , and he Eu opean Union,
unde p ojec RTC-2017-5965-6, p ojec PGC2018-098719-B-I00
(MCIU/ AEI/FEDER, UE), esea ch g oup DGA-T45 17R, schol-
a ship FPU14/04171, and p og am FSE A ag´on 2014-2020.
an equi alen and a swi ching con ol e m (Lee e al.,
2015; Zhao e al., 2016). Al e na i ely, Eyabi and Wash-
ing on (2006) p oposed a SMC wi h only a swi ching
e m, which is hen app oxima ed o a p opo ional one.
One impo an aspec ha is omi ed in hese wo ks is
he de ini ion o he acking ajec o y, which di ec ly
a ec s he obus ness condi ions o he SMC. Ano he
impo an issue is he in luence o he sampling a e. In
gene al, he sliding accu acy is p opo ional o he squa e
o he swi ching delay (Le an , 1993). S ill, i s e ec on
he esul ing impac eloci ies needs o be e alua ed.
This pape p esen s a obus SMC con olle o single-
coil eluc ance ac ua o s. I is pu ely a swi ching con-
olle , which esul s in a e y simple and compu a ionally
inexpensi e implemen a ion. Al hough he esul ing con-
olle is model- ee—i.e. i does no depend on any model
unc ions o pa ame e s—a dynamic model is equi ed
du ing he design p ocess o gua an ee i s obus ness. The
gene alized sys em, which p esen s bo h con inuous and
disc e e dynamic beha io , is modeled wi h a hyb id au-
oma on. A obus ness condi ion is de i ed, which depends
on he sys em dynamics and he posi ion ajec o y. I is
hen e alua ed o a speci ic dynamic model, based on a
comme cial solenoid al e, and se e al ajec o ies. The
i s con ibu ion o he pape is he p oposal o a swi ching
model- ee SMC, which wo ks o e e y disc e e mode o
he sys em. The second con ibu ion is he analysis o he
in luence o he sampling a e on he impac eloci ies.
2. SYSTEM DYNAMICS
A gene al single-coil eluc ance ac ua o is ep esen ed in
Fig. 1. The magne ic co e is di ided in o wo pa s: a ixed
Fig. 1. Schema ic ep esen a ion o a single-coil eluc ance
ac ua o .
pa (s a o ) and a mo able pa (mo e o a ma u e). The
ai gap be ween hem is dependen on he posi ion o he
mo e . The e a e wo ypes o ope a ions depending on
he di ec ion o he mo emen : in a making ope a ion, he
magne ic o ce is la ge enough o a ac he mo e owa d
he s a o ; whe eas in a b eaking ope a ion, he magne ic
o ce is educed and he passi e o ces (e.g. elas ic o
g a i y) mo e he a ma u e in he opposi e di ec ion.
Mo eo e , he posi ion o he mo e is es ic ed be ween
a lowe and an uppe limi .
The mo ion dynamics is gi en by New on’s second law,
wi h wo o ces,
˙ = (z, ,φ)= 1
mFpas(z, )+Fmag(z,φ),(1)
whe e z, , and ˙ a e he posi ion, eloci y and accele a ion
o he mo e ; Fpas, and Fmag a e he passi e and magne ic
o ces; φis he magne ic lux; and mis he mo ing
mass. No e ha he dynamic unc ion o is exp essed
compac ly as . The o ce ha can be con olled—albei
indi ec ly—is Fmag, which is de ined as (Rami ez-Labo eo
e al., 2016)
Fmag =−1
2R
g(z)φ2,R
g(z)=∂Rg(z)
∂z ,(2)
whe e Rgis he gap eluc ance. No e ha R
g>0, and
he e o e Fmag ≤0, o all z∈[zmin,z
max] (i.e. he
magne ic o ce is always a ac i e).
