An Audio-Based I e a i e Con olle o
So Landing o Elec omechanical Relays
Eloy Se ano-Seco, Edga Rami ez-Labo eo, Edua do Moya-Lashe as, and Ca los Sagues
Abs ac —Elec omechanical elays and con ac o s su -
e om s ong collisions a he end o he swi ching ope a-
ions. This causes se e al undesi able phenomena, such as
clicking, mechanical wea and con ac bounce. Thus, he e
is g ea in e es in mi iga ing hese swi ching impac s while
keeping he ad an ageous ea u es o hese de ices. This
pape p oposes a comple e con ol s a egy o so land-
ing. The con ol s uc u e includes h ee main componen s.
The i s one is a eal- ime lux- acking eedback con olle ,
which p esen s se e al ad an ages o e ol age o cu en
con ol. The second one is a eed o wa d con olle , which
compu es he lux e e ence signal based on a p oposed
dynamical model and he desi ed posi ion ajec o y o he
swi ching ope a ions. Las ly, he hi d con ol componen
is a lea ning- ype un- o- un adap a ion law ha i e a i ely
adap s he model pa ame e s based on an audio signal.
I exploi s he epe i i e na u e o hese de ices in o de
o ci cum en modeling disc epancies due o uni - o-uni
a iabili y o small changes be ween ope a ions. The e ec-
i eness o he p oposed con ol is demons a ed h ough
a ious expe imen s.
Index Te ms—Adap i e con ol, Elec omechanical de-
ices, I e a i e me hods, Nonlinea dynamical sys ems, Op-
imiza ion, Relays, So landing, Swi ches
I. INTRODUCTION
ELECTROMECHANICAL elays a e used in many ap-
plica ions, e.g. d i e by wi e [1], induc ion hea ing [2],
ba e y cha ging [3] o wi eless powe ans e [4]. They o e
many ad an ages compa ed o solid s a e swi ches: hey a e
gene ally cheape and mo e e icien , a e able o conduc and
block cu en in bo h di ec ions, p o ide elec ical isola ion
be ween he ac i a ion ci cui and he powe e minals, and
ha e a simple ac i a ion mode. Addi ionally, whe eas al e -
na i e semiconduc o de ices can only p o ide single-pole
single- h ow a angemen s, elays can be designed as mul iple-
pole and mul iple- h ow, which is eally use ul in a ious
applica ions.
This wo k was suppo ed in pa by he Spanish Go e nmen /EU,
unde p ojec s RTC-2017-5965-6, PGC2018-098719-B-I00, PID2021-
124137OB-I00, TED2021-130224B-I00, and CPP2021-008938, in pa
by he A ag´
on Go e nmen /EU, unde p ojec DGA FSE T45 20R, and
in pa by he Eu opean Union - Nex Gene a ion EU. (Co esponding
au ho : Eloy Se ano-Seco.)
The au ho s a e wi h he Depa amen o de In o ma ica e Ingenie ia
de Sis emas (DIIS) and he 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: ese anoseco@uniza .es; ami lab@uniza .es; emoya@uniza .es,
csagues@uniza .es).
This is he accep ed e sion o he manusc ip : E. Se ano-Seco,
E. Rami ez-Labo eo, E. Moya-Lashe as and C. Sagues, “An Audio-
Based I e a i e Con olle o So Landing o Elec omechanical Relays,”
in IEEE T ansac ions on Indus ial Elec onics, ol. 70, no. 12, pp. 12730-
12738, Dec. 2023, doi: 10.1109/TIE.2022.3231254. Please ci e he
publishe ’s e sion. Fo he publishe ’s e sion and ull ci a ion de ails
see: h ps://doi.o g/10.1109/TIE.2022.3231254.
Elec omechanical elays, howe e , also p esen some d aw-
backs. One o hei mos known and undesi able p oblems
is he s ong swi ching impac s be ween he mo able and
ixed componen s, which cause wea , con ac bouncing and an
acous ic noise ha is undesi able in ce ain domes ic applica-
ions. Addi ionally, bounces may p o oke con ac welding [5]
o a cing ha exace ba es he e osion [6]. These p oblems
lead o a educ ion in he se ice li e o hese de ices and
he equipmen in which hey a e embedded, and limi he
ange o applica ions in which elec omechanical elays a e he
bes swi ching solu ion. The e o e, he e is a g ea in e es in
mi iga ing he swi ching impac s o elec omechanical elays
while keeping hei ad an ages o e solid-s a e al e na i es.
