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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.2023.10.181
10.1016/j.i acol.2023.10.181 2405-8963
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2023 The Au ho s. This is an open access a icle unde he CC BY-NC-ND license
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)
Run- o-Run Adap i e Nonlinea
Feed o wa d Con ol o
Elec omechanical Swi ching De ices ⋆
Edua do Moya-Lashe as ∗Edga Rami ez-Labo eo ∗
Eloy Se ano-Seco ∗
∗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, 50018 Za agoza, Spain,
(e-mail: {emoya, ami lab, ese anoseco}@uniza .es)
Abs ac : Feed o wa d con ol can g ea ly imp o e he esponse ime and con ol accu acy o
any mecha onic sys em. Howe e , in o de o compensa e o he e ec s o modeling e o s
o dis u bances, i is impe a i e ha his ype o con ol wo ks in conjunc ion wi h some
o m o eedback. In his pape , we p esen a new adap i e eed o wa d con ol scheme o
elec omechanical sys ems in which eal- ime measu emen s o es ima es o he posi ion and
i s de i a i es a e no echnically o economically easible. This is he case, o example,
o comme cial elec omechanical swi ching de ices such as solenoid ac ua o s. Ou p oposal
consis s o wo blocks: on he one hand, a eed o wa d con olle based on di e en ial la ness
heo y; on he o he , an i e a i e adap a ion law ha exploi s he epe i i e ope a ion o hese
de ices o modi y he con olle pa ame e s cycle by cycle. As shown, his law can be ed wi h
any a ailable measu emen o he sys em, wi h he only equi emen ha i can be p ocessed
and con e ed in o an indica o o he pe o mance o any gi en ope a ion. Simula ed and
expe imen al esul s show ha ou p oposal is e ec i e in dealing wi h a long-s anding con ol
p oblem in elec omechanics: he so -landing con ol o elec omechanical swi ching de ices.
Keywo ds: Adap i e con ol, Di e en ial la ness, Elec omechanical de ices, Feed o wa d
con ol, I e a i e me hods, Mecha onic sys ems, So landing, Swi ches
1. INTRODUCTION
Feed o wa d con ol is widely used o acking applica-
ions because i signi ican ly ou pe o ms o he con ol
schemes in e ms o esponse ime and acking accu acy.
Since his ype o con ol does no equi e s a e in o -
ma ion, i is pa icula ly use ul o applica ions in which
sensing o es ima ion o he a iables o be con olled is
inaccu a e o una ailable. Speci ically, o di e en ially la
sys ems, i is possible o design a eed o wa d law ha
p o ides he con ol signal om he desi ed ou pu ajec-
o y wi hou need o sol ing any di e en ial equa ion. An
ad an ageous aspec o la ness-based eed o wa d con ol
(also known as exac eed o wa d linea iza ion) is ha i
does no su e om he well-known obus ness p oblems
o i s eedback coun e pa (exac eedback linea iza ion)
due o model pa ame e unce ain y (Hagenmeye and
Delaleau, 2003). In ecen yea s, his ype o eed o wa d
con ol has been p oposed o con olling he mo ion o
a wide ange o mecha onic sys ems, such as c ane o a-
o s (Baue e al., 2014), elec ical d i es (S umpe e al.,
⋆This wo k was suppo ed in pa ia g an s PID2021-124137OB-
I00, TED2021-130224B-I00, and CPP2021-008938, unded by
MCIN/AEI/10.13039/501100011033, by ERDF A way o making
Eu ope, and by he Eu opean Union Nex Gene a ionEU/PRTR, in
pa by g an T45 20R unded by he Go e nmen o A ag´on and
in pa by he “P og ama In es igo” unded by he Eu opean Union
Nex Gene a ionEU.
2015), elec ohyd aulic sys ems (Kim e al., 2015), quad o-
o s (G ee and Schoellig, 2018), and elec os a ic quasi-
s a ic mic oscanne s (Sch oed e e al., 2018).
Despi e he ad an ages o eed o wa d con olle s, i is well
known ha a con ol scheme based solely on a eed o wa d
e m, i.e., open loop, is qui e sensi i e o dis u bances and
modeling e o s. The e o e, i is mos usual o eed o wa d
con ol o be complemen ed by some o m o eedback, as
in he p e iously ci ed e e ences. This o cou se implies
ha he s a e o ou pu a iables can be measu ed o
es ima ed in eal ime wi h su icien accu acy.
Al e na i ely, some wo ks p opose un- o- un adap a ion
laws, which a e use ul o de ices unde epe i i e ope a-
ion. The key idea o hese app oaches is ha , ins ead o
using equen measu emen s o he s a e o ou pu (which
may no e en be a ailable), he adap a ion law makes use
o o he auxilia y a iables ela ed o he o e all con ol
pe o mance o each ope a ion. Fo example, Blanken e al.
(2017) has p esen ed ecen ly a uni ying amewo k o
un- o- un adap a ion o eed o wa d con ol based on ba-
sis unc ions. Howe e , his me hodology is no applicable
o la ness-based eed o wa d con olle s.
Ce ain mecha onic sys ems canno inco po a e he e-
qui ed senso s o eal- ime eedback con ol due o di e -
en easons, such as economic o space limi a ions. Among
hese de ices a e swi ch- ype elec omechanical de ices,
Run- o-Run Adap i e Nonlinea
Feed o wa d Con ol o
Elec omechanical Swi ching De ices ⋆
Edua do Moya-Lashe as ∗Edga Rami ez-Labo eo ∗
Eloy Se ano-Seco ∗
∗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, 50018 Za agoza, Spain,
(e-mail: {emoya, ami lab, ese anoseco}@uniza .es)
Abs ac : Feed o wa d con ol can g ea ly imp o e he esponse ime and con ol accu acy o
any mecha onic sys em. Howe e , in o de o compensa e o he e ec s o modeling e o s
o dis u bances, i is impe a i e ha his ype o con ol wo ks in conjunc ion wi h some
o m o eedback. In his pape , we p esen a new adap i e eed o wa d con ol scheme o
elec omechanical sys ems in which eal- ime measu emen s o es ima es o he posi ion and
i s de i a i es a e no echnically o economically easible. This is he case, o example,
o comme cial elec omechanical swi ching de ices such as solenoid ac ua o s. Ou p oposal
consis s o wo blocks: on he one hand, a eed o wa d con olle based on di e en ial la ness
heo y; on he o he , an i e a i e adap a ion law ha exploi s he epe i i e ope a ion o hese
de ices o modi y he con olle pa ame e s cycle by cycle. As shown, his law can be ed wi h
any a ailable measu emen o he sys em, wi h he only equi emen ha i can be p ocessed
and con e ed in o an indica o o he pe o mance o any gi en ope a ion. Simula ed and
expe imen al esul s show ha ou p oposal is e ec i e in dealing wi h a long-s anding con ol
p oblem in elec omechanics: he so -landing con ol o elec omechanical swi ching de ices.
Keywo ds: Adap i e con ol, Di e en ial la ness, Elec omechanical de ices, Feed o wa d
con ol, I e a i e me hods, Mecha onic sys ems, So landing, Swi ches
1. INTRODUCTION
Feed o wa d con ol is widely used o acking applica-
ions because i signi ican ly ou pe o ms o he con ol
schemes in e ms o esponse ime and acking accu acy.
Since his ype o con ol does no equi e s a e in o -
ma ion, i is pa icula ly use ul o applica ions in which
sensing o es ima ion o he a iables o be con olled is
inaccu a e o una ailable. Speci ically, o di e en ially la
sys ems, i is possible o design a eed o wa d law ha
p o ides he con ol signal om he desi ed ou pu ajec-
o y wi hou need o sol ing any di e en ial equa ion. An
ad an ageous aspec o la ness-based eed o wa d con ol
(also known as exac eed o wa d linea iza ion) is ha i
does no su e om he well-known obus ness p oblems
o i s eedback coun e pa (exac eedback linea iza ion)
due o model pa ame e unce ain y (Hagenmeye and
Delaleau, 2003). In ecen yea s, his ype o eed o wa d
con ol has been p oposed o con olling he mo ion o
a wide ange o mecha onic sys ems, such as c ane o a-
o s (Baue e al., 2014), elec ical d i es (S umpe e al.,
⋆This wo k was suppo ed in pa ia g an s PID2021-124137OB-
I00, TED2021-130224B-I00, and CPP2021-008938, unded by
MCIN/AEI/10.13039/501100011033, by ERDF A way o making
Eu ope, and by he Eu opean Union Nex Gene a ionEU/PRTR, in
pa by g an T45 20R unded by he Go e nmen o A ag´on and
in pa by he “P og ama In es igo” unded by he Eu opean Union
Nex Gene a ionEU.
2015), elec ohyd aulic sys ems (Kim e al., 2015), quad o-
o s (G ee and Schoellig, 2018), and elec os a ic quasi-
s a ic mic oscanne s (Sch oed e e al., 2018).
Despi e he ad an ages o eed o wa d con olle s, i is well
known ha a con ol scheme based solely on a eed o wa d
e m, i.e., open loop, is qui e sensi i e o dis u bances and
modeling e o s. The e o e, i is mos usual o eed o wa d
con ol o be complemen ed by some o m o eedback, as
in he p e iously ci ed e e ences. This o cou se implies
ha he s a e o ou pu a iables can be measu ed o
es ima ed in eal ime wi h su icien accu acy.
Al e na i ely, some wo ks p opose un- o- un adap a ion
laws, which a e use ul o de ices unde epe i i e ope a-
ion. The key idea o hese app oaches is ha , ins ead o
using equen measu emen s o he s a e o ou pu (which
may no e en be a ailable), he adap a ion law makes use
o o he auxilia y a iables ela ed o he o e all con ol
pe o mance o each ope a ion. Fo example, Blanken e al.
(2017) has p esen ed ecen ly a uni ying amewo k o
un- o- un adap a ion o eed o wa d con ol based on ba-
sis unc ions. Howe e , his me hodology is no applicable
o la ness-based eed o wa d con olle s.
Ce ain mecha onic sys ems canno inco po a e he e-
qui ed senso s o eal- ime eedback con ol due o di e -
en easons, such as economic o space limi a ions. Among
hese de ices a e swi ch- ype elec omechanical de ices,
Run- o-Run Adap i e Nonlinea
Feed o wa d Con ol o
Elec omechanical Swi ching De ices ⋆
Edua do Moya-Lashe as ∗Edga Rami ez-Labo eo ∗
Eloy Se ano-Seco ∗
∗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, 50018 Za agoza, Spain,
(e-mail: {emoya, ami lab, ese anoseco}@uniza .es)
Abs ac : Feed o wa d con ol can g ea ly imp o e he esponse ime and con ol accu acy o
any mecha onic sys em. Howe e , in o de o compensa e o he e ec s o modeling e o s
o dis u bances, i is impe a i e ha his ype o con ol wo ks in conjunc ion wi h some
o m o eedback. In his pape , we p esen a new adap i e eed o wa d con ol scheme o
elec omechanical sys ems in which eal- ime measu emen s o es ima es o he posi ion and
i s de i a i es a e no echnically o economically easible. This is he case, o example,
o comme cial elec omechanical swi ching de ices such as solenoid ac ua o s. Ou p oposal
consis s o wo blocks: on he one hand, a eed o wa d con olle based on di e en ial la ness
heo y; on he o he , an i e a i e adap a ion law ha exploi s he epe i i e ope a ion o hese
de ices o modi y he con olle pa ame e s cycle by cycle. As shown, his law can be ed wi h
any a ailable measu emen o he sys em, wi h he only equi emen ha i can be p ocessed
and con e ed in o an indica o o he pe o mance o any gi en ope a ion. Simula ed and
expe imen al esul s show ha ou p oposal is e ec i e in dealing wi h a long-s anding con ol
p oblem in elec omechanics: he so -landing con ol o elec omechanical swi ching de ices.
Keywo ds: Adap i e con ol, Di e en ial la ness, Elec omechanical de ices, Feed o wa d
con ol, I e a i e me hods, Mecha onic sys ems, So landing, Swi ches
1. INTRODUCTION
Feed o wa d con ol is widely used o acking applica-
ions because i signi ican ly ou pe o ms o he con ol
schemes in e ms o esponse ime and acking accu acy.