Then, φcan be ela ed o he cu en h ough he coil icoil
in e ms o he o al eluc ance, gi en Amp`e e’s ci cui al
law,
Ni
coil +ieddy =Rc(φ)+Rg(z)φ, (3)
whe e Nis he numbe o coil u ns, Rcis he co e
eluc ance, and ieddy is he ne eddy cu en h ough he
co e. Assuming ha he magne ic lux densi y is cons an
ac oss he sec ion, ieddy is p opo ional o he magne ic
lux de i a i e (Rami ez-Labo eo e al., 2019),
ieddy =−ke˙
φ. (4)
Mos commonly, he ol age is ea ed as he sys em inpu
u, because i can be di ec ly supplied o he de ice. The
dynamics o he magne ic lux is gi en by he elec ical
ci cui equa ion,
u=Ri
coil +N˙
φ, (5)
whe e Ris he coil esis ance. Then, subs i u ing (3) in o
(5) and sol ing o ˙
φ, he dynamic unc ion is de i ed as
˙
φ= φ(z,φ)+Bφu
=−RRg(z)+Rc(φ)φ
N2+Rk
e
+N
N2+Rk
e
u, (6)
whe e he unc ion φdepends on he posi ion and mag-
ne ic lux, and Bφis a cons an .
˙z=
˙ = (z, ,φ)
˙
φ= φ(z,φ)+Bφu
˙z=0
˙ =0
˙
φ= φ(z,φ)+Bφu
˙z=0
˙ =0
˙
φ= φ(z,φ)+Bφu
Mo ion (q= 2 ):
Lowe limi (q= 1):
Uppe limi (q= 3):
z=zmin ∧ <0⇒ +=0
z=zmax ∧ >0⇒ +=0
(z, ,φ)≥0
(z, ,φ)≤0
Fig. 2. Hyb id au oma on ha models he dynamics o
eluc ance ac ua o s wi h a limi ed ange o mo ion.
The comple e sys em dynamics can be desc ibed h ough
a s a e-space ep esen a ion wi h h ee s a e a iables (z,
, and φ). As he mo ion is cons ained, zand mus be
s a ic i he mo e eaches one o he wo limi s. Thus, he
sys em is modeled wi h a hyb id au oma on, wi h h ee
disc e e modes, as illus a ed in Fig. 2. Each ansi ion is
accompanied by i s gua d condi ion. The e is also a ese
unc ion when ansi ioning o one o he posi ion limi s:
+= 0.
3. CONTROL DESIGN
3.1 T ajec o y planning
The i s c i ical aspec o he so -landing acking con ol
is he de ini ion o he posi ion ajec o y z e , o all ime
∈[ 0,
]. Fo a gi en ope a ion, he ini ial posi ion is z0
and he desi ed inal posi ion is z .
Le land be he in ended ins an in which he a ma u e
eaches he inal posi ion. Fo a pe ec so landing, z e
should sa is y
z e ( land)=z ,
e ( land)=0,a
e ( land)=0,(7)
whe e
e ( )= ˙z e ( ),a
e ( )=˙ e ( ).(8)
Equi alen ly, in o de o s a he mo ion smoo hly, z e
should sa is y
z e ( akeo )=z0,
e ( akeo )=0,a
e ( akeo )=0,(9)
whe e akeo is he ake-o ins an .
Du ing mo ion (q= 2), he accele a ion is de e mined om
(1). The e o e, in o de o s a mo ing immedia ely a =
akeo ,φshould be φ akeo , such ha (z0,0,φ
akeo )=
0. No e ha he e a e wo symme ical solu ions o φ akeo
(posi i e and nega i e), because is an e en unc ion wi h
espec o φ. No e also ha he con olle should dec ease
|φ|i q= 1, o inc ease i i q= 3, un il i eaches φ akeo .
Thus, p io o mo ing, a s a ic in e al is de ined,
z e ( )=z0,∀ ∈[ 0,
akeo ].(10)
whe e akeo − 0should be la ge enough o le he
magne ic lux each φ akeo be o e akeo .