The s a e-o - he-a so -landing app oaches ound in he
li e a u e ocus almos solely on educing he con ac bounces
and hei du a ion. In addi ion, he as majo i y o wo ks
ocus on con ac o s, which a e elec omechanical elays o
high powe swi ching. In 1996, Da ies e al. [7] p esen ed
one o he i s con ol-o ien ed app oaches o con ac bounce
educ ion. The p oposed open-loop con olle was howe e
no obus because i did no ha e he abili y o sel -adap
o changing condi ions o pe u ba ions. Then, based on his
wo k, imp o emen s we e made o inc ease obus ness. Fo
ins ance, a cu en con olle was p oposed in [8], which de-
ec s he s a o he closing p ocess and acco dingly modi ies
he coil ene giza ion. As a di e en app oach o inc ease
obus ness, a un- o- un con ol was p oposed in [9], which
i e a i ely adap s ol age pulse du a ions. The main limi a ion
o his app oach is i s sensi i i y o esis ance a ia ions due
o empe a u e changes. Ano he ecen wo k [10] p oposed
a cu en acking solu ion o educe con ac bounce, whose
exci a ion ime is i e a i ely adap ed wi h a uzzy con olle .
The cu en acking con olle ci cum en s he a iabili y due
o empe a u e changes, bu he cu en e e ence is cons an ,
which heo e ically canno gua an ee pe ec so landing [11].
In pa allel, displacemen o posi ion eedback con ol has
also been in es iga ed. In 1999, Ca se e al. [12] showed ia
simula ion ha a simple uzzy con olle can educe con ac
bouncing. The main obs acle o his app oach is he need o
accu a e measu emen s o he mechanism posi ion. In mos
cases, posi ion senso s a e oo expensi e, o he mo able pa s
a e no accessible. As wo ka ounds, some wo ks p esen posi-
ion es ima o s, e.g. model-based [13] o neu al ne wo ks [10].
Howe e , hese es ima o s a e e y sensi i e o disc epancies
be ween he dynamics o di e en elays.
In sho , he so -landing con ol p oblem is s ill no sa -
is ac o ily sol ed o elec omechanical elays. Some wo ks
p opose posi ion eedback con olle s [12]–[14] ha may
be used o posi ion acking and ul ima ely so landing.
Howe e , hey ely on posi ion senso s—which canno be
implemen ed in mos applica ions—o es ima o s—which a e
oo compu a ionally expensi e, inaccu a e o sensi i e o mod-
eling and measu emen e o s. As a possible solu ion, o he
wo ks p esen open-loop o i e a i e con olle s ha do no
ely on posi ion measu emen s o es ima es [7]–[10], [15],
[16]. Howe e , hey ocus on educing impac bounces, which
can be coun e p oduc i e o he so landing objec i e (e.g.,
s ong swi ching o ces may educe he numbe o bounces
bu inc ease he impac s). Fu he mo e, mos solu ions a e
sensi i e o empe a u e a ia ions [7], [9] o modeling dis-
c epancies [7], [10], [13], [16].
This pape p esen s a new con ol s a egy o impac
educ ion in elec omechanical elays. The main con ibu ion is
he combina ion o h ee dis inc componen s: a eal- ime lux-
acking con olle , a la ness-based eed o wa d con olle ,
and a lea ning- ype un- o- un adap a ion law. The lux-
acking con olle elies on a eal- ime lux es ima o which
is ed wi h easily measu able elec ical signals, i.e., ol age
and cu en . The main ad an age o a lux con olle o e a
ol age o cu en con olle is ha his app oach depends on
ewe pa ame e s, which a e a main sou ce o unce ain y and
con ol a iabili y.
The eed o wa d con olle , which gene a es he desi ed lux
ajec o y, is based on a dynamical model also p esen ed in he
pape . The main con ibu ion o his model is ha i sepa a es
he mo ion o he mechanism in o di e en s ages, aking in o
accoun he de o ma ion o se e al componen s o he de ice.