Since his ype o con ol does no equi e s a e in o -
ma ion, i is pa icula ly use ul o applica ions in which
sensing o es ima ion o he a iables o be con olled is
inaccu a e o una ailable. Speci ically, o di e en ially la
sys ems, i is possible o design a eed o wa d law ha
p o ides he con ol signal om he desi ed ou pu ajec-
o y wi hou need o sol ing any di e en ial equa ion. An
ad an ageous aspec o la ness-based eed o wa d con ol
(also known as exac eed o wa d linea iza ion) is ha i
does no su e om he well-known obus ness p oblems
o i s eedback coun e pa (exac eedback linea iza ion)
due o model pa ame e unce ain y (Hagenmeye and
Delaleau, 2003). In ecen yea s, his ype o eed o wa d
con ol has been p oposed o con olling he mo ion o
a wide ange o mecha onic sys ems, such as c ane o a-
o s (Baue e al., 2014), elec ical d i es (S umpe e al.,
⋆This wo k was suppo ed in pa ia g an s PID2021-124137OB-
I00, TED2021-130224B-I00, and CPP2021-008938, unded by
MCIN/AEI/10.13039/501100011033, by ERDF A way o making
Eu ope, and by he Eu opean Union Nex Gene a ionEU/PRTR, in
pa by g an T45 20R unded by he Go e nmen o A ag´on and
in pa by he “P og ama In es igo” unded by he Eu opean Union
Nex Gene a ionEU.
2015), elec ohyd aulic sys ems (Kim e al., 2015), quad o-
o s (G ee and Schoellig, 2018), and elec os a ic quasi-
s a ic mic oscanne s (Sch oed e e al., 2018).
Despi e he ad an ages o eed o wa d con olle s, i is well
known ha a con ol scheme based solely on a eed o wa d
e m, i.e., open loop, is qui e sensi i e o dis u bances and
modeling e o s. The e o e, i is mos usual o eed o wa d
con ol o be complemen ed by some o m o eedback, as
in he p e iously ci ed e e ences. This o cou se implies
ha he s a e o ou pu a iables can be measu ed o
es ima ed in eal ime wi h su icien accu acy.
Al e na i ely, some wo ks p opose un- o- un adap a ion
laws, which a e use ul o de ices unde epe i i e ope a-
ion. The key idea o hese app oaches is ha , ins ead o
using equen measu emen s o he s a e o ou pu (which
may no e en be a ailable), he adap a ion law makes use
o o he auxilia y a iables ela ed o he o e all con ol
pe o mance o each ope a ion. Fo example, Blanken e al.
(2017) has p esen ed ecen ly a uni ying amewo k o
un- o- un adap a ion o eed o wa d con ol based on ba-
sis unc ions. Howe e , his me hodology is no applicable
o la ness-based eed o wa d con olle s.
Ce ain mecha onic sys ems canno inco po a e he e-
qui ed senso s o eal- ime eedback con ol due o di e -
en easons, such as economic o space limi a ions. Among
hese de ices a e swi ch- ype elec omechanical de ices,
Run- o-Run Adap i e Nonlinea
Feed o wa d Con ol o
Elec omechanical Swi ching De ices ⋆
Edua do Moya-Lashe as ∗Edga Rami ez-Labo eo ∗
Eloy Se ano-Seco ∗
∗
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, 50018 Za agoza, Spain,
(e-mail:
{
emoya, ami lab, ese anoseco
}
@uniza .es)
Abs ac : Feed o wa d con ol can g ea ly imp o e he esponse ime and con ol accu acy o
any mecha onic sys em. Howe e , in o de o compensa e o he e ec s o modeling e o s
o dis u bances, i is impe a i e ha his ype o con ol wo ks in conjunc ion wi h some
o m o eedback. In his pape , we p esen a new adap i e eed o wa d con ol scheme o
elec omechanical sys ems in which eal- ime measu emen s o es ima es o he posi ion and
i s de i a i es a e no echnically o economically easible. This is he case, o example,
o comme cial elec omechanical swi ching de ices such as solenoid ac ua o s. Ou p oposal
consis s o wo blocks: on he one hand, a eed o wa d con olle based on di e en ial la ness
heo y; on he o he , an i e a i e adap a ion law ha exploi s he epe i i e ope a ion o hese
de ices o modi y he con olle pa ame e s cycle by cycle. As shown, his law can be ed wi h
any a ailable measu emen o he sys em, wi h he only equi emen ha i can be p ocessed
and con e ed in o an indica o o he pe o mance o any gi en ope a ion. Simula ed and
expe imen al esul s show ha ou p oposal is e ec i e in dealing wi h a long-s anding con ol
p oblem in elec omechanics: he so -landing con ol o elec omechanical swi ching de ices.
Keywo ds: Adap i e con ol, Di e en ial la ness, Elec omechanical de ices, Feed o wa d
con ol, I e a i e me hods, Mecha onic sys ems, So landing, Swi ches
1. INTRODUCTION
Feed o wa d con ol is widely used o acking applica-
ions because i signi ican ly ou pe o ms o he con ol
schemes in e ms o esponse ime and acking accu acy.
Since his ype o con ol does no equi e s a e in o -
ma ion, i is pa icula ly use ul o applica ions in which
sensing o es ima ion o he a iables o be con olled is
inaccu a e o una ailable. Speci ically, o di e en ially la
sys ems, i is possible o design a eed o wa d law ha
p o ides he con ol signal om he desi ed ou pu ajec-
o y wi hou need o sol ing any di e en ial equa ion. An
ad an ageous aspec o la ness-based eed o wa d con ol
(also known as exac eed o wa d linea iza ion) is ha i
does no su e om he well-known obus ness p oblems
o i s eedback coun e pa (exac eedback linea iza ion)
due o model pa ame e unce ain y (Hagenmeye and
Delaleau, 2003). In ecen yea s, his ype o eed o wa d
con ol has been p oposed o con olling he mo ion o
a wide ange o mecha onic sys ems, such as c ane o a-
o s (Baue e al., 2014), elec ical d i es (S umpe e al.,
⋆This wo k was suppo ed in pa ia g an s PID2021-124137OB-
I00, TED2021-130224B-I00, and CPP2021-008938, unded by
MCIN/AEI/10.13039/501100011033, by ERDF A way o making
Eu ope, and by he Eu opean Union Nex Gene a ionEU/PRTR, in
pa by g an T45 20R unded by he Go e nmen o A ag´on and
in pa by he “P og ama In es igo” unded by he Eu opean Union
Nex Gene a ionEU.
2015), elec ohyd aulic sys ems (Kim e al., 2015), quad o-
o s (G ee and Schoellig, 2018), and elec os a ic quasi-
s a ic mic oscanne s (Sch oed e e al., 2018).
Despi e he ad an ages o eed o wa d con olle s, i is well
known ha a con ol scheme based solely on a eed o wa d
e m, i.e., open loop, is qui e sensi i e o dis u bances and
modeling e o s. The e o e, i is mos usual o eed o wa d
con ol o be complemen ed by some o m o eedback, as
in he p e iously ci ed e e ences. This o cou se implies
ha he s a e o ou pu a iables can be measu ed o
es ima ed in eal ime wi h su icien accu acy.
Al e na i ely, some wo ks p opose un- o- un adap a ion
laws, which a e use ul o de ices unde epe i i e ope a-
ion. The key idea o hese app oaches is ha , ins ead o
using equen measu emen s o he s a e o ou pu (which
may no e en be a ailable), he adap a ion law makes use
o o he auxilia y a iables ela ed o he o e all con ol
pe o mance o each ope a ion. Fo example, Blanken e al.
(2017) has p esen ed ecen ly a uni ying amewo k o
un- o- un adap a ion o eed o wa d con ol based on ba-
sis unc ions. Howe e , his me hodology is no applicable
o la ness-based eed o wa d con olle s.
Ce ain mecha onic sys ems canno inco po a e he e-
qui ed senso s o eal- ime eedback con ol due o di e -
en easons, such as economic o space limi a ions. Among
hese de ices a e swi ch- ype elec omechanical de ices,
Run- o-Run Adap i e Nonlinea
Feed o wa d Con ol o
Elec omechanical Swi ching De ices ⋆
Edua do Moya-Lashe as
∗
Edga Rami ez-Labo eo
∗
Eloy Se ano-Seco ∗
∗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, 50018 Za agoza, Spain,
(e-mail: {emoya, ami lab, ese anoseco}@uniza .es)
Abs ac : Feed o wa d con ol can g ea ly imp o e he esponse ime and con ol accu acy o
any mecha onic sys em. Howe e , in o de o compensa e o he e ec s o modeling e o s
o dis u bances, i is impe a i e ha his ype o con ol wo ks in conjunc ion wi h some
o m o eedback. In his pape , we p esen a new adap i e eed o wa d con ol scheme o
elec omechanical sys ems in which eal- ime measu emen s o es ima es o he posi ion and
i s de i a i es a e no echnically o economically easible. This is he case, o example,
o comme cial elec omechanical swi ching de ices such as solenoid ac ua o s. Ou p oposal
consis s o wo blocks: on he one hand, a eed o wa d con olle based on di e en ial la ness
heo y; on he o he , an i e a i e adap a ion law ha exploi s he epe i i e ope a ion o hese
de ices o modi y he con olle pa ame e s cycle by cycle. As shown, his law can be ed wi h
any a ailable measu emen o he sys em, wi h he only equi emen ha i can be p ocessed
and con e ed in o an indica o o he pe o mance o any gi en ope a ion. Simula ed and
expe imen al esul s show ha ou p oposal is e ec i e in dealing wi h a long-s anding con ol
p oblem in elec omechanics: he so -landing con ol o elec omechanical swi ching de ices.
Keywo ds: Adap i e con ol, Di e en ial la ness, Elec omechanical de ices, Feed o wa d
con ol, I e a i e me hods, Mecha onic sys ems, So landing, Swi ches
1. INTRODUCTION
Feed o wa d con ol is widely used o acking applica-
ions because i signi ican ly ou pe o ms o he con ol
schemes in e ms o esponse ime and acking accu acy.
Since his ype o con ol does no equi e s a e in o -
ma ion, i is pa icula ly use ul o applica ions in which
sensing o es ima ion o he a iables o be con olled is
inaccu a e o una ailable. Speci ically, o di e en ially la
sys ems, i is possible o design a eed o wa d law ha
p o ides he con ol signal om he desi ed ou pu ajec-
o y wi hou need o sol ing any di e en ial equa ion. An
ad an ageous aspec o la ness-based eed o wa d con ol
(also known as exac eed o wa d linea iza ion) is ha i
does no su e om he well-known obus ness p oblems
o i s eedback coun e pa (exac eedback linea iza ion)
due o model pa ame e unce ain y (Hagenmeye and
Delaleau, 2003). In ecen yea s, his ype o eed o wa d
con ol has been p oposed o con olling he mo ion o
a wide ange o mecha onic sys ems, such as c ane o a-
o s (Baue e al., 2014), elec ical d i es (S umpe e al.,
⋆This wo k was suppo ed in pa ia g an s PID2021-124137OB-
I00, TED2021-130224B-I00, and CPP2021-008938, unded by
MCIN/AEI/10.13039/501100011033, by ERDF A way o making
Eu ope, and by he Eu opean Union Nex Gene a ionEU/PRTR, in
pa by g an T45 20R unded by he Go e nmen o A ag´on and
in pa by he “P og ama In es igo” unded by he Eu opean Union
Nex Gene a ionEU.
2015), elec ohyd aulic sys ems (Kim e al., 2015), quad o-
o s (G ee and Schoellig, 2018), and elec os a ic quasi-
s a ic mic oscanne s (Sch oed e e al., 2018).
Despi e he ad an ages o eed o wa d con olle s, i is well
known ha a con ol scheme based solely on a eed o wa d
e m, i.e., open loop, is qui e sensi i e o dis u bances and
modeling e o s. The e o e, i is mos usual o eed o wa d
con ol o be complemen ed by some o m o eedback, as
in he p e iously ci ed e e ences. This o cou se implies
ha he s a e o ou pu a iables can be measu ed o
es ima ed in eal ime wi h su icien accu acy.
Al e na i ely, some wo ks p opose un- o- un adap a ion
laws, which a e use ul o de ices unde epe i i e ope a-
ion. The key idea o hese app oaches is ha , ins ead o
using equen measu emen s o he s a e o ou pu (which
may no e en be a ailable), he adap a ion law makes use
o o he auxilia y a iables ela ed o he o e all con ol
pe o mance o each ope a ion. Fo example, Blanken e al.