In he second in e al ( om akeo o land), a posi ion
ajec o y mus be de ined o each z smoo hly. Thus, z e
should be a unc ion o ime , o all ∈[ akeo ,
land],
6258 Edua do Moya-Lashe as e al. / IFAC Pape sOnLine 53-2 (2020) 6256–6261
wi h bounda y condi ions (9) and (7). In he hi d in e al
( om land o ), he mo e mus be kep in he desi ed
inal posi ion, so
z e ( )=z ,∀ ∈[ land,
].(11)
Once a ajec o y is de ined, i s easibili y should be
checked. Fi s , he posi ion mus be kep inside i s bounds,
z e ( )∈[zmin,z
max],∀ . (12)
Secondly, he equi ed magne ic o ce F∗
mag should be cal-
cula ed, and ensu e ha i is always nonposi i e, because
epelling magne ic o ces a e no physically possible (see
(2)),
F∗
mag( )=ma
e ( )−Fpas(z e ( ),
e ( )) ≤0,∀ . (13)
Mo eo e , magne ic sa u a ion mus also be aken in o
accoun . Gi en he sa u a ed alue o he magne ic lux
φsa , he equi ed magne ic o ce should also sa is y
F∗
mag( )≥−
1
2R
g(z e ( )) φ2
sa ,∀ . (14)
3.2 Con ol o mo ion dynamics
The con olle is ini ially designed based on he dynamic
equa ions o he mo ion mode (q= 2), which can be
exp essed compac ly as
˙
x= (x)+Bu, (15)
whe e
x=z
φT
, (x)=
(z, ,φ)
φ(z,φ),B=0
0
Bφ.(16)
As s a ed in he in oduc ion, ou p oposal elies on
an SMC. I is assumed ha he posi ion z, eloci y
, and accele a ion acan be ob ained ei he h ough
measu emen o es ima ion. The p oposed sliding su ace
is de ined in e ms o hei e o s,
s=λ1+d
d λ2+d
d ˜z
=˜a+(λ1+λ2)˜ +λ1λ2˜z, (17)
whe e λ1and λ2a e posi i e cons an s; and ˜z,˜ , and ˜a
a e he posi ion e o and i s de i a i es,
˜z=z−z e ,˜ = − e ,˜a=a−a e .(18)
To analyze he con e gence o he sliding su ace s= 0,
he ollowing Lyapuno unc ion is de ined,
V=1
2s2.(19)
Thus, o ensu e ha scon e ges o ze o in ini e ime, we
impose he ollowing condi ion,
˙
V=s˙s≤−η|s|,(20)
whe e ηis a s ic ly posi i e cons an ha de e mines he
con e gence speed (|˙s|≥η).
Then, by de i ing (17) and subs i u ing in o (20),
˙
V=sj−j e +(λ1+λ2)˜a+λ1λ2˜ ,(21)
whe e j=˙ais he je k and j e =˙a e is he e e ence
je k. The je k jcan be de i ed om and he dynamic
equa ion (15) as
j=d (x)
d =∂ (x)
∂x (x)+∂ (x)
∂xBu. (22)
No e ha jdepends on u. Thus, he con e gence condi ion
(20) can be exp essed in e ms o he con ol u,
s( j−j e +ε−Bju)≤−η|s|,(23)
whe e
ε=ε(˜ ,˜a)=(λ1+λ2)˜a+λ1λ2˜ , (24)
j= j(x)=∂ (x)
∂x (x)
=1
m∂Fpas
∂z (z, ) +∂Fpas
∂ (z, )a
−1
2R
g(z)φ2 −R
g(z)φ
φ(z,φ),(25)
Bj=Bj(x)=−∂ (x)
∂xB=R
gφB
φ
m.(26)
Then, wi h some manipula ions,
sgn(s)Bju≥sgn(s)( j−j e +ε)+η. (27)
No e ha , assuming R
g>0 o all z∈[zmin,z
max], i is
ob ained ha sgn(Bj) = sgn(φ). Thus, he con ol umus
sa is y he ollowing condi ion,
sgn(s) sgn(φ)u≥sgn(s)( j−j e +ε)+η
|Bj|.(28)
We p opose his model- ee con ol,
umo ion =umax sgn(s) sgn(φ),(29)
whe e umax is a cons an ha , in o de o ensu e he
con e gence o s= 0, mus sa is y
umax ≥max | j−j e +ε|+η
|Bj|.(30)
3.3 Con ol o hyb id dynamics
In he p e ious sec ion, we ha e p oposed a con olle and
p o ed i s con e gence o he mo ion dynamics (q= 2).
Now, we ensu e ha i wo ks o he comple e hyb id
sys em. Fo ha , we p opose a sligh modi ica ion o he
Lyapuno unc ion,
V=1
2σ2,(31)
whe e σis a gene aliza ion o s. I is de ined as
σ= (z, ,φ)−a e +(λ1+λ2)˜ +λ1λ2˜z. (32)
No e ha σis equal o sin he case o mo ion, because
a= . On he o he hand, be o e he s a o mo ion
( ≤ akeo ), σ= . The e o e, σ= 0 implies ha
φ=φ akeo . As a esul , i σ= 0, he sys em beha es
as desi ed bo h be o e and a e he s a o mo ion.