The la ness p ope y o he model p o ides a me hod o
ob ain he lux ajec o y used by he lux- acking con olle
om a desi ed posi ion ajec o y o he a ma u e, which is in
u n designed conside ing he di e en phases o he mo ion.
Las ly, he un- o- un adap a ion law i e a i ely adap s he
model pa ame e s— equi ed by he eed o wa d con olle —
making he o e all con ol mo e obus o any ype o
pa ame e a iabili y. The adap a ion law is o mula ed as an
op imiza ion p oblem ha i e a i ely modi ies he pa ame e s
o he model in o de o maximize he sys em pe o mance.
This pe o mance is compu ed based on a mic ophone signal
o he noise gene a ed du ing he swi chings. Thus, ins ead
o ocusing on educing con ac bounce—as is mos common
o elays—ou so -landing app oach consis s in educing
impac sounds and, indi ec ly, impac eloci ies.
II. MODEL
The elec omechanical elay used in his wo k is shown in
Fig. 1. I is a single-pole double- h ow elay wi h a U-shaped
magne ic co e and a mo ing co e, o a ma u e. The e a e wo
dis inc ope a ions depending on he di ec ion o he a ma u e
mo ion: making and b eaking, when he coil ene gizes o de-
ene gizes, espec i ely. The a ma u e is connec ed wi h he
mo able elec ical con ac ia a plas ic componen . In he
making ope a ions, he con ac is pushed owa d he no mally
open posi ion, whe eas in he b eaking ope a ions, he con ac
is pulled owa d he no mally closed posi ion.
In o de o design he con olle , i is necessa y o cha -
ac e ize he dynamics o he sys em. Fo cla i y, he model
(a) (b)
Fig. 1: Elec omechanical elay. (a) Pho o. (b) Schema ic diag am.
is di ided in o wo in e connec ed sys ems. One o hem is
he elec omagne ic sys em, which is go e ned by wo main
equa ions. The i s one is he elec ical ci cui equa ion,
u=R i +˙
λ, (1)
whe e u,R,iand λa e he ol age be ween he coil e minals,
he coil in e nal esis ance, he coil cu en , and he magne ic
lux linkage, espec i ely. The second equa ion e e s o he
magne ic equi alen ci cui ,
N2i=Rc(λ) + Rg(θ)λ, (2)
whe e Nis he numbe o coil u ns, and Rcand Rga e
espec i ely he magne ic eluc ances o he i on co e and he
ai gap. These eluc ances can be modeled as unc ions o he
lux, λ, and he angula posi ion o he a ma u e, θ, aking in o
accoun he magne ic sa u a ion and lux inging phenomena,
espec i ely [17]. In o de o simpli y hese exp essions and
he consequen model-based con olle , he ollowing auxilia y
unc ions a e de ined as scaled e sions o he eluc ances,
R∗
c(λ) = Rc(λ)
N2=R∗
c0
1− |λ|/λsa
,(3)
R∗
g(θ) = Rg(θ)
N2=R∗
g0 +R′ ∗
g0 θ
1 + κ1θln(κ2/θ),(4)
being R∗
c0,λsa ,R∗
g0,R′ ∗
g0,κ1and κ2posi i e cons an s.
Using (1)–(4), he lux linkage dynamics is de i ed as
˙
λ=−RR∗
c(λ) + R∗
g(θ)λ+u. (5)
Rega ding he mechanical sys em, i mus be no ed ha ,
despi e i s small size and appa en simplici y, he elay mech-
anism shown in Fig. 1 is in ac a he complex. The e a e
ac ually h ee mo ing componen s: he a ma u e, he plas ic
pa a he op and he mo ing con ac . Due o mechanical play,
howe e , hese h ee componen s do no always mo e oge he .
When a es , he a ma u e emains a a posi ion de ined by
θ=θmax. In he i s phase o he mo ion, he a ma u e and
he plas ic componen mo e solidly wi hou making con ac
wi h he mo ing con ac , which is ouching he no mally
closed (NC) con ac (see Fig. 2a). Then, when he a ma u e
eaches he posi ion θ=θNC, he plas ic pa ouches he
mo ing con ac and he h ee componen s s a o mo e solidly
(Fig. 2b). The e is an a ma u e posi ion, θ=θNO, a which
he mo ing con ac eaches he no mally open (NO) con ac .