(2017) has p esen ed ecen ly a uni ying amewo k o
un- o- un adap a ion o eed o wa d con ol based on ba-
sis unc ions. Howe e , his me hodology is no applicable
o la ness-based eed o wa d con olle s.
Ce ain mecha onic sys ems canno inco po a e he e-
qui ed senso s o eal- ime eedback con ol due o di e -
en easons, such as economic o space limi a ions. Among
hese de ices a e swi ch- ype elec omechanical de ices,
Run- o-Run Adap i e Nonlinea
Feed o wa d Con ol o
Elec omechanical Swi ching De ices ⋆
Edua do Moya-Lashe as ∗Edga Rami ez-Labo eo ∗
Eloy Se ano-Seco ∗
∗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, 50018 Za agoza, Spain,
(e-mail: {emoya, ami lab, ese anoseco}@uniza .es)
Abs ac : Feed o wa d con ol can g ea ly imp o e he esponse ime and con ol accu acy o
any mecha onic sys em. Howe e , in o de o compensa e o he e ec s o modeling e o s
o dis u bances, i is impe a i e ha his ype o con ol wo ks in conjunc ion wi h some
o m o eedback. In his pape , we p esen a new adap i e eed o wa d con ol scheme o
elec omechanical sys ems in which eal- ime measu emen s o es ima es o he posi ion and
i s de i a i es a e no echnically o economically easible. This is he case, o example,
o comme cial elec omechanical swi ching de ices such as solenoid ac ua o s. Ou p oposal
consis s o wo blocks: on he one hand, a eed o wa d con olle based on di e en ial la ness
heo y; on he o he , an i e a i e adap a ion law ha exploi s he epe i i e ope a ion o hese
de ices o modi y he con olle pa ame e s cycle by cycle. As shown, his law can be ed wi h
any a ailable measu emen o he sys em, wi h he only equi emen ha i can be p ocessed
and con e ed in o an indica o o he pe o mance o any gi en ope a ion. Simula ed and
expe imen al esul s show ha ou p oposal is e ec i e in dealing wi h a long-s anding con ol
p oblem in elec omechanics: he so -landing con ol o elec omechanical swi ching de ices.
Keywo ds: Adap i e con ol, Di e en ial la ness, Elec omechanical de ices, Feed o wa d
con ol, I e a i e me hods, Mecha onic sys ems, So landing, Swi ches
1. INTRODUCTION
Feed o wa d con ol is widely used o acking applica-
ions because i signi ican ly ou pe o ms o he con ol
schemes in e ms o esponse ime and acking accu acy.
Since his ype o con ol does no equi e s a e in o -
ma ion, i is pa icula ly use ul o applica ions in which
sensing o es ima ion o he a iables o be con olled is
inaccu a e o una ailable. Speci ically, o di e en ially la
sys ems, i is possible o design a eed o wa d law ha
p o ides he con ol signal om he desi ed ou pu ajec-
o y wi hou need o sol ing any di e en ial equa ion. An
ad an ageous aspec o la ness-based eed o wa d con ol
(also known as exac eed o wa d linea iza ion) is ha i
does no su e om he well-known obus ness p oblems
o i s eedback coun e pa (exac eedback linea iza ion)
due o model pa ame e unce ain y (Hagenmeye and
Delaleau, 2003). In ecen yea s, his ype o eed o wa d
con ol has been p oposed o con olling he mo ion o
a wide ange o mecha onic sys ems, such as c ane o a-
o s (Baue e al., 2014), elec ical d i es (S umpe e al.,
⋆This wo k was suppo ed in pa ia g an s PID2021-124137OB-
I00, TED2021-130224B-I00, and CPP2021-008938, unded by
MCIN/AEI/10.13039/501100011033, by ERDF A way o making
Eu ope, and by he Eu opean Union Nex Gene a ionEU/PRTR, in
pa by g an T45 20R unded by he Go e nmen o A ag´on and
in pa by he “P og ama In es igo” unded by he Eu opean Union
Nex Gene a ionEU.
2015), elec ohyd aulic sys ems (Kim e al., 2015), quad o-
o s (G ee and Schoellig, 2018), and elec os a ic quasi-
s a ic mic oscanne s (Sch oed e e al., 2018).
Despi e he ad an ages o eed o wa d con olle s, i is well
known ha a con ol scheme based solely on a eed o wa d
e m, i.e., open loop, is qui e sensi i e o dis u bances and
modeling e o s. The e o e, i is mos usual o eed o wa d
con ol o be complemen ed by some o m o eedback, as
in he p e iously ci ed e e ences. This o cou se implies
ha he s a e o ou pu a iables can be measu ed o
es ima ed in eal ime wi h su icien accu acy.
Al e na i ely, some wo ks p opose un- o- un adap a ion
laws, which a e use ul o de ices unde epe i i e ope a-
ion. The key idea o hese app oaches is ha , ins ead o
using equen measu emen s o he s a e o ou pu (which
may no e en be a ailable), he adap a ion law makes use
o o he auxilia y a iables ela ed o he o e all con ol
pe o mance o each ope a ion. Fo example, Blanken e al.
(2017) has p esen ed ecen ly a uni ying amewo k o
un- o- un adap a ion o eed o wa d con ol based on ba-
sis unc ions. Howe e , his me hodology is no applicable
o la ness-based eed o wa d con olle s.
Ce ain mecha onic sys ems canno inco po a e he e-
qui ed senso s o eal- ime eedback con ol due o di e -
en easons, such as economic o space limi a ions. Among
hese de ices a e swi ch- ype elec omechanical de ices,
Edua do Moya-Lashe as e al. / IFAC Pape sOnLine 56-2 (2023) 5358–5363 5359
Copy igh ©
2023 The Au ho s. This is an open access a icle unde he CC BY-NC-ND license
(
h ps://c ea i ecommons.o g/licenses/by-nc-nd/4.0/
)
Run- o-Run Adap i e Nonlinea
Feed o wa d Con ol o
Elec omechanical Swi ching De ices ⋆
Edua do Moya-Lashe as ∗Edga Rami ez-Labo eo ∗
Eloy Se ano-Seco ∗
∗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, 50018 Za agoza, Spain,
(e-mail: {emoya, ami lab, ese anoseco}@uniza .es)
Abs ac : Feed o wa d con ol can g ea ly imp o e he esponse ime and con ol accu acy o
any mecha onic sys em. Howe e , in o de o compensa e o he e ec s o modeling e o s
o dis u bances, i is impe a i e ha his ype o con ol wo ks in conjunc ion wi h some
o m o eedback. In his pape , we p esen a new adap i e eed o wa d con ol scheme o
elec omechanical sys ems in which eal- ime measu emen s o es ima es o he posi ion and
i s de i a i es a e no echnically o economically easible. This is he case, o example,
o comme cial elec omechanical swi ching de ices such as solenoid ac ua o s. Ou p oposal
consis s o wo blocks: on he one hand, a eed o wa d con olle based on di e en ial la ness
heo y; on he o he , an i e a i e adap a ion law ha exploi s he epe i i e ope a ion o hese
de ices o modi y he con olle pa ame e s cycle by cycle. As shown, his law can be ed wi h
any a ailable measu emen o he sys em, wi h he only equi emen ha i can be p ocessed
and con e ed in o an indica o o he pe o mance o any gi en ope a ion. Simula ed and
expe imen al esul s show ha ou p oposal is e ec i e in dealing wi h a long-s anding con ol
p oblem in elec omechanics: he so -landing con ol o elec omechanical swi ching de ices.
Keywo ds: Adap i e con ol, Di e en ial la ness, Elec omechanical de ices, Feed o wa d
con ol, I e a i e me hods, Mecha onic sys ems, So landing, Swi ches
1. INTRODUCTION
Feed o wa d con ol is widely used o acking applica-
ions because i signi ican ly ou pe o ms o he con ol
schemes in e ms o esponse ime and acking accu acy.
Since his ype o con ol does no equi e s a e in o -
ma ion, i is pa icula ly use ul o applica ions in which
sensing o es ima ion o he a iables o be con olled is
inaccu a e o una ailable. Speci ically, o di e en ially la
sys ems, i is possible o design a eed o wa d law ha
p o ides he con ol signal om he desi ed ou pu ajec-
o y wi hou need o sol ing any di e en ial equa ion. An
ad an ageous aspec o la ness-based eed o wa d con ol
(also known as exac eed o wa d linea iza ion) is ha i
does no su e om he well-known obus ness p oblems
o i s eedback coun e pa (exac eedback linea iza ion)
due o model pa ame e unce ain y (Hagenmeye and
Delaleau, 2003). In ecen yea s, his ype o eed o wa d
con ol has been p oposed o con olling he mo ion o
a wide ange o mecha onic sys ems, such as c ane o a-
o s (Baue e al., 2014), elec ical d i es (S umpe e al.,
⋆This wo k was suppo ed in pa ia g an s PID2021-124137OB-
I00, TED2021-130224B-I00, and CPP2021-008938, unded by
MCIN/AEI/10.13039/501100011033, by ERDF A way o making
Eu ope, and by he Eu opean Union Nex Gene a ionEU/PRTR, in
pa by g an T45 20R unded by he Go e nmen o A ag´on and
in pa by he “P og ama In es igo” unded by he Eu opean Union
Nex Gene a ionEU.
2015), elec ohyd aulic sys ems (Kim e al., 2015), quad o-
o s (G ee and Schoellig, 2018), and elec os a ic quasi-
s a ic mic oscanne s (Sch oed e e al., 2018).
Despi e he ad an ages o eed o wa d con olle s, i is well
known ha a con ol scheme based solely on a eed o wa d
e m, i.e., open loop, is qui e sensi i e o dis u bances and
modeling e o s. The e o e, i is mos usual o eed o wa d
con ol o be complemen ed by some o m o eedback, as
in he p e iously ci ed e e ences. This o cou se implies
ha he s a e o ou pu a iables can be measu ed o
es ima ed in eal ime wi h su icien accu acy.
Al e na i ely, some wo ks p opose un- o- un adap a ion
laws, which a e use ul o de ices unde epe i i e ope a-
ion. The key idea o hese app oaches is ha , ins ead o
using equen measu emen s o he s a e o ou pu (which
may no e en be a ailable), he adap a ion law makes use
o o he auxilia y a iables ela ed o he o e all con ol
pe o mance o each ope a ion. Fo example, Blanken e al.
(2017) has p esen ed ecen ly a uni ying amewo k o
un- o- un adap a ion o eed o wa d con ol based on ba-
sis unc ions. Howe e , his me hodology is no applicable
o la ness-based eed o wa d con olle s.
Ce ain mecha onic sys ems canno inco po a e he e-
qui ed senso s o eal- ime eedback con ol due o di e -
en easons, such as economic o space limi a ions. Among
hese de ices a e swi ch- ype elec omechanical de ices,
Run- o-Run Adap i e Nonlinea
Feed o wa d Con ol o
Elec omechanical Swi ching De ices ⋆
Edua do Moya-Lashe as ∗Edga Rami ez-Labo eo ∗
Eloy Se ano-Seco ∗
∗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, 50018 Za agoza, Spain,
(e-mail: {emoya, ami lab, ese anoseco}@uniza .es)
Abs ac : Feed o wa d con ol can g ea ly imp o e he esponse ime and con ol accu acy o
any mecha onic sys em. Howe e , in o de o compensa e o he e ec s o modeling e o s
o dis u bances, i is impe a i e ha his ype o con ol wo ks in conjunc ion wi h some
o m o eedback. In his pape , we p esen a new adap i e eed o wa d con ol scheme o
elec omechanical sys ems in which eal- ime measu emen s o es ima es o he posi ion and
i s de i a i es a e no echnically o economically easible. This is he case, o example,
o comme cial elec omechanical swi ching de ices such as solenoid ac ua o s. Ou p oposal
consis s o wo blocks: on he one hand, a eed o wa d con olle based on di e en ial la ness
heo y; on he o he , an i e a i e adap a ion law ha exploi s he epe i i e ope a ion o hese
de ices o modi y he con olle pa ame e s cycle by cycle. As shown, his law can be ed wi h
any a ailable measu emen o he sys em, wi h he only equi emen ha i can be p ocessed
and con e ed in o an indica o o he pe o mance o any gi en ope a ion. Simula ed and
expe imen al esul s show ha ou p oposal is e ec i e in dealing wi h a long-s anding con ol
p oblem in elec omechanics: he so -landing con ol o elec omechanical swi ching de ices.