Following he same line o easoning as in Sec ion 3.2,
con e gence o σ= 0 equi es
sgn(σ) sgn(φ)u≥sgn(σ)( j−j e +ε)+η
|Bj|.(33)
To keep he con olle model- ee, he p oposal canno
depend on . Ins ead, i should be a unc ion o s. We
gene alize he p oposed con ol (29),
uhyb id =umax sgn(φ) sgn∗(s),(34)
Edua do Moya-Lashe as e al. / IFAC Pape sOnLine 53-2 (2020) 6256–6261 6259
wi h bounda y condi ions (9) and (7). In he hi d in e al
( om land o ), he mo e mus be kep in he desi ed
inal posi ion, so
z e ( )=z ,∀ ∈[ land,
].(11)
Once a ajec o y is de ined, i s easibili y should be
checked. Fi s , he posi ion mus be kep inside i s bounds,
z e ( )∈[zmin,z
max],∀ . (12)
Secondly, he equi ed magne ic o ce F∗
mag should be cal-
cula ed, and ensu e ha i is always nonposi i e, because
epelling magne ic o ces a e no physically possible (see
(2)),
F∗
mag( )=ma
e ( )−Fpas(z e ( ),
e ( )) ≤0,∀ . (13)
Mo eo e , magne ic sa u a ion mus also be aken in o
accoun . Gi en he sa u a ed alue o he magne ic lux
φsa , he equi ed magne ic o ce should also sa is y
F∗
mag( )≥−
1
2R
g(z e ( )) φ2
sa ,∀ . (14)
3.2 Con ol o mo ion dynamics
The con olle is ini ially designed based on he dynamic
equa ions o he mo ion mode (q= 2), which can be
exp essed compac ly as
˙
x= (x)+Bu, (15)
whe e
x=z
φT
, (x)=
(z, ,φ)
φ(z,φ),B=0
0
Bφ.(16)
As s a ed in he in oduc ion, ou p oposal elies on
an SMC. I is assumed ha he posi ion z, eloci y
, and accele a ion acan be ob ained ei he h ough
measu emen o es ima ion. The p oposed sliding su ace
is de ined in e ms o hei e o s,
s=λ1+d
d λ2+d
d ˜z
=˜a+(λ1+λ2)˜ +λ1λ2˜z, (17)
whe e λ1and λ2a e posi i e cons an s; and ˜z,˜ , and ˜a
a e he posi ion e o and i s de i a i es,
˜z=z−z e ,˜ = − e ,˜a=a−a e .(18)
To analyze he con e gence o he sliding su ace s= 0,
he ollowing Lyapuno unc ion is de ined,
V=1
2s2.(19)
Thus, o ensu e ha scon e ges o ze o in ini e ime, we
impose he ollowing condi ion,
˙
V=s˙s≤−η|s|,(20)
whe e ηis a s ic ly posi i e cons an ha de e mines he
con e gence speed (|˙s|≥η).
Then, by de i ing (17) and subs i u ing in o (20),
˙
V=sj−j e +(λ1+λ2)˜a+λ1λ2˜ ,(21)
whe e j=˙ais he je k and j e =˙a e is he e e ence
je k. The je k jcan be de i ed om and he dynamic
equa ion (15) as
j=d (x)
d =∂ (x)
∂x (x)+∂ (x)
∂xBu. (22)
No e ha jdepends on u. Thus, he con e gence condi ion
(20) can be exp essed in e ms o he con ol u,
s( j−j e +ε−Bju)≤−η|s|,(23)
whe e
ε=ε(˜ ,˜a)=(λ1+λ2)˜a+λ1λ2˜ , (24)
j= j(x)=∂ (x)
∂x (x)
=1
m∂Fpas
∂z (z, ) +∂Fpas
∂ (z, )a
−1
2R
g(z)φ2 −R
g(z)φ
φ(z,φ),(25)
Bj=Bj(x)=−∂ (x)
∂xB=R
gφB
φ
m.(26)
Then, wi h some manipula ions,
sgn(s)Bju≥sgn(s)( j−j e +ε)+η. (27)
No e ha , assuming R
g>0 o all z∈[zmin,z
max], i is
ob ained ha sgn(Bj) = sgn(φ). Thus, he con ol umus
sa is y he ollowing condi ion,
sgn(s) sgn(φ)u≥sgn(s)( j−j e +ε)+η
|Bj|.(28)
We p opose his model- ee con ol,
umo ion =umax sgn(s) sgn(φ),(29)
whe e umax is a cons an ha , in o de o ensu e he
con e gence o s= 0, mus sa is y
umax ≥max | j−j e +ε|+η
|Bj|.(30)
3.3 Con ol o hyb id dynamics
In he p e ious sec ion, we ha e p oposed a con olle and
p o ed i s con e gence o he mo ion dynamics (q= 2).