Howe e , due o he geome y and elas ici y o his la e
componen , he a ma u e and he plas ic componen can s ill
con inue o mo e o wa d by de o ming he mo ing con ac
(Fig. 2c).
(a) (b) (c)
Fig. 2: Schema ic diag am o he di e en s ages du ing he a ma u e
mo emen . (a) Only he a ma u e and he plas ic componen a e in
mo ion. (b) The plas ic componen pushes and de o ms he mo ing
con ac . (c) The mo ing con ac is al eady ouching he no mally open
con ac , bu can s ill be de o med.
Some assump ions a e aken in o de o cha ac e ize he
mo ion o his mechanism in a simple bu de ailed way. Fi s ly,
i is assumed ha all he mass is concen a ed in he mo ing
a ma u e, which is by a he hea ies componen o he h ee
p e iously men ioned. Fu he mo e, i is also assumed ha
he h ee phases o he mo ion desc ibed abo e a e he same
ega dless o whe he he a ma u e is mo ing in one di ec ion
o he o he . Tha is, i is assumed ha he posi ion o he
h ee componen s can be unambiguously de ined based solely
on he posi ion o he a ma u e θ. Unde hese simpli ica ions,
he mo ion o he mechanism can be desc ibed by New on’s
second law,
I¨
θ= Γmag(θ, λ)+Γe(θ) + c˙
θ, (6)
whe e Iis he a ma u e momen o ine ia, Γmag and Γea e
espec i ely he magne ic and elas ic o ques, and cis a damp-
ing coe icien . The magne ic o que depends on he de i a i e
o he gap eluc ance and he lux [18] o , equi alen ly, on he
de i a i e o he auxilia y unc ion R∗
gand he lux linkage,
Γmag(θ, λ) = −R′ ∗
g(θ)λ2
2,R′ ∗
g(θ) = ∂R∗
g(θ)
∂θ .(7)
The elas ic o que, on he o he hand, is modeled as he
ollowing piecewise unc ion,
Γe(θ) =
Γe,1,i θ > θNC
Γe,2,i θNO ≤θ≤θNC
Γe,3,i θ < θNO
,(8)
which de ines a speci ic exp ession o each s age o he
mo ion. In he i s s age (Fig. 2a), he e is a single elas ic
o que ha opposes he magne ic o ce and ends o keep he
a ma u e posi ion a θ=θmax. I is modeled as an ideal o sion
sp ing,
Γe,1=k1(θmax −θ),(9)
whe e k1is he s i ness cons an . In he second s age (Fig. 2b),
he elas ic o que is inc eased due o he de o ma ion o
he con ac . Acco ding o he Eule –Be noulli beam heo y,
and assuming small de o ma ion and angles, his can be
app oxima ed as an addi ional p opo ional elas ic o que,
Γe,2=k1(θmax −θ) + k2(θNC −θ),(10)
whe e k2is he s i ness cons an due o he mo ing con ac
de o ma ion. Las ly, in he hi d s age (Fig. 2c), he mo ing
con ac has eached i s limi , bu pa o i s ill de o ms.
Following he same easoning as in (10), he o al o que is
Γe,3=k1(θmax −θ)+k2(θNC −θNO)+k3(θNO −θ),(11)
whe e k3is he s i ness cons an due o he pa o he mo ing
con ac ha is s ill wa ping.
III. CONTROL
The p oposed con olle is schema ized in Fig. 3. I can be
in e p e ed as a cascade con olle wi h wo eedback loops.
The inne loop is an online lux- acking con olle which is
ed wi h he es ima ed lux linkage ˆ
λ. On he o he hand, he
ou e loop is a lea ning- ype con olle ha i e a i ely adap s
he e e ence lux linkage λd, based on he model pa ame e s
p. This e e ence lux is ob ained h ough a eed o wa d
con olle based on he desi ed posi ion ajec o y θd.