Keywo ds: Adap i e con ol, Di e en ial la ness, Elec omechanical de ices, Feed o wa d
con ol, I e a i e me hods, Mecha onic sys ems, So landing, Swi ches
1. INTRODUCTION
Feed o wa d con ol is widely used o acking applica-
ions because i signi ican ly ou pe o ms o he con ol
schemes in e ms o esponse ime and acking accu acy.
Since his ype o con ol does no equi e s a e in o -
ma ion, i is pa icula ly use ul o applica ions in which
sensing o es ima ion o he a iables o be con olled is
inaccu a e o una ailable. Speci ically, o di e en ially la
sys ems, i is possible o design a eed o wa d law ha
p o ides he con ol signal om he desi ed ou pu ajec-
o y wi hou need o sol ing any di e en ial equa ion. An
ad an ageous aspec o la ness-based eed o wa d con ol
(also known as exac eed o wa d linea iza ion) is ha i
does no su e om he well-known obus ness p oblems
o i s eedback coun e pa (exac eedback linea iza ion)
due o model pa ame e unce ain y (Hagenmeye and
Delaleau, 2003). In ecen yea s, his ype o eed o wa d
con ol has been p oposed o con olling he mo ion o
a wide ange o mecha onic sys ems, such as c ane o a-
o s (Baue e al., 2014), elec ical d i es (S umpe e al.,
⋆This wo k was suppo ed in pa ia g an s PID2021-124137OB-
I00, TED2021-130224B-I00, and CPP2021-008938, unded by
MCIN/AEI/10.13039/501100011033, by ERDF A way o making
Eu ope, and by he Eu opean Union Nex Gene a ionEU/PRTR, in
pa by g an T45 20R unded by he Go e nmen o A ag´on and
in pa by he “P og ama In es igo” unded by he Eu opean Union
Nex Gene a ionEU.
2015), elec ohyd aulic sys ems (Kim e al., 2015), quad o-
o s (G ee and Schoellig, 2018), and elec os a ic quasi-
s a ic mic oscanne s (Sch oed e e al., 2018).
Despi e he ad an ages o eed o wa d con olle s, i is well
known ha a con ol scheme based solely on a eed o wa d
e m, i.e., open loop, is qui e sensi i e o dis u bances and
modeling e o s. The e o e, i is mos usual o eed o wa d
con ol o be complemen ed by some o m o eedback, as
in he p e iously ci ed e e ences. This o cou se implies
ha he s a e o ou pu a iables can be measu ed o
es ima ed in eal ime wi h su icien accu acy.
Al e na i ely, some wo ks p opose un- o- un adap a ion
laws, which a e use ul o de ices unde epe i i e ope a-
ion. The key idea o hese app oaches is ha , ins ead o
using equen measu emen s o he s a e o ou pu (which
may no e en be a ailable), he adap a ion law makes use
o o he auxilia y a iables ela ed o he o e all con ol
pe o mance o each ope a ion. Fo example, Blanken e al.
(2017) has p esen ed ecen ly a uni ying amewo k o
un- o- un adap a ion o eed o wa d con ol based on ba-
sis unc ions. Howe e , his me hodology is no applicable
o la ness-based eed o wa d con olle s.
Ce ain mecha onic sys ems canno inco po a e he e-
qui ed senso s o eal- ime eedback con ol due o di e -
en easons, such as economic o space limi a ions. Among
hese de ices a e swi ch- ype elec omechanical de ices,
Run- o-Run Adap i e Nonlinea
Feed o wa d Con ol o
Elec omechanical Swi ching De ices ⋆
Edua do Moya-Lashe as ∗Edga Rami ez-Labo eo ∗
Eloy Se ano-Seco ∗
∗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, 50018 Za agoza, Spain,
(e-mail: {emoya, ami lab, ese anoseco}@uniza .es)
Abs ac : Feed o wa d con ol can g ea ly imp o e he esponse ime and con ol accu acy o
any mecha onic sys em. Howe e , in o de o compensa e o he e ec s o modeling e o s
o dis u bances, i is impe a i e ha his ype o con ol wo ks in conjunc ion wi h some
o m o eedback. In his pape , we p esen a new adap i e eed o wa d con ol scheme o
elec omechanical sys ems in which eal- ime measu emen s o es ima es o he posi ion and
i s de i a i es a e no echnically o economically easible. This is he case, o example,
o comme cial elec omechanical swi ching de ices such as solenoid ac ua o s. Ou p oposal
consis s o wo blocks: on he one hand, a eed o wa d con olle based on di e en ial la ness
heo y; on he o he , an i e a i e adap a ion law ha exploi s he epe i i e ope a ion o hese
de ices o modi y he con olle pa ame e s cycle by cycle. As shown, his law can be ed wi h
any a ailable measu emen o he sys em, wi h he only equi emen ha i can be p ocessed
and con e ed in o an indica o o he pe o mance o any gi en ope a ion. Simula ed and
expe imen al esul s show ha ou p oposal is e ec i e in dealing wi h a long-s anding con ol
p oblem in elec omechanics: he so -landing con ol o elec omechanical swi ching de ices.
Keywo ds: Adap i e con ol, Di e en ial la ness, Elec omechanical de ices, Feed o wa d
con ol, I e a i e me hods, Mecha onic sys ems, So landing, Swi ches
1. INTRODUCTION
Feed o wa d con ol is widely used o acking applica-
ions because i signi ican ly ou pe o ms o he con ol
schemes in e ms o esponse ime and acking accu acy.
Since his ype o con ol does no equi e s a e in o -
ma ion, i is pa icula ly use ul o applica ions in which
sensing o es ima ion o he a iables o be con olled is
inaccu a e o una ailable. Speci ically, o di e en ially la
sys ems, i is possible o design a eed o wa d law ha
p o ides he con ol signal om he desi ed ou pu ajec-
o y wi hou need o sol ing any di e en ial equa ion. An
ad an ageous aspec o la ness-based eed o wa d con ol
(also known as exac eed o wa d linea iza ion) is ha i
does no su e om he well-known obus ness p oblems
o i s eedback coun e pa (exac eedback linea iza ion)
due o model pa ame e unce ain y (Hagenmeye and
Delaleau, 2003). In ecen yea s, his ype o eed o wa d
con ol has been p oposed o con olling he mo ion o
a wide ange o mecha onic sys ems, such as c ane o a-
o s (Baue e al., 2014), elec ical d i es (S umpe e al.,
⋆This wo k was suppo ed in pa ia g an s PID2021-124137OB-
I00, TED2021-130224B-I00, and CPP2021-008938, unded by
MCIN/AEI/10.13039/501100011033, by ERDF A way o making
Eu ope, and by he Eu opean Union Nex Gene a ionEU/PRTR, in
pa by g an T45 20R unded by he Go e nmen o A ag´on and
in pa by he “P og ama In es igo” unded by he Eu opean Union
Nex Gene a ionEU.
2015), elec ohyd aulic sys ems (Kim e al., 2015), quad o-
o s (G ee and Schoellig, 2018), and elec os a ic quasi-
s a ic mic oscanne s (Sch oed e e al., 2018).
Despi e he ad an ages o eed o wa d con olle s, i is well
known ha a con ol scheme based solely on a eed o wa d
e m, i.e., open loop, is qui e sensi i e o dis u bances and
modeling e o s. The e o e, i is mos usual o eed o wa d
con ol o be complemen ed by some o m o eedback, as
in he p e iously ci ed e e ences. This o cou se implies
ha he s a e o ou pu a iables can be measu ed o
es ima ed in eal ime wi h su icien accu acy.
Al e na i ely, some wo ks p opose un- o- un adap a ion
laws, which a e use ul o de ices unde epe i i e ope a-
ion. The key idea o hese app oaches is ha , ins ead o
using equen measu emen s o he s a e o ou pu (which
may no e en be a ailable), he adap a ion law makes use
o o he auxilia y a iables ela ed o he o e all con ol
pe o mance o each ope a ion. Fo example, Blanken e al.
(2017) has p esen ed ecen ly a uni ying amewo k o
un- o- un adap a ion o eed o wa d con ol based on ba-
sis unc ions. Howe e , his me hodology is no applicable
o la ness-based eed o wa d con olle s.
Ce ain mecha onic sys ems canno inco po a e he e-
qui ed senso s o eal- ime eedback con ol due o di e -
en easons, such as economic o space limi a ions. Among
hese de ices a e swi ch- ype elec omechanical de ices,
Run- o-Run Adap i e Nonlinea
Feed o wa d Con ol o
Elec omechanical Swi ching De ices ⋆
Edua do Moya-Lashe as ∗Edga Rami ez-Labo eo ∗
Eloy Se ano-Seco ∗
∗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, 50018 Za agoza, Spain,
(e-mail: {emoya, ami lab, ese anoseco}@uniza .es)
Abs ac : Feed o wa d con ol can g ea ly imp o e he esponse ime and con ol accu acy o
any mecha onic sys em. Howe e , in o de o compensa e o he e ec s o modeling e o s
o dis u bances, i is impe a i e ha his ype o con ol wo ks in conjunc ion wi h some
o m o eedback. In his pape , we p esen a new adap i e eed o wa d con ol scheme o
elec omechanical sys ems in which eal- ime measu emen s o es ima es o he posi ion and
i s de i a i es a e no echnically o economically easible. This is he case, o example,
o comme cial elec omechanical swi ching de ices such as solenoid ac ua o s. Ou p oposal
consis s o wo blocks: on he one hand, a eed o wa d con olle based on di e en ial la ness
heo y; on he o he , an i e a i e adap a ion law ha exploi s he epe i i e ope a ion o hese
de ices o modi y he con olle pa ame e s cycle by cycle. As shown, his law can be ed wi h
any a ailable measu emen o he sys em, wi h he only equi emen ha i can be p ocessed
and con e ed in o an indica o o he pe o mance o any gi en ope a ion. Simula ed and
expe imen al esul s show ha ou p oposal is e ec i e in dealing wi h a long-s anding con ol
p oblem in elec omechanics: he so -landing con ol o elec omechanical swi ching de ices.
Keywo ds: Adap i e con ol, Di e en ial la ness, Elec omechanical de ices, Feed o wa d
con ol, I e a i e me hods, Mecha onic sys ems, So landing, Swi ches
1. INTRODUCTION
Feed o wa d con ol is widely used o acking applica-
ions because i signi ican ly ou pe o ms o he con ol
schemes in e ms o esponse ime and acking accu acy.
Since his ype o con ol does no equi e s a e in o -
ma ion, i is pa icula ly use ul o applica ions in which
sensing o es ima ion o he a iables o be con olled is
inaccu a e o una ailable. Speci ically, o di e en ially la
sys ems, i is possible o design a eed o wa d law ha
p o ides he con ol signal om he desi ed ou pu ajec-
o y wi hou need o sol ing any di e en ial equa ion. An
ad an ageous aspec o la ness-based eed o wa d con ol
(also known as exac eed o wa d linea iza ion) is ha i
does no su e om he well-known obus ness p oblems
o i s eedback coun e pa (exac eedback linea iza ion)
due o model pa ame e unce ain y (Hagenmeye and
Delaleau, 2003). In ecen yea s, his ype o eed o wa d
con ol has been p oposed o con olling he mo ion o
a wide ange o mecha onic sys ems, such as c ane o a-
o s (Baue e al., 2014), elec ical d i es (S umpe e al.,
⋆This wo k was suppo ed in pa ia g an s PID2021-124137OB-
I00, TED2021-130224B-I00, and CPP2021-008938, unded by
MCIN/AEI/10.13039/501100011033, by ERDF A way o making
Eu ope, and by he Eu opean Union Nex Gene a ionEU/PRTR, in
pa by g an T45 20R unded by he Go e nmen o A ag´on and
in pa by he “P og ama In es igo” unded by he Eu opean Union
Nex Gene a ionEU.
2015), elec ohyd aulic sys ems (Kim e al., 2015), quad o-
o s (G ee and Schoellig, 2018), and elec os a ic quasi-
s a ic mic oscanne s (Sch oed e e al., 2018).