Now, we ensu e ha i wo ks o he comple e hyb id
sys em. Fo ha , we p opose a sligh modi ica ion o he
Lyapuno unc ion,
V=1
2σ2,(31)
whe e σis a gene aliza ion o s. I is de ined as
σ= (z, ,φ)−a e +(λ1+λ2)˜ +λ1λ2˜z. (32)
No e ha σis equal o sin he case o mo ion, because
a= . On he o he hand, be o e he s a o mo ion
( ≤ akeo ), σ= . The e o e, σ= 0 implies ha
φ=φ akeo . As a esul , i σ= 0, he sys em beha es
as desi ed bo h be o e and a e he s a o mo ion.
Following he same line o easoning as in Sec ion 3.2,
con e gence o σ= 0 equi es
sgn(σ) sgn(φ)u≥sgn(σ)( j−j e +ε)+η
|Bj|.(33)
To keep he con olle model- ee, he p oposal canno
depend on . Ins ead, i should be a unc ion o s. We
gene alize he p oposed con ol (29),
uhyb id =umax sgn(φ) sgn∗(s),(34)
whe e
sgn∗(s)=
−1,i s<0,
+1,i s>0,
−1,i s=0∧q=1,
+1,o he wise.
(35)
Unde he assump ion ha (30) is sa is ied, a su icien
condi ion o con e gence is
sgn∗(s) = sgn(σ).(36)
Then, con e gence is s udied in h ee sepa a e cases. Fi s ,
i q= 2, he con e gence condi ion is di ec ly gua an eed
because s=σ.
Secondly, i z=z e =zmax o z=z e =zmin,sis always
ze o, bu σmay be no . No e ha a e =˜ =˜z= 0. Then,
sgn(σ) = sgn( ).(37)
No e also ha <0 i q= 1 and >0i q= 3,
o he wise he hyb id sys em would make a ansi ion o
q= 2 (see gua d condi ions in Fig. 2). The e o e,
sgn(σ) = sgn( )=−1,i q=1,
+1,i q=3.(38)
Then, gi en he p oposed de ini ion o sgn∗(s), condi ion
(36) holds, so con e gence is s ill gua an eed.
Thi dly, we s ill need o check he con e gence o he
con olle in he case ha he posi ion is in one o he
limi s (q= 2), bu he e e ence is no . In ha e en , (32)
is simpli ied in o
σ= +s, (39)
whe e
s=−a e −(λ1+λ2) e −λ1λ2(z−z e ).(40)
Assuming ha he posi ion ajec o y is de ined smoo hly
a he s a o he mo emen , condi ion (36) is sa is ied
because, when z0=zmin (b eaking ope a ion),
<0and(z e −z0),
e ,a
e ≥0.(41)
Equi alen ly, when z0=zmax (making ope a ion),
>0,and (z e −z0),
e ,a
e ≤0.(42)
On he o he hand, a he end o mo emen , i zhas
eached he limi bu z e no ye , he condi ion is no
necessa ily sa is ied. This may seem like a limi a ion bu ,
i he mo e has eached he inal posi ion p ema u ely, i
is ac ually p e e able o ix i ins ead o sepa a ing i o
con inue ollowing he ajec o y. Thus, expe ules a e
added o he con olle so he mo e is kep a z =zmin
(making ope a ion) o z =zmax (b eaking ope a ion),
u=umax sgn(φ) i z=z =zmin,
0,i z=z =zmax,
umax sgn∗(s) sgn(φ),o he wise.