A. Posi ion 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 noise. Mos
commonly, so -landing ajec o ies a e de ined such ha he
inal eloci y and accele a ion a e minimized [11]. Howe e ,
o he s udied elay, he e a e wo main impac e en s ha
cause noise and wea : he s oke ends o he mo ing con ac
and o he a ma u e. Taking his in o accoun , his pape p o-
poses a conca ena ion o wo so -landing ajec o ies. O e all,
he combined posi ion ajec o y has he ollowing bounda y
condi ions:
θd( 0) = θ0,˙
θd( 0)=0,¨
θd( 0)=0,(12)
θd( c) = θc,˙
θd( c)=0,¨
θd( c)=0,(13)
θd( ) = θ ,˙
θd( )=0,¨
θd( )=0,(14)
whe e 0, cand a e he ini ial, con ac and inal use -de ined
ins an s and θ0,θcand θ a e he ini ial, con ac and inal po-
si ions, which espec i ely co espond o θmax,θNO and 0 o
he making (i.e. closing) ope a ions, o o 0,θNC and θmax o
he b eaking (i.e. opening) ope a ions. The wo conca ena ing
ajec o ies a e designed as 5 h-deg ee polynomials. The i s
one de ines he ajec o y o ∈[ 0, c], while he second
one applies o he in e al ∈[ c, ]. Thus, he bounda y
condi ions (12) and (13) a e used o sol e he six polynomial
coe icien s co esponding o he ajec o y o he i s in e al,
whe eas (13) and (14) a e used o sol e he six polynomial
coe icien s o he second in e al. The gene al o m o he
designed posi ion ajec o y is ep esen ed in Fig. 4.
B. Feed o wa d con olle
The eed o wa d con olle o his pape exploi s he di e -
en ial la ness p ope y o he model. In sho , an n h-o de
sys em is di e en ially la i he n h de i a i e o he ou pu
is he i s one whe e he inpu appea s explici ly [19]. In his
case, i can be shown ha he angula posi ion is a la ou pu
o he dynamical model p esen ed in he p e ious sec ion. As
a consequence, he desi ed lux linkage λdcan be calcula ed
om he desi ed posi ion θdand i s de i a i es as
λd=q2 (Γe(θd)−c˙
θd−I¨
θd)/R′ ∗
g(θd).(15)
θd( )Feed o wa d
con olle
Posi ion
ajec o y
design
Flux- acking con olle Plan
Flux es ima o
Run- o- un
adap a ion law
Hold
λn
d( )un( )
in( )
ˆ
λn( )
υn
audio( )
pn+1
pn
+
−
Fig. 3: Con ol diag am. The supe sc ip nis used o deno e he a iables o he n h ope a ion. The inne loop (blocks in o ange) is he eal- ime
lux- acking con olle . The lux linkage e e ence λdis p o ided in eal ime by he eed o wa d con olle (in g een) which is ed wi h he desi ed
posi ion ajec o y θd. The un- o- un adap a ion law (in blue) uses he audio signal audio o upda e he pa ame e se po he eed o wa d con olle
only once pe ope a ion.
Fig. 4: Desi ed posi ion ajec o y based on wo conca ena ed 5 h-
deg ee polynomial ajec o ies.
This exp ession, which esul s om (6)–(11), depends on he
pa ame e ec o p, gi en by
p=I k1k2k3cR′ ∗
g0 κ1κ2.(16)
The desi ed cu en idmay also be ob ained om (2)–(4) as
id=R∗
c(λd) + R∗
g(θd)λd,(17)
whe e λdis gi en by (15). This exp ession, which is sligh ly
mo e complex han he p e ious one, depends on h ee ad-
di ional pa ame e s: R∗
c0,λ∗
sa and R∗
g0. Finally, he desi ed
ol age udmigh also be compu ed using (1) as
ud=˙
λd+R id,(18)
whe e ˙
λdis he ime de i a i e o (15) and idis gi en by (17).
Again, his exp ession depends on an addi ional pa ame e : he
esis ance R.
A his poin , i is ob ious ha i all he pa ame e s we e
known, i would be p ac ically he same o ac ua e using
ol age, cu en o lux linkage. Howe e , i he e is any
unce ain y wi h he pa ame e alues, in e ms o obus ness
i is clea ly ad an ageous o con ol he lux linkage, whose
desi ed alue depends on ewe pa ame e s han he o he
a iables. Fo his eason, he eed o wa d con olle p esen ed
in his pape de ines he desi ed ajec o y o he lux linkage.