Despi e he ad an ages o eed o wa d con olle s, i is well
known ha a con ol scheme based solely on a eed o wa d
e m, i.e., open loop, is qui e sensi i e o dis u bances and
modeling e o s. The e o e, i is mos usual o eed o wa d
con ol o be complemen ed by some o m o eedback, as
in he p e iously ci ed e e ences. This o cou se implies
ha he s a e o ou pu a iables can be measu ed o
es ima ed in eal ime wi h su icien accu acy.
Al e na i ely, some wo ks p opose un- o- un adap a ion
laws, which a e use ul o de ices unde epe i i e ope a-
ion. The key idea o hese app oaches is ha , ins ead o
using equen measu emen s o he s a e o ou pu (which
may no e en be a ailable), he adap a ion law makes use
o o he auxilia y a iables ela ed o he o e all con ol
pe o mance o each ope a ion. Fo example, Blanken e al.
(2017) has p esen ed ecen ly a uni ying amewo k o
un- o- un adap a ion o eed o wa d con ol based on ba-
sis unc ions. Howe e , his me hodology is no applicable
o la ness-based eed o wa d con olle s.
Ce ain mecha onic sys ems canno inco po a e he e-
qui ed senso s o eal- ime eedback con ol due o di e -
en easons, such as economic o space limi a ions. Among
hese de ices a e swi ch- ype elec omechanical de ices,
Run- o-Run Adap i e Nonlinea
Feed o wa d Con ol o
Elec omechanical Swi ching De ices ⋆
Edua do Moya-Lashe as ∗Edga Rami ez-Labo eo ∗
Eloy Se ano-Seco ∗
∗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, 50018 Za agoza, Spain,
(e-mail: {emoya, ami lab, ese anoseco}@uniza .es)
Abs ac : Feed o wa d con ol can g ea ly imp o e he esponse ime and con ol accu acy o
any mecha onic sys em. Howe e , in o de o compensa e o he e ec s o modeling e o s
o dis u bances, i is impe a i e ha his ype o con ol wo ks in conjunc ion wi h some
o m o eedback. In his pape , we p esen a new adap i e eed o wa d con ol scheme o
elec omechanical sys ems in which eal- ime measu emen s o es ima es o he posi ion and
i s de i a i es a e no echnically o economically easible. This is he case, o example,
o comme cial elec omechanical swi ching de ices such as solenoid ac ua o s. Ou p oposal
consis s o wo blocks: on he one hand, a eed o wa d con olle based on di e en ial la ness
heo y; on he o he , an i e a i e adap a ion law ha exploi s he epe i i e ope a ion o hese
de ices o modi y he con olle pa ame e s cycle by cycle. As shown, his law can be ed wi h
any a ailable measu emen o he sys em, wi h he only equi emen ha i can be p ocessed
and con e ed in o an indica o o he pe o mance o any gi en ope a ion. Simula ed and
expe imen al esul s show ha ou p oposal is e ec i e in dealing wi h a long-s anding con ol
p oblem in elec omechanics: he so -landing con ol o elec omechanical swi ching de ices.
Keywo ds: Adap i e con ol, Di e en ial la ness, Elec omechanical de ices, Feed o wa d
con ol, I e a i e me hods, Mecha onic sys ems, So landing, Swi ches
1. INTRODUCTION
Feed o wa d con ol is widely used o acking applica-
ions because i signi ican ly ou pe o ms o he con ol
schemes in e ms o esponse ime and acking accu acy.
Since his ype o con ol does no equi e s a e in o -
ma ion, i is pa icula ly use ul o applica ions in which
sensing o es ima ion o he a iables o be con olled is
inaccu a e o una ailable. Speci ically, o di e en ially la
sys ems, i is possible o design a eed o wa d law ha
p o ides he con ol signal om he desi ed ou pu ajec-
o y wi hou need o sol ing any di e en ial equa ion. An
ad an ageous aspec o la ness-based eed o wa d con ol
(also known as exac eed o wa d linea iza ion) is ha i
does no su e om he well-known obus ness p oblems
o i s eedback coun e pa (exac eedback linea iza ion)
due o model pa ame e unce ain y (Hagenmeye and
Delaleau, 2003). In ecen yea s, his ype o eed o wa d
con ol has been p oposed o con olling he mo ion o
a wide ange o mecha onic sys ems, such as c ane o a-
o s (Baue e al., 2014), elec ical d i es (S umpe e al.,
⋆This wo k was suppo ed in pa ia g an s PID2021-124137OB-
I00, TED2021-130224B-I00, and CPP2021-008938, unded by
MCIN/AEI/10.13039/501100011033, by ERDF A way o making
Eu ope, and by he Eu opean Union Nex Gene a ionEU/PRTR, in
pa by g an T45 20R unded by he Go e nmen o A ag´on and
in pa by he “P og ama In es igo” unded by he Eu opean Union
Nex Gene a ionEU.
2015), elec ohyd aulic sys ems (Kim e al., 2015), quad o-
o s (G ee and Schoellig, 2018), and elec os a ic quasi-
s a ic mic oscanne s (Sch oed e e al., 2018).
Despi e he ad an ages o eed o wa d con olle s, i is well
known ha a con ol scheme based solely on a eed o wa d
e m, i.e., open loop, is qui e sensi i e o dis u bances and
modeling e o s. The e o e, i is mos usual o eed o wa d
con ol o be complemen ed by some o m o eedback, as
in he p e iously ci ed e e ences. This o cou se implies
ha he s a e o ou pu a iables can be measu ed o
es ima ed in eal ime wi h su icien accu acy.
Al e na i ely, some wo ks p opose un- o- un adap a ion
laws, which a e use ul o de ices unde epe i i e ope a-
ion. The key idea o hese app oaches is ha , ins ead o
using equen measu emen s o he s a e o ou pu (which
may no e en be a ailable), he adap a ion law makes use
o o he auxilia y a iables ela ed o he o e all con ol
pe o mance o each ope a ion. Fo example, Blanken e al.
(2017) has p esen ed ecen ly a uni ying amewo k o
un- o- un adap a ion o eed o wa d con ol based on ba-
sis unc ions. Howe e , his me hodology is no applicable
o la ness-based eed o wa d con olle s.
Ce ain mecha onic sys ems canno inco po a e he e-
qui ed senso s o eal- ime eedback con ol due o di e -
en easons, such as economic o space limi a ions. Among
hese de ices a e swi ch- ype elec omechanical de ices,
Run- o-Run Adap i e Nonlinea
Feed o wa d Con ol o
Elec omechanical Swi ching De ices ⋆
Edua do Moya-Lashe as ∗Edga Rami ez-Labo eo ∗
Eloy Se ano-Seco ∗
∗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, 50018 Za agoza, Spain,
(e-mail: {emoya, ami lab, ese anoseco}@uniza .es)
Abs ac : Feed o wa d con ol can g ea ly imp o e he esponse ime and con ol accu acy o
any mecha onic sys em. Howe e , in o de o compensa e o he e ec s o modeling e o s
o dis u bances, i is impe a i e ha his ype o con ol wo ks in conjunc ion wi h some
o m o eedback. In his pape , we p esen a new adap i e eed o wa d con ol scheme o
elec omechanical sys ems in which eal- ime measu emen s o es ima es o he posi ion and
i s de i a i es a e no echnically o economically easible. This is he case, o example,
o comme cial elec omechanical swi ching de ices such as solenoid ac ua o s. Ou p oposal
consis s o wo blocks: on he one hand, a eed o wa d con olle based on di e en ial la ness
heo y; on he o he , an i e a i e adap a ion law ha exploi s he epe i i e ope a ion o hese
de ices o modi y he con olle pa ame e s cycle by cycle. As shown, his law can be ed wi h
any a ailable measu emen o he sys em, wi h he only equi emen ha i can be p ocessed
and con e ed in o an indica o o he pe o mance o any gi en ope a ion. Simula ed and
expe imen al esul s show ha ou p oposal is e ec i e in dealing wi h a long-s anding con ol
p oblem in elec omechanics: he so -landing con ol o elec omechanical swi ching de ices.
Keywo ds: Adap i e con ol, Di e en ial la ness, Elec omechanical de ices, Feed o wa d
con ol, I e a i e me hods, Mecha onic sys ems, So landing, Swi ches
1. INTRODUCTION
Feed o wa d con ol is widely used o acking applica-
ions because i signi ican ly ou pe o ms o he con ol
schemes in e ms o esponse ime and acking accu acy.
Since his ype o con ol does no equi e s a e in o -
ma ion, i is pa icula ly use ul o applica ions in which
sensing o es ima ion o he a iables o be con olled is
inaccu a e o una ailable. Speci ically, o di e en ially la
sys ems, i is possible o design a eed o wa d law ha
p o ides he con ol signal om he desi ed ou pu ajec-
o y wi hou need o sol ing any di e en ial equa ion. An
ad an ageous aspec o la ness-based eed o wa d con ol
(also known as exac eed o wa d linea iza ion) is ha i
does no su e om he well-known obus ness p oblems
o i s eedback coun e pa (exac eedback linea iza ion)
due o model pa ame e unce ain y (Hagenmeye and
Delaleau, 2003). In ecen yea s, his ype o eed o wa d
con ol has been p oposed o con olling he mo ion o
a wide ange o mecha onic sys ems, such as c ane o a-
o s (Baue e al., 2014), elec ical d i es (S umpe e al.,
⋆This wo k was suppo ed in pa ia g an s PID2021-124137OB-
I00, TED2021-130224B-I00, and CPP2021-008938, unded by
MCIN/AEI/10.13039/501100011033, by ERDF A way o making
Eu ope, and by he Eu opean Union Nex Gene a ionEU/PRTR, in
pa by g an T45 20R unded by he Go e nmen o A ag´on and
in pa by he “P og ama In es igo” unded by he Eu opean Union
Nex Gene a ionEU.
2015), elec ohyd aulic sys ems (Kim e al., 2015), quad o-
o s (G ee and Schoellig, 2018), and elec os a ic quasi-
s a ic mic oscanne s (Sch oed e e al., 2018).
Despi e he ad an ages o eed o wa d con olle s, i is well
known ha a con ol scheme based solely on a eed o wa d
e m, i.e., open loop, is qui e sensi i e o dis u bances and
modeling e o s. The e o e, i is mos usual o eed o wa d
con ol o be complemen ed by some o m o eedback, as
in he p e iously ci ed e e ences. This o cou se implies
ha he s a e o ou pu a iables can be measu ed o
es ima ed in eal ime wi h su icien accu acy.
Al e na i ely, some wo ks p opose un- o- un adap a ion
laws, which a e use ul o de ices unde epe i i e ope a-
ion. The key idea o hese app oaches is ha , ins ead o
using equen measu emen s o he s a e o ou pu (which
may no e en be a ailable), he adap a ion law makes use
o o he auxilia y a iables ela ed o he o e all con ol
pe o mance o each ope a ion. Fo example, Blanken e al.
(2017) has p esen ed ecen ly a uni ying amewo k o
un- o- un adap a ion o eed o wa d con ol based on ba-
sis unc ions. Howe e , his me hodology is no applicable
o la ness-based eed o wa d con olle s.
Ce ain mecha onic sys ems canno inco po a e he e-
qui ed senso s o eal- ime eedback con ol due o di e -
en easons, such as economic o space limi a ions. Among
hese de ices a e swi ch- ype elec omechanical de ices,
such as solenoid al es and elec omagne ic elays. Thus,
o some o hese cases in which eedback con ol is no ea-
sible, un- o- un lea ning- ype adap a ion laws ha e been
p oposed. Howe e , mos o hem s ill equi e some kind
o eal- ime eedback (Pe e son and S e anopoulou, 2004;
Benosman and A ın¸c, 2015; Moya-Lashe as and Sagues,
2020), o use model- ee inpu pa ame e iza ion ins ead
o eed o wa d con olle s (Yang e al., 2013; Me co elli,
2012; Di Gae a e al., 2015).
In his pape , we p esen a new con ol scheme o elec-
omechanical sys ems ha ope a e in a epe i i e manne .