(43)
4. ANALYSIS AND DISCUSSION
4.1 Robus ness analysis
Fo he gi en dynamic model, i is impossible o gua an ee
obus ness in gene al, o any easible s a e. As a clea
coun e example, se ing φ= 0 makes Bj= 0, and umax ≥
∞(see (30)). The e o e, he obus ness mus be s udied
o a gi en ajec o y. To illus a e his, he obus ness is
analyzed o h ee di e en scena ios.
Fig. 3. Solenoid al e: schema ic ep esen a ion (le ) and
pho o ( igh ).
Table 1. Pa ame e s o he solenoid al e.
Pa am. Value
m0.0016 kg
ks61.8N/m
zs0.019 m
c0.8 Ns/m
zmin 0m
zmax 0.001 m
Pa am. Value
N1200
R50 Ω
ke1630 Ω−1
Rc,04.41 ×106H−1
φsa 2.6×10−5Wb
Then, h ee posi ion ajec o ies a e de ined. Each one o
hem consis s o a making ope a ion, ollowed by a b eak-
ing ope a ion. The mo ion in e als a e de ined wi h a 5 h
deg ee polynomial, sa is ying he bounda y condi ions (7),
(9). Mo eo e , o he sake o simplici y, he ime in e als
a e de ined in e ms o he mo ion du a ion (τmo ),
land − akeo =τmo ,(44)
akeo − 0= − land =τmo /4,(45)
whe e τmo is 3, 4 o 5 ms, o each case.
The ac ua o model is pa icula ized o a comme cial
solenoid al e, depic ed in Fig. 3, whose es ima ed pa-
ame e s a e p esen ed in Table 1. The passi e o ce is
gene a ed by he sp ing and ic ion. I is modeled as a
mass-sp ing-dampe sys em,
Fpas =ks(zs−z)−c , (46)
whe e ksis he sp ing cons an , zsis he sp ing es ing
posi ion, and cis he damping coe icien . Mo eo e , he
co e eluc ance is gi en by a pa ame ic exp ession ha
akes in o accoun magne ic sa u a ion (Moya-Lashe as
e al., 2017),
Rc=Rc,0
1−φ/φsa
,(47)
whe e Rc,0is he co e eluc ance o φ= 0, and φsa is he
sa u a ed alue o he magne ic lux.
The gap eluc ance, on he o he hand, is highly nonlin-
ea wi h espec o he posi ion. Ins ead o a pa ame ic
exp ession, a look-up able is used (see Fig. 4). I s da a
has been ob ained om ini e elemen analysis and expe -
imen a ion (Rami ez-Labo eo and Sagues, 2018).
In Fig. 5, he desi ed posi ion and i s de i a i es (z e ,
e ,a e ) a e displayed. Th ee addi ional use ul signals
a e calcula ed and shown in Fig. 5: he equi ed magne ic
o ce (as desc ibed in Sec ion 3.1), he equi ed ac ion u∗,
and Bj. The equi ed ac ion is he absolu e alue o u o
be able o ack z e in an ideal scena io (no pe u ba ions
o e o s),
u∗=| j−j e |
|Bj|.(48)
6260 Edua do Moya-Lashe as e al. / IFAC Pape sOnLine 53-2 (2020) 6256–6261
Fig. 4. Gap eluc ance and i s de i a i e wi h espec o
he gap leng h.
No e ha , i τmo = 3 ms, F∗
mag is posi i e in a small
in e al in he b eaking ope a ion (a ound /τmo = 2).
Thus, his ajec o y is in easible. This can be checked as
well in u∗, which ends o in ini y as F∗
mag app oaches ze o.
Then, as he mo ion du a ion inc eases, he equi emen s
a e less demanding, because e and a e a e educed.
The e o e, he maximum alues o u∗a e also educed.
A necessa y condi ion o con e gence o s= 0 is umax >
max(u∗( )). This condi ion is su icien o pe ec acking
in he ideal case, in which ε= 0. O he wise, in gene al,
a su icien condi ion o con e gence can be de i ed om
(30),
umax ≥max(u∗)+max |ε|+η
min |Bj|,(49)
whe e εis bounded, assuming ha ˜ and ˜aa e bounded,
max |ε|≤εmax(|˜ |),max(|˜a|).(50)
Some assump ions mus be made abou he bounds o
e o s ˜ and ˜a o sa is y he p e ious condi ion. As an
example, he con olle cons an s a e se as
λ1=λ2= 2000,η= 105.(51)
And, o he sake o simplici y, e y conse a i e assump-
ions a e made abou he e o bounds,
|˜ |≤0.2 max(| e |),|˜a|≤0.2 max(|a e |).(52)
Thus, om (49) and (50), he obus ness condi ion is
umax ≥39.13 V (i τmo = 4 ms), o umax ≥31.29 V (i
τmo = 5 ms). As expec ed, he condi ion is less es ic i e
when he mo ion du a ion is inc eased.