The ollowing sec ions explain how his desi ed ajec o y is
acked and how he unce ain y o he pa ame e ec o pis
managed.
C. Flux- acking con olle
The lux linkage λis a a iable ha canno be easily
measu ed. Howe e , i is possible o ob ain an es ima e based
only on measu emen s o he elec ical a iables, i.e., ol age
and cu en . In his wo k, we es ima e λby means o he ese
es ima o p esen ed and discussed in [20]. An impo an issue
wi h his es ima o is he esis ance alue, which is equi ed
by he algo i hm bu a ies wi h empe a u e. Ou p oposal is
o es ima e his pa ame e jus be o e each ope a ion. This can
be achie ed h ough Ohm’s law, simply by applying a small
cons an ol age and measu ing he s eady-s a e elec ical
cu en . No e ha he he mal dynamics o he de ice is much
slowe han he elec omechanical one, so i is easonable o
assume ha R emains cons an du ing each ope a ion.
Once we ha e an es ima e o he lux linkage, we can
p oceed o he design o he acking con olle . Ou p oposal is
based only on he dynamics desc ibed by (5). In ha equa ion,
he key is o no e ha RR∗
c(λ) + R∗
g(θ)>0 o all λand
θ[see (3) and (4)]. Thus, he lux linkage dynamics can be
ega ded as a i s -o de linea ime- a ian sys em,
˙
λ( ) = −a( )λ( ) + u( ),(19)
whe e a( ) = RR∗
c(λ( )) + R∗
g(θ( ))is s ic ly posi i e,
which gua an ees he sys em s abili y. Then, o con ol λwe
use a p opo ional-in eg al (PI) con olle in pa allel o m,
u( ) = Kpe( )+KiZ
0
e(τ) dτ, e( ) = λd( )−ˆ
λ( ),(20)
whe e ˆ
λis he es ima e o he lux linkage. Gi en ha (19)
is s ic ly s able, i can be easily shown ha he closed-loop
sys em is also s able o any posi i e Kpand Ki(see p oo
in Appendix A).
D. Run- o- un adap a ion law
The pe o mance o he lux- acking con olle depends
on he accu acy o he model, which is di ec ly used by
he eed o wa d e m. The e a e a ious sou ces o modeling
disc epancies, such as a iabili y be ween uni s due o la ge
manu ac u ing ole ances, o a iabili y be ween ope a ions
o each uni due o mechanical wea . Thus, o inc ease he
con ol obus ness, a lea ning- ype un- o- un adap a ion law
is inco po a ed o i e a i ely adap he model pa ame e s.
Ideally, he e e ence o his con ol loop would be he
angula posi ion, bu posi ion sensing is usually un easible—
he mo ing componen s may no be physically eachable—o
una o dable—posi ion senso s a e much mo e expensi e han
he elays hemsel es. Ne e heless, i is possible o ob ain
auxilia y measu emen s o e alua ing he pe o mance o he
inne -loop lux- acking con olle , such as he impac sound
o he bouncing du a ion. As s a ed in he in oduc ion, many
con ol p oposals o elays a e aimed a educing con ac
bounce, bu ano he undesi able p oblem, he acous ic noise,
is o en neglec ed. Bo h undesi able p oblems a e no en i ely
ela ed, i.e. minimizing con ac bouncing does no necessa ily
imply ha he acous ic noise is also educed. Fu he mo e,
educing he noise in ol es u he di icul ies, e.g. he con ol
should be applied o he en i e a ma u e s oke, e en a e
he elec ical con ac has eached he inal posi ion. Fo hese
easons, his pape p oposes o measu e an audio signal,
υaudio, cap u ed wi h a mic ophone du ing each swi ching
ope a ion. Then, o each i e a ion, a cos Jis compu ed
om his signal in o de o e alua e he ope a ion. This pape
p oposes
J=Z 0+∆
0
υaudio2( ) d , (21)
whe e ∆ should be la ge enough o cap u e all he acous ic
noise gene a ed du ing he swi ching.