I is pa icula ly sui able o cases in which eal- ime
measu emen s o es ima es o he posi ion a e no echni-
cally o economically easible, i.e., sys ems whe e eal- ime
eedback con ol canno be implemen ed. I consis s o wo
sepa a e bu in e connec ed blocks. Fi s ly, a eed o wa d
con olle based on di e en ial la ness heo y. Thanks
o he la ness p ope y—which is sa is ied by a la ge
numbe o elec omechanical sys ems— his block can be
pa ame e ized in e ms o he physical pa ame e s o he
model. Secondly, an i e a i e adap a ion law ha exploi s
he epe i i e ope a ion o hese de ices o modi y he
pa ame e s o he eed o wa d block cycle by cycle. As i
is shown, his un- o- un law can be ed wi h any a ailable
measu emen o he sys em, wi h he only equi emen
ha i can be p ocessed and con e ed in o an indica o o
he pe o mance o any gi en ope a ion. In he pape , we
desc ibe and apply his me hod o a long-s anding con ol
p oblem in elec omechanics: he so -landing con ol o
elec omechanical swi ching de ices. Simula ion and ex-
pe imen al esul s a e p esen ed in o de o alida e he
p oposal.
2. CONTROL-ORIENTED DYNAMICAL MODEL
Elec omechanical swi ch- ype de ices a e based on a
single-coil eluc ance ac ua o . Schema ic diag ams o his
kind o ac ua o s wi h di e en shapes a e ep esen ed in
Fig. 1. They all ha e a ixed co e, which is magne ized
by a coil cu en , and a mo able co e, which is gene ally
a ached o a sp ing o o he elas ic componen s. The
pu pose o he ixed magne ic co e is o ac as an elec-
omagne ha a ac s he mo able magne ic co e (i.e.,
a ma u e), closing he gap be ween hem. Gi en ha he
single coil is only able o gene a e magne ic o ce in one
di ec ion, elas ic and o he passi e uncon ollable o ces
a e necessa y o mo e he a ma u e away om he ixed
co e when he coil cu en is educed and he elec omag-
ne is de-ene gized. In he case o swi ch- ype ac ua o s,
he a ma u e posi ion is cons ained be ween wo limi s.
The sys em dynamics is as ollows. Fi s ly, he dynamics
o he magne ic lux linkage λis de i ed om he elec ical
and magne ic equi alen ci cui s. On he one hand, he coil
elec ical ci cui equa ion is gi en by Ohm’s law, Fa aday’s
law and Ki chho ’s ol age law, esul ing in
u=Ri+˙
λ, (1)
whe e u,Rand ia e espec i ely he coil ol age, in e nal
esis ance and cu en . On he o he hand, he cu en is
ela ed o he lux linkage h ough he magne ic equi alen
ci cui equa ion gi en by Hopkinson’s law and Amp`e e’s
ci cui al law,
i=ˆ
Rλ, (2)
Fig. 1. Schema ic ep esen a ion o single-coil eluc ance
ac ua o s
whe e ˆ
Ris he magne ic eluc ance pe u n squa ed o ,
equi alen ly, he in e se o he induc ance o he coil.
Gene ally, he eluc ance can be de ined as a unc ion o
he lux linkage (due o magne ic sa u a ion in he co e)
and he a ma u e posi ion z(due o i s dependence on he
ai gap leng h in he lux pa h). A con enien app oach is
o sepa a e he eluc ance in o wo unc ions,
ˆ
R=ˆ
Rc(λ)+ ˆ
Rg(z),(3)
co esponding o he eluc ance con ibu ions o he co e,
ˆ
Rc, and he gap, ˆ
Rg. No e ha he e a e many possible
de ini ions o hese unc ions, conside ing di e en ac ua-
o s and elec omagne ic phenomena. Since hei speci ic
exp essions a e no needed o explain he con olle design,
in his and he ollowing sec ion we conside hem o be
a bi a y unc ions.
Then, he lux linkage dynamics can be de i ed om (1)–
(3), esul ing in he ollowing equa ion:
˙
λ=−Rˆ
Rc(λ)+ ˆ
Rg(z)λ+u. (4)
Secondly, he posi ion dynamics du ing mo ion is de-
sc ibed by New on’s second law,
m¨z=Fpas(z, ˙z)+Fmag(z,λ),(5)
whe e mis he a ma u e mass, and Fpas and Fmag a e
he o ce unc ions. The passi e o ces a e encompassed in
he unc ion Fpas which, in a gene alized manne , depends
on he posi ion (e.g. elas ic o ces) and he eloci y (e.g.
iscous ic ion). The magne ic o ce, on he o he hand,
depends on he magne ic lux, which can be indi ec ly
con olled om he coil ol age u, as seen in (4). Mo e
speci ically, he magne ic o ce is gi en by he unc ion
Fmag, de ined as ollows (Rami ez-Labo eo e al., 2016):
Fmag(z,λ)=−1
2
∂ˆ
Rg
∂z λ2.(6)
To summa ize: he mo ion dynamics can be desc ibed wi h
a s a e-space ep esen a ion in which he s a e a iables
a e he posi ion, z, i s de i a i e, ˙z, and he lux linkage,
λ; he inpu is he coil ol age, u; and he s a e dynamics
is gi en by he di e en ial equa ions (4) and (5). These
equa ions a e alid in he domain gi en by
z∈[zmin,z
max],˙z∈R,λ∈(−λsa ,λ
sa ),(7)
whe e zmin and zmax co espond o he physical limi s o
he mechanism and λsa is he maximum lux linkage due
o magne ic sa u a ion.
5360 Edua do Moya-Lashe as e al. / IFAC Pape sOnLine 56-2 (2023) 5358–5363
zd( )Fla ness-based
eed o wa d con olle
Posi ion
ajec o y design Ac ua o
Hold Cos
Run- o- un
adap a ion law
un
d( )
yn( )
Jn
pn+1
pn
Fig. 2. Con ol diag am. The supe sc ip ndeno es a iables o he n h ope a ion. The ol age signal udis compu ed
by he eed o wa d con olle as a unc ion o he desi ed posi ion ajec o y zdand i s de i a i es. The un- o-
un adap a ion law uses he ope a ion cos J—compu ed using he sys em measu able ou pu y— o upda e he
pa ame e ec o po he eed o wa d con olle only once pe ope a ion
3. CONTROLLER DESIGN
The p oposed con olle is schema ized in Fig. 2. I in-
cludes a la ness-based eed o wa d con olle ha com-
pu es he ol age signal udbased on he desi ed posi ion
ajec o y zd. The ol age signal is ed o he ac ua o o
pe o m an ope a ion, and a cos o pe o mance index J
is calcula ed. The un- o- un adap a ion law adap s he
model pa ame e s used in he eed o wa d con olle in
o de o educe he cos in he subsequen ope a ions.
3.1 T ajec o y design
Fi s ly, he desi ed posi ion ajec o y is de ined, consid-
e ing ha he objec i e is o educe he impac eloci ies.
This pape p oposes a so -landing ajec o y wi h he
ollowing bounda y condi ions:
zd( 0)=z0,˙zd( 0)=0,¨zd( 0)=0,
zd( )=z ,˙zd( )=0,¨zd( )=0,(8)
whe e 0and a e he ini ial and inal use -de ined imes
o he swi ching ope a ions, and z0and z a e he ini ial
and inal posi ions, which espec i ely co espond o zmax
and zmin o he closing ope a ions, o o zmin and zmax o
he opening ope a ions. The ajec o y is hen designed as
a 5 h-deg ee polynomial, because i s six coe icien s can be
i ed om he six bounda y condi ions (8). The esul ing
ajec o y is ep esen ed in a gene alized manne in Fig. 3.
3.2 Fla ness-based eed o wa d con olle
The nex s ep is he design o he eed o wa d con olle ,
which uses he desi ed posi ion ajec o y o compu e
a ol age con ol signal applicable o he ac ua o . The
p oposed con olle is based on he la ness p ope y o he
nonlinea dynamical sys em. This s uc u al p ope y can
be in e p e ed as an ex ension o Kalman’s con ollabili y
o linea sys ems (a linea sys em is la i and only i i
is con ollable). P o ing ha a sys em is la simpli ies
g ea ly he design o many ypes o con olle s. Rega ding
eed o wa d con ol, he la ness p ope y implies ha
bo h he s a e and inpu a iables can be exp essed
as unc ions o he ou pu and a ini e numbe o i s
de i a i es (Fliess e al., 1995). This is ex emely use ul i
he a iable o be con olled is a la ou pu , because he
inpu can be calcula ed wi hou sol ing any di e en ial
equa ion. Fo una ely, he posi ion is a di e en ially la
ou pu in a la ge numbe o elec omechanical sys ems.
Thus, we belie e ha ou p oposal migh be o gene al
use in he design o con olle s in he mecha onic ield.
I can be easily e i ied ha he posi ion zis a la ou pu
o he p esen ed model: he posi ion and eloci y e iden ly
Fig. 3. Desi ed so -landing ajec o y (gene al o m)
depend on he ou pu and i s de i a i e, whe eas he lux
and ol age can be exp essed as unc ions o he ou pu
and i s de i a i es by simple manipula ions o he model
equa ions. Speci ically, he lux can be de i ed om (5) as
λ=λ(z, ˙z, ¨z)=±2Fpas(z, ˙z)−m¨z
∂ˆ
Rg/∂z .(9)
Then, he sys em inpu ucan be ob ained in a simila
manne om (4),
u=u(z, ˙z, ¨z, ...
z)=Rˆ
Rg(z)+ ˆ
Rc(λ)λ+˙
λ, (10)
whe e λmus be eplaced by he exp ession (9). The
unc ion o ˙
λcan be ob ained by calcula ing he ime
de i a i e o (5) and sol ing o ˙
λ.
˙
λ=˙
λz, ˙z, ¨z, ...
z=∂ˆ
Rg
∂z −1
∂Fpas
∂z ˙z+∂Fpas
∂˙z¨z
−m...
z−1
2
∂2ˆ
Rg
∂z2˙zλ
2(11)
No e ha , o he sys em o be la , Fpas mus be di e -
en iable and ˆ
Rg wice di e en iable.
The p e ious equa ions, which demons a e he la ness
p ope y o he p esen ed sys em, also se e o design he
eed o wa d con ol e m. In pa icula , he inpu signal
ud o achie e he desi ed ajec o y is ob ained simply by
eplacing zwi h zdin (10).
3.3 Run- o- un adap a ion
As al eady s a ed, he eed o wa d con olle is e y sen-
si i e o modeling e o s i i is no accompanied by some
o m o eedback. Ideally, eal- ime measu emen s o he
posi ion would be used as eedback o close he loop, bu in
many cases his is echnically o economically un easible.
Run- o- un con ol can be seen as an al e na i e app oach
when eal- ime eedback loops a e no an op ion. I is
use ul o de ices ha ope a e in a epe i i e manne ,
Edua do Moya-Lashe as e al. / IFAC Pape sOnLine 56-2 (2023) 5358–5363 5361
zd( )Fla ness-based
eed o wa d con olle
Posi ion
ajec o y design Ac ua o
Hold Cos
Run- o- un
adap a ion law
un
d( )
yn( )
Jn
pn+1
pn
Fig. 2. Con ol diag am. The supe sc ip ndeno es a iables o he n h ope a ion. The ol age signal udis compu ed
by he eed o wa d con olle as a unc ion o he desi ed posi ion ajec o y zdand i s de i a i es. The un- o-
un adap a ion law uses he ope a ion cos J—compu ed using he sys em measu able ou pu y— o upda e he
pa ame e ec o po he eed o wa d con olle only once pe ope a ion
3. CONTROLLER DESIGN
The p oposed con olle is schema ized in Fig. 2. I in-
cludes a la ness-based eed o wa d con olle ha com-
pu es he ol age signal udbased on he desi ed posi ion
ajec o y zd. The ol age signal is ed o he ac ua o o
pe o m an ope a ion, and a cos o pe o mance index J
is calcula ed. The un- o- un adap a ion law adap s he
model pa ame e s used in he eed o wa d con olle in
o de o educe he cos in he subsequen ope a ions.
3.1 T ajec o y design
Fi s ly, he desi ed posi ion ajec o y is de ined, consid-
e ing ha he objec i e is o educe he impac eloci ies.
This pape p oposes a so -landing ajec o y wi h he
ollowing bounda y condi ions:
zd( 0)=z0,˙zd( 0)=0,¨zd( 0)=0,
zd( )=z ,˙zd( )=0,¨zd( )=0,(8)
whe e 0and a e he ini ial and inal use -de ined imes
o he swi ching ope a ions, and z0and z a e he ini ial
and inal posi ions, which espec i ely co espond o zmax
and zmin o he closing ope a ions, o o zmin and zmax o
he opening ope a ions. The ajec o y is hen designed as
a 5 h-deg ee polynomial, because i s six coe icien s can be
i ed om he six bounda y condi ions (8). The esul ing
ajec o y is ep esen ed in a gene alized manne in Fig. 3.