No e ha , in o de o he con olle o be obus o mod-
eling dis u bances, he model pa ame e s used o de i e
he obus ness c i e ia (49) should ep esen he wo s -case
scena io, assuming he bounds o each model pa ame e
a e known. In p ac ice, howe e , de e mining he combi-
na ion o pa ame e s ha esul s in he wo se-case scena io
may be oo cumbe some, due o he immense numbe
o possibili ies. Al e na i ely, a Mon e-Ca lo e alua ion
could be pe o med, pe mu ing all pa ame e s inside hei
bounds, and hen selec ing umax such ha (49) holds o
e e y case.
4.2 Sampling a e analysis
We ha e p o ed ha obus ness can be gua an eed unde
some easonable ope a ing condi ions. S ill, he sampling
a e may be a limi ing ac o , and i s in luence should
be analyzed. Thus, he p oposed con olle is es ed wi h
di e en sampling pe iods Ts. As e e ence, he second
posi ion ajec o y om Sec ion 4.1 is used (τmo = 4 ms).
The con olle cons an s a e se as in Sec ion 4.1, wi h
Fig. 5. Simula ion esul s. No e ha he ime axis is
no malized wi h espec o τmo .
umax = 40 V. The dynamic sys em is simula ed using he
hyb id au oma on om Fig. 2 and he model pa ame e s
om Table 1.
The impac eloci ies a e calcula ed o di e en sampling
pe iods and depic ed in Fig. 6, sepa a ing he making and
b eaking ope a ions. Wi h a sampling a e o 100 kHz,
he esul s a e e y good, specially in he making ope a-
ion. Fo la ge sampling pe iods, he esul s inc easingly
wo sen. S ill, wi h a sampling a e o only 10 kHz, he im-
pac eloci ies a e be e han he ones in a non-con olled
scena io. Fo e e ence, using a squa e ol age o 40 V and
0 V, he impac eloci ies a e −2.2 and 0.9 m/s, o he
making and b eaking ope a ions espec i ely (which a e
beyond he g aph limi s).
Fig. 7 p esen s he esul ing s a e a iables o h ee
ep esen a i e sampling pe iods. Wi h a sampling a e o
Edua do Moya-Lashe as e al. / IFAC Pape sOnLine 53-2 (2020) 6256–6261 6261
Fig. 4. Gap eluc ance and i s de i a i e wi h espec o
he gap leng h.
No e ha , i τmo = 3 ms, F∗
mag is posi i e in a small
in e al in he b eaking ope a ion (a ound /τmo = 2).
Thus, his ajec o y is in easible. This can be checked as
well in u∗, which ends o in ini y as F∗
mag app oaches ze o.
Then, as he mo ion du a ion inc eases, he equi emen s
a e less demanding, because e and a e a e educed.
The e o e, he maximum alues o u∗a e also educed.
A necessa y condi ion o con e gence o s= 0 is umax >
max(u∗( )). This condi ion is su icien o pe ec acking
in he ideal case, in which ε= 0. O he wise, in gene al,
a su icien condi ion o con e gence can be de i ed om
(30),
umax ≥max(u∗)+max |ε|+η
min |Bj|,(49)
whe e εis bounded, assuming ha ˜ and ˜aa e bounded,
max |ε|≤εmax(|˜ |),max(|˜a|).(50)
Some assump ions mus be made abou he bounds o
e o s ˜ and ˜a o sa is y he p e ious condi ion. As an
example, he con olle cons an s a e se as
λ1=λ2= 2000,η= 105.(51)
And, o he sake o simplici y, e y conse a i e assump-
ions a e made abou he e o bounds,
|˜ |≤0.2 max(| e |),|˜a|≤0.2 max(|a e |).(52)
Thus, om (49) and (50), he obus ness condi ion is
umax ≥39.13 V (i τmo = 4 ms), o umax ≥31.29 V (i
τmo = 5 ms). As expec ed, he condi ion is less es ic i e
when he mo ion du a ion is inc eased.