Then, he con ol ask is o mula ed as an op imiza ion
p oblem ha sea ches he space o he model pa ame e s ha
a e in ol ed in he eed o wa d e m. The pa ame e ec o p
[see (16)] is modi ied in each swi ching ope a ion in o de o
minimize he cos Jcalcula ed om he audio signal. No e ha
his is a black-box op imiza ion p oblem, because he unc ion
ha ela es he model pa ame e s o he cos is unknown.
As such, he con e gence o an op imal pe o mance poin
depends on he shape o he cos unc ion, he dimension-
ali y o he sea ch space and he con e gence p ope ies o
he op imiza ion algo i hm selec ed o he ask. No e ha ,
in his aspec , he lux-based eed o wa d con olle is also
ad an ageous o e he ol age- o cu en -based ones because
i depends on ewe pa ame e s. The op imiza ion me hod
used in his wo k is based on he Nelde –Mead me hod [21],
which is a widely used and s udied di ec -sea ch op imiza ion
me hod [22]. Al hough i was designed o o line op imiza ion
o de e minis ic unc ions, i has been adap ed o online
op imiza ion o s ochas ic unc ions in se e al wo ks. In hese
cases, howe e , he e is no heo e ical p oo o con e gence
o an op imal poin , so i equi es simula ed o expe imen al
alida ion. Speci ically, his pape uses a modi ied e sion ha
has been p e iously p oposed and alida ed o he con ol o
a di e en class o swi ch- ype ac ua o s [23].
As a inal ema k, i should be no ed 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 acous ic noise due o swi ching impac s. Thus,
ega dless o he op imiza ion me hod, 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 noise.
IV. EXPERIMENTAL VALIDATION
A. P o o ype and expe imen al se up
This subsec ion desc ibes he expe imen al se up (Fig. 5)
used o analyze and alida e he p oposed con ol. The en i e
con ol algo i hm is implemen ed on a low-cos C2000 32-bi
mic ocon olle and es ed wi h a ious TE Connec i i y RZ
elays. I uns he eed o wa d and lux- acking con olle s in
Fig. 5: Expe imen al se up.
Fig. 6: PCB Shield o adap ing he signals be ween he mic ocon olle
and he elec omechancial elay (in o ange).
eal ime (100 kHz). On he o he hand, he un- o- un adap-
a ion law, which is execu ed be ween swi ching ope a ions
wi hou eal- ime cons ain s, equi es only a ew milliseconds
o each i e a ion.
A PCB shield (Fig. 6) has been designed o adap he signals
om he mic ocon olle o he elay and ice e sa. The
shield includes an ADG5436 analog swi ch as powe con e e ,
which supplies a pulse-wid h modula ed signal be ween 0 V
and 35 V a 100 kHz. I also includes a 10 Ω shun esis o
o measu ing he cu en , as well as i s co esponding signal
condi ioning ci cui . The cu en analog signal is il e ed
h ough a 16 kHz low-pass il e , whose ampli ude is se
o he mic ocon olle ADC eading ange. I consis s o 2
ope a ional ampli ie s in Sallen-Key opology, which esul s
in a 4 h o de il e . A Bessel ypology has been selec ed
because his ypology bes p ese es he signal shape. The
shield also includes a low-cos comme cial sound de ec o
(Spa k un BOB-12758 Elec e mic ophone) o measu ing he
impac noise.
The wo kbench is complemen ed by a Keyence lase sen-
so (LK-H082 senso head and LK-G5001P con olle ). The
measu emen o e ed by his senso is ne e used as con ol
eedback, bu only o p elimina y es ima ion o nominal
pa ame e alues. Las ly, all impo an signals a e cap u ed
by an eigh -channel USB oscilloscope (PicoScope 4824), and
Fig. 7: Sound signal in a con en ional making and b eaking swi ching
cycle. The noise is highe du ing he making ope a ion.
sen o a pe sonal compu e o s o age and analysis.
B. Expe imen s, esul s and discussion
As p e iously s a ed, elec omechanical elays pe o m wo
di e en ope a ions: making and b eaking. Al hough he p e-
sen ed con olle is able o imp o e he pe o mance o bo h,
he swi ching sound is much highe in making ope a ions (see
Fig. 7). Fo his eason, he expe imen al analysis p esen ed
in his sec ion is ocused on he making ope a ions.
In o de o ini ialize