3.2 Fla ness-based eed o wa d con olle
The nex s ep is he design o he eed o wa d con olle ,
which uses he desi ed posi ion ajec o y o compu e
a ol age con ol signal applicable o he ac ua o . The
p oposed con olle is based on he la ness p ope y o he
nonlinea dynamical sys em. This s uc u al p ope y can
be in e p e ed as an ex ension o Kalman’s con ollabili y
o linea sys ems (a linea sys em is la i and only i i
is con ollable). P o ing ha a sys em is la simpli ies
g ea ly he design o many ypes o con olle s. Rega ding
eed o wa d con ol, he la ness p ope y implies ha
bo h he s a e and inpu a iables can be exp essed
as unc ions o he ou pu and a ini e numbe o i s
de i a i es (Fliess e al., 1995). This is ex emely use ul i
he a iable o be con olled is a la ou pu , because he
inpu can be calcula ed wi hou sol ing any di e en ial
equa ion. Fo una ely, he posi ion is a di e en ially la
ou pu in a la ge numbe o elec omechanical sys ems.
Thus, we belie e ha ou p oposal migh be o gene al
use in he design o con olle s in he mecha onic ield.
I can be easily e i ied ha he posi ion zis a la ou pu
o he p esen ed model: he posi ion and eloci y e iden ly
Fig. 3. Desi ed so -landing ajec o y (gene al o m)
depend on he ou pu and i s de i a i e, whe eas he lux
and ol age can be exp essed as unc ions o he ou pu
and i s de i a i es by simple manipula ions o he model
equa ions. Speci ically, he lux can be de i ed om (5) as
λ=λ(z, ˙z, ¨z)=±2Fpas(z, ˙z)−m¨z
∂ˆ
Rg/∂z .(9)
Then, he sys em inpu ucan be ob ained in a simila
manne om (4),
u=u(z, ˙z, ¨z, ...
z)=Rˆ
Rg(z)+ ˆ
Rc(λ)λ+˙
λ, (10)
whe e λmus be eplaced by he exp ession (9). The
unc ion o ˙
λcan be ob ained by calcula ing he ime
de i a i e o (5) and sol ing o ˙
λ.
˙
λ=˙
λz, ˙z, ¨z, ...
z=∂ˆ
Rg
∂z −1
∂Fpas
∂z ˙z+∂Fpas
∂˙z¨z
−m...
z−1
2
∂2ˆ
Rg
∂z2˙zλ
2(11)
No e ha , o he sys em o be la , Fpas mus be di e -
en iable and ˆ
Rg wice di e en iable.
The p e ious equa ions, which demons a e he la ness
p ope y o he p esen ed sys em, also se e o design he
eed o wa d con ol e m. In pa icula , he inpu signal
ud o achie e he desi ed ajec o y is ob ained simply by
eplacing zwi h zdin (10).
3.3 Run- o- un adap a ion
As al eady s a ed, he eed o wa d con olle is e y sen-
si i e o modeling e o s i i is no accompanied by some
o m o eedback. Ideally, eal- ime measu emen s o he
posi ion would be used as eedback o close he loop, bu in
many cases his is echnically o economically un easible.
Run- o- un con ol can be seen as an al e na i e app oach
when eal- ime eedback loops a e no an op ion. I is
use ul o de ices ha ope a e in a epe i i e manne ,
and o which auxilia y measu emen s can be ob ained o
e alua ing he pe o mance o each ope a ion.
Fo hese easons, we p opose o inco po a e a un- o-
un adap a ion law o he con ol scheme. I s pu pose
is o modi y he model-based eed o wa d pa ame e s so
as o minimize a ce ain cos , J, which is calcula ed in
each ope a ion. In pa icula , o he gi en example, he
con ol objec i e is o achie e so -landing ajec o ies
when swi ching he de ices. Thus, he ideal cos would
be he absolu e alue o he impac eloci y c.
J=| c|(12)
In p ac ical scena ios whe e he impac eloci ies canno
be di ec ly measu ed o es ima ed, o he concep s ela ed
o hese bu mo e easily ob ainable can be used, e.g. he
impac sound o he bouncing du a ion.
The un- o- un adap a ion ask can he e o e be ega ded
as a black-box op imiza ion p oblem. Since he eed o -
wa d e m is pa ame e ized as a unc ion o he physical
pa ame e s o he model, he unde lying idea is o ind
alues o hese pa ame e s ha minimize he cos J. In his
ega d, no e ha he goal o he adap a ion law is no o
es ima e he model pa ame e s, bu o educe he impac
eloci y. Thus, he e is no gua an ee ha he adap ed
alues con e ge o he ue ones, bu simply o a se o
alues ha minimize he cos . The selec ed op imiza ion
me hod is he one p esen ed in Rami ez-Labo eo e al.
(2017). I is based on he Pa e n Sea ch me hod, which
is a widely used and s udied di ec -sea ch op imiza ion
me hod (Lewis and To czon, 2000). I s con e gence p op-
e ies a e di ec ly applicable, so i can be gua an eed ha
a leas a local minimum will be eached p o ided ha he
cos unc ion is de e minis ic.
4. SIMULATION RESULTS
In his sec ion, we p esen esul s ob ained by simula ion,
using a speci ic model p esen ed below. The main eason
o pe o ming model-based simula ions is o be able
o analyze he con e gence o he adap ed pa ame e s
wi h espec o hei nominal alues. In o he wo ds,
he simula ions p o ide a way o es ing whe he he
adap a ion law iden i ies he ue pa ame e alues o i ,
on he con a y, i only inds a se o alues ha achie e
he p oposed con ol objec i e.
4.1 Ac ua o and simula ion model
The con olle p esen ed in he p e ious sec ion has been
de i ed o a model wi h a bi a y eluc ance unc ions
ˆ
Rc(λ) and ˆ
Rg(z). In o de o pe o m he simula ions, he
ollowing exp essions ha e been used:
ˆ
Rc(λ)= κ1
1−|λ|/κ2
,(13)
ˆ
Rg(z)=κ3+κ4z
1+κ5zln(κ6/z),(14)
whe e he pa ame e s κ1, ..., κ6a e posi i e cons an s.
This model, which has been ex ac ed om Moya-Lashe as
and Sagues (2020), includes magne ic sa u a ion in he
co e and lux inging in he ai gap. These a e he wo
mos signi ican magne ic phenomena ha appea in his
class o ac ua o s.
Table 1. Model pa ame e alues.
Pa ame e Value Pa ame e Value
κ11.35 H−1zmin 0
κ20.0229 Wb zmax 10−3m
κ33.88 H−1m1.6×10−3kg
κ47.67 H−1/mks55 N/m
κ51320 m−1zs0.15 m
κ69.73 ·10−3mR50 Ω
Fu he mo e, he unc ion Fpas, which encompasses he
passi e o ces, is de ined assuming an ideal sp ing and
negligible ic ion and g a i y o ces,
Fpas =−ks(z−zs),(15)
whe e ksand zsa e espec i ely he sp ing s i ness con-
s an and es ing posi ion.
The nominal model pa ame e s used in he simula ions a e
p esen ed in Table 1.
4.2 Desc ip ion o he simula ed expe imen s
In he simula ions, i is assumed ha he dynamics o he
sys em is comple ely desc ibed by he model equa ions p e-
iously p esen ed. Tha is, he model ac ing in he ole o
he eal sys em and he eed o wa d con olle a e based on
exac ly he same equa ions. Howe e , in o de o analyze
he con e gence o he algo i hm, i is assumed ha he e
is some unce ain y in he pa ame e alues ini ially used
by he con olle . Mo e speci ically, he magne ic pa ame-
e s, κ1, ..., κ6, a e pe u bed ollowing con inuous uni o m
p obabili y dis ibu ions whose bounds a e ±5 % o he
nominal alues. In essence, hese dis ibu ions model he
e o s ha migh ypically be encoun e ed in a pa ame ic
es ima ion p ocess. On he o he hand, he mechanical
pa ame e s, i.e., zmin,zmax,m,ksand zs,aswellas he
elec ical esis ance R, a e assumed o be pe ec ly known,
as hese can usually be measu ed o es ima ed wi h ea-
sonable accu acy. Acco dingly, he pa ame e ec o ha
is i e a i ely adap ed is p=[κ1··· κ6], while he es o
he pa ame e s a e kep cons an .
Fo he sake o b e i y, he simula ions ha e been ocused
on he closing ope a ion, i.e., he mo ion om z=zmax o
z=zmin. None heless, he con ol p ocess o he opening
ope a ion is comple ely equi alen . The du a ion o he
so landing ajec o y, − 0, is a pa ame e ha allows
o con ol how agg essi e he con olle is. In his case i
has been se o 3.5 ms, which, o he nominal alues o
he pa ame e s, ensu es ha he magne ic lux does no
sa u a e and ha he ol age le els a e no oo demanding.
A o al o 10 000 di e en expe imen s ha e been simu-
la ed. Fo each one o hem, he ini ial pa ame e ec o
used by he eed o wa d con olle has been andomly
gene a ed acco ding o he ules desc ibed abo e. This
way, we analyze he con e gence om a la ge numbe o
ini ial si ua ions, he eby a oiding inco ec conclusions
associa ed o he esul s o ce ain pa icula cases. Then,
he con ol algo i hm has been un o 250 swi ching ope -
a ions, sea ching o a pa ame e ec o ha minimizes he
impac eloci ies. As a consequence, a o al o 2.5 million
swi ching ope a ions ha e been simula ed, allowing us o
deeply analyze he pe o mance o he me hod.
5362 Edua do Moya-Lashe as e al. / IFAC Pape sOnLine 56-2 (2023) 5358–5363
4.3 Resul s and discussion
The con ol esul s a e summa ized in Fig. 4, which ep-
esen s he ob ained dis ibu ion o cos s, J=| c|, wi h
espec o he swi ching ope a ion, n. To show he con ol
e ec i eness, he g aph also displays he cos o a con en-
ional swi ching ope a ion (speci ically, wi h a 30 V con-
s an ac i a ion). As shown, he andomness in oduced
in he ini ial pa ame e ec o esul s in a la ge a iabili y
in he impac eloci ies o he i s swi ching ope a ions.
Despi e ha , no e ha he ini ial impac eloci ies a e
in all cases lowe han ha o he uncon olled scena io.
Then, he con ol esul s imp o e g ea ly as he numbe
o i e a ions inc eases, which shows he impo ance o he
adap a ion law. I akes abou 100 swi ching ope a ions o
achie e a cos ha is hal ha o he uncon olled scena io,
and abou 200 o make i one en h.
As explained, i is he pa ame e adap a ion law which
causes he impac eloci ies o be educed o e he cou se
o he ope a ions. The e olu ion o he pa ame e alues
as a unc ion o he numbe o swi ching ope a ions is
ep esen ed in Fig. 5. This g aph shows he dis ibu ion
o alues o he six pa ame e s ha a e modi ied by he
adap i e law, as well as hei nominal alues. I can be
seen ha he pa ame e s a e ini ially in he ±5 % ange
de ined ea lie . Then, o each simula ed expe imen , he
adap a ion law sea ches he space o pa ame e s un il i
eaches i ually s able alues om i e a ion 200 onwa d.
The poin o no e is ha he pa ame e s do no con e ge o
hei nominal alues, bu ins ead each expe imen esul s
in a di e en inal se o pa ame e s. In his sense, ou un-
o- un adap i e law beha es like mos eal- ime adap i e
echniques. Tha is, he algo i hm manages o con ol he
sys em (in his case, o minimize he impac eloci ies)
as i he ue pa ame e s we e known. Howe e , i does
no gua an ee ha he adap ed pa ame e s con e ge o
he ue alues. This dis inc ion is impo an because, as
al eady men ioned, ou algo i hm should no be seen as a
pa ame ic es ima ion me hod.