No e ha , in o de o he con olle o be obus o mod-
eling dis u bances, he model pa ame e s used o de i e
he obus ness c i e ia (49) should ep esen he wo s -case
scena io, assuming he bounds o each model pa ame e
a e known. In p ac ice, howe e , de e mining he combi-
na ion o pa ame e s ha esul s in he wo se-case scena io
may be oo cumbe some, due o he immense numbe
o possibili ies. Al e na i ely, a Mon e-Ca lo e alua ion
could be pe o med, pe mu ing all pa ame e s inside hei
bounds, and hen selec ing umax such ha (49) holds o
e e y case.
4.2 Sampling a e analysis
We ha e p o ed ha obus ness can be gua an eed unde
some easonable ope a ing condi ions. S ill, he sampling
a e may be a limi ing ac o , and i s in luence should
be analyzed. Thus, he p oposed con olle is es ed wi h
di e en sampling pe iods Ts. As e e ence, he second
posi ion ajec o y om Sec ion 4.1 is used (τmo = 4 ms).
The con olle cons an s a e se as in Sec ion 4.1, wi h
Fig. 5. Simula ion esul s. No e ha he ime axis is
no malized wi h espec o τmo .
umax = 40 V. The dynamic sys em is simula ed using he
hyb id au oma on om Fig. 2 and he model pa ame e s
om Table 1.
The impac eloci ies a e calcula ed o di e en sampling
pe iods and depic ed in Fig. 6, sepa a ing he making and
b eaking ope a ions. Wi h a sampling a e o 100 kHz,
he esul s a e e y good, specially in he making ope a-
ion. Fo la ge sampling pe iods, he esul s inc easingly
wo sen. S ill, wi h a sampling a e o only 10 kHz, he im-
pac eloci ies a e be e han he ones in a non-con olled
scena io. Fo e e ence, using a squa e ol age o 40 V and
0 V, he impac eloci ies a e −2.2 and 0.9 m/s, o he
making and b eaking ope a ions espec i ely (which a e
beyond he g aph limi s).
Fig. 7 p esen s he esul ing s a e a iables o h ee
ep esen a i e sampling pe iods. Wi h a sampling a e o
Fig. 6. Impac eloci ies in making (le ) and b eaking
( igh ) ope a ions, as unc ions o he sampling pe-
iod.
Fig. 7. Simula ed s a e a iables using he con olle wi h
h ee di e en sampling pe iods Ts.
1 MHz, he acking is almos pe ec . Wi h a sampling
a e o 100 kHz, he e is a sligh e o in he posi ion
(almos impe cep ible in he g aphic), bu he impac
eloci ies a e app eciably la ge . S ill, he pe o mance is
e y good. Wi h a sampling a e o 10 kHz, he esul s a e
much wo se. The high ipple o he magne ic lux is il e ed,
bu leads o signi ican acking e o s. E en hough he
posi ion e o s may seem small, he eloci y e o s and,
mo e impo an ly, he impac eloci ies a e much la ge
han in he o he cases.
5. CONCLUSIONS
We ha e add essed he so -landing con ol o single-coil
eluc ance ac ua o s, p esen ing a sliding-mode con olle
which does no use any in o ma ion abou he dynamic
sys em. We ha e also de i ed he con e gence c i e ia,
based on a gene alized dynamical model. This con olle
equi es o know he posi ion and i s de i a i es, as well
as he sign o φ. Al e na i ely, he cu en h ough he
coil can be es ic ed o nonnega i e alues. Tha way,
he magne ic lux is always nonnega i e, simpli ying he
con ol.
Due o he as dynamics, he sampling a e mus be la ge
enough o ack he p ede ined posi ion and achie e low
impac eloci ies. No e ha he posi ion e o s may be
small wi h easonably low sampling a es, bu he esul ing
impac eloci ies a e signi ican . Anyway, i a as e con ol
canno be implemen ed, he esul s a e s ill be e han in
non-con olled scena ios.
In many de ices he posi ion is no measu able in eal
ime. Thus, u he in es iga ion is equi ed conce ning
he eal- ime posi ion es ima ion om easily measu able
signals, e.g. ol age and cu en , and he e alua ion o he
con olle obus ness wi h espec o es ima ion e o s.
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