5. EXPERIMENTAL RESULTS
Las ly, expe imen al es s ha e been also pe o med o
alida e he p oposal in a eal sys em. In pa icula , he
con ol has been applied o en comme cial single-pole
double- h ow powe elays o he same amily. Ten expe -
imen s ha e been pe o med wi h each elay, esul ing in
a o al o a hund ed ials. No e ha hese de ices a e
based on a small swi ching ac ua o and, hus, hey su e
om he a o emen ioned p oblems, i.e., impac s, noise and
p ema u e ailu e. The con ol objec i e is hus iden ical
o he simula ions: o achie e a so landing ajec o y.
Howe e , gi en ha he impac eloci ies canno be easily
measu ed o es ima ed in hese de ices, an al e na i e
cos is calcula ed using an audio signal om a low-cos
mic ophone placed nea he de ice. The key is o ealize
ha he highe he impac eloci y, he g ea e he amoun
o sound gene a ed in he swi ching. Mo e speci ically, le
mic be he ol age signal gene a ed by he mic ophone.
Then, he pe o mance index o a gi en ope a ion can be
compu ed as
J= +∆
0
mic2( )d , (16)
Fig. 4. Simula ion esul s. Cos as a unc ion o he numbe
o swi ching ope a ions. The g aph shows he median
(p50) and he 10 h and 90 h pe cen iles (p10 and p90,
espec i ely) o he dis ibu ion o alues ob ained o
he 10 000 simula ed expe imen s. The cos wi hou
con ol is also ep esen ed
Fig. 5. Simula ion esul s. Pa ame e alues as a unc ion
o he numbe o swi ching ope a ions. The g aphs
show he median (p50) and he 10 h and 90 h pe -
cen iles (p10 and p90, espec i ely) o he dis ibu ion
o alues ob ained o he 10 000 simula ed expe i-
men s. The nominal alues a e also ep esen ed
Edua do Moya-Lashe as e al. / IFAC Pape sOnLine 56-2 (2023) 5358–5363 5363
4.3 Resul s and discussion
The con ol esul s a e summa ized in Fig. 4, which ep-
esen s he ob ained dis ibu ion o cos s, J=| c|, wi h
espec o he swi ching ope a ion, n. To show he con ol
e ec i eness, he g aph also displays he cos o a con en-
ional swi ching ope a ion (speci ically, wi h a 30 V con-
s an ac i a ion). As shown, he andomness in oduced
in he ini ial pa ame e ec o esul s in a la ge a iabili y
in he impac eloci ies o he i s swi ching ope a ions.
Despi e ha , no e ha he ini ial impac eloci ies a e
in all cases lowe han ha o he uncon olled scena io.
Then, he con ol esul s imp o e g ea ly as he numbe
o i e a ions inc eases, which shows he impo ance o he
adap a ion law. I akes abou 100 swi ching ope a ions o
achie e a cos ha is hal ha o he uncon olled scena io,
and abou 200 o make i one en h.
As explained, i is he pa ame e adap a ion law which
causes he impac eloci ies o be educed o e he cou se
o he ope a ions. The e olu ion o he pa ame e alues
as a unc ion o he numbe o swi ching ope a ions is
ep esen ed in Fig. 5. This g aph shows he dis ibu ion
o alues o he six pa ame e s ha a e modi ied by he
adap i e law, as well as hei nominal alues. I can be
seen ha he pa ame e s a e ini ially in he ±5 % ange
de ined ea lie . Then, o each simula ed expe imen , he
adap a ion law sea ches he space o pa ame e s un il i
eaches i ually s able alues om i e a ion 200 onwa d.
The poin o no e is ha he pa ame e s do no con e ge o
hei nominal alues, bu ins ead each expe imen esul s
in a di e en inal se o pa ame e s. In his sense, ou un-
o- un adap i e law beha es like mos eal- ime adap i e
echniques. Tha is, he algo i hm manages o con ol he
sys em (in his case, o minimize he impac eloci ies)
as i he ue pa ame e s we e known. Howe e , i does
no gua an ee ha he adap ed pa ame e s con e ge o
he ue alues. This dis inc ion is impo an because, as
al eady men ioned, ou algo i hm should no be seen as a
pa ame ic es ima ion me hod.
5. EXPERIMENTAL RESULTS
Las ly, expe imen al es s ha e been also pe o med o
alida e he p oposal in a eal sys em. In pa icula , he
con ol has been applied o en comme cial single-pole
double- h ow powe elays o he same amily. Ten expe -
imen s ha e been pe o med wi h each elay, esul ing in
a o al o a hund ed ials. No e ha hese de ices a e
based on a small swi ching ac ua o and, hus, hey su e
om he a o emen ioned p oblems, i.e., impac s, noise and
p ema u e ailu e. The con ol objec i e is hus iden ical
o he simula ions: o achie e a so landing ajec o y.
Howe e , gi en ha he impac eloci ies canno be easily
measu ed o es ima ed in hese de ices, an al e na i e
cos is calcula ed using an audio signal om a low-cos
mic ophone placed nea he de ice. The key is o ealize
ha he highe he impac eloci y, he g ea e he amoun
o sound gene a ed in he swi ching. Mo e speci ically, le
mic be he ol age signal gene a ed by he mic ophone.
Then, he pe o mance index o a gi en ope a ion can be
compu ed as
J= +∆
0
mic2( )d , (16)
Fig. 4. Simula ion esul s. Cos as a unc ion o he numbe
o swi ching ope a ions. The g aph shows he median
(p50) and he 10 h and 90 h pe cen iles (p10 and p90,
espec i ely) o he dis ibu ion o alues ob ained o
he 10 000 simula ed expe imen s. The cos wi hou
con ol is also ep esen ed
Fig. 5. Simula ion esul s. Pa ame e alues as a unc ion
o he numbe o swi ching ope a ions. The g aphs
show he median (p50) and he 10 h and 90 h pe -
cen iles (p10 and p90, espec i ely) o he dis ibu ion
o alues ob ained o he 10 000 simula ed expe i-
men s. The nominal alues a e also ep esen ed
Fig. 6. Expe imen al esul s. Cos as a unc ion o he
numbe o swi ching ope a ions. The g aph shows he
median (p50) and he 10 h and 90 h pe cen iles (p10
and p90, espec i ely) o he dis ibu ion o alues ob-
ained o he 100 eal expe imen s. The cos wi hou
con ol is also ep esen ed
whe e ∆ is la ge enough o cap u e all he acous ic noise
gene a ed du ing (and a e ) he swi ching.
The expe imen al esul s a e p esen ed in Fig. 6. Fo
cla i y, he con ol cos s a e no malized wi h espec o he
cos o an uncon olled scena io. Simila ly o he simula ed
case, i can be seen ha he ini ial cos s a y signi ican ly.
Howe e , all he expe imen s ha e used ini ially he same
con olle pa ame e s, so in his case he a iabili y mus
be due o di e ences be ween he en elays o a ia ions
in en i onmen al condi ions. Then, he adap a ion law
manages o educe he cos s as he numbe o ope a ions
inc eases. No e ha , al hough he con e gence is slowe
han in he simula ed expe imen s, all elays beha e be e
han in he uncon olled case a e he 250 ope a ions.
The e o e, we can conclude ha ou p oposal is also
e ec i e in a p ac ical scena io.
6. CONCLUSION
In his pape , we ha e p esen ed a new eed o wa d con-
ol scheme o elec omechanical sys ems in which eal-
ime measu emen s o es ima es a e di icul , expensi e o
simply impossible o ob ain. The e o e, he undamen al
di e ence wi h o he adap i e o eed o wa d me hods in
he li e a u e is ha i does no use eal- ime eedback.
Ins ead, i includes a un- o- un adap a ion law ha can
use i ually any ype o measu emen . As has been shown
h ough simula ions and expe imen s, he only necessa y
condi ion is ha hese measu emen s can be p ocessed o
ob ain a pe o mance index o any gi en ope a ion.
As an applica ion example, he scheme has been applied
o he so -landing con ol o elec omechanical swi ching
de ices. Howe e , he p oposal is su icien ly e sa ile
o be applied o nea ly any mecha onic sys em ha
pe o ms a epe i i e ask. Fu u e wo k will be ocused
on imp o ing he con e gence speed o he me hod, o
example by modi ying he sea ch algo i hm o by applying
dimensionali y educ ion echniques o he pa ame e se .
REFERENCES
Baue , D., Schape , U., Schneide , K., and Sawodny, O.
(2014). Obse e design and la ness-based eed o wa d
con ol wi h model p edic i e ajec o y planning o a
c ane o a o . In Ame . Con ol Con ., 4020–4025.
Benosman, M. and A ın¸c, G.M. (2015). Ex emum
seeking-based adap i e con ol o elec omagne ic ac-
ua o s. In . J. Con ol, 88(3), 517–530.
Blanken, L., Boe en, F., B uijnen, D., and Oomen, T.
(2017). Ba ch- o-Ba ch Ra ional Feed o wa d Con ol:
F om I e a i e Lea ning o Iden i ica ion App oaches,
Wi h Applica ion o a Wa e S age. IEEE/ASME
T ans. Mecha onics, 22(2), 826–837.
Di Gae a, A., Hoyos Velasco, C.I., and Mon ana o, U.
(2015). Cycle-by-cycle adap i e o ce compensa ion o
he so -landing con ol o an elec o-mechanical engine
al e ac ua o . Asian J. Con ol, 17(5), 1707–1724.
Fliess, M., L´e ine, J., Ma in, P., and Rouchon, P. (1995).
Fla ness and de ec o non-linea sys ems: In oduc o y
heo y and examples. In . J. Con ol, 61(6), 1327–1361.
G ee , M. and Schoellig, A.P. (2018). Fla ness-Based
Model P edic i e Con ol o Quad o o T ajec o y
T acking. In IEEE/RSJ In . Con . In ell. Robo s Sys .
(IROS), 6740–6745.
Hagenmeye , V. and Delaleau, E. (2003). Exac eed o -
wa d linea iza ion based on di e en ial la ness. In . J.
Con ol, 76(6), 537–556.
Kim, W., Won, D., and Tomizuka, M. (2015). Fla ness-
Based Nonlinea Con ol o Posi ion T acking o Elec-
ohyd aulic Sys ems. IEEE/ASME T ans. Mecha on-
ics, 20(1), 197–206.
Lewis, R.M. and To czon, V. (2000). Pa e n sea ch
me hods o linea ly cons ained minimiza ion. SIAM
J. Op imiza ion, 10(3), 917–941.
Me co elli, P. (2012). An An isa u a ing Adap i e P eac-
ion and a Slide Su ace o Achie e So Landing Con ol
o Elec omagne ic Ac ua o s. IEEE/ASME T ans.
Mecha onics, 17(1), 76–85.
Moya-Lashe as, E. and Sagues, C. (2020). Run- o-Run
Con ol Wi h Bayesian Op imiza ion o So Landing
o Sho -S oke Reluc ance Ac ua o s. IEEE/ASME
T ans. Mecha onics, 25(6), 2645–2656.
Pe e son, K.S. and S e anopoulou, A.G. (2004). Ex-
emum seeking con ol o so landing o an elec ome-
chanical al e ac ua o . Au oma ica, 40(6), 1063–1069.
Rami ez-Labo eo, E., Sagues, C., and Llo en e, S. (2016).
A new model o elec omechanical elays o p edic ing
he mo ion and elec omagne ic dynamics. IEEE T ans.
Ind. Appl., 52(3), 2545–2553.
Rami ez-Labo eo, E., Sagues, C., and Llo en e, S. (2017).
A new un- o- un app oach o educing con ac bounce
in elec omagne ic swi ches. IEEE T ans. Ind. Elec on.,
64(1), 535–543.
Sch oed e , R., Ro h, M., Janschek, K., and Sandne ,
T. (2018). Fla ness-based open-loop and closed-loop
con ol o elec os a ic quasi-s a ic mic oscanne s using
je k-limi ed ajec o y design. Mecha onics, 56, 318–
331.
S umpe , J.F., Hagenmeye , V., Kuehl, S., and Kennel,
R. (2015). Deadbea Con ol o Elec ical D i es: A
Robus and Pe o man Design Based on Di e en ial
Fla ness. IEEE T ans. Powe Elec on., 30(8), 4585–
4596.
Yang, Y.P., Liu, J.J., Ye, D.H., Chen, Y.R., and Lu,
P.H. (2013). Mul iobjec i e op imal design and so
landing con ol o an elec omagne ic al e ac ua o o
a camless engine. IEEE/ASME T ans. Mecha onics,
18(3), 963–972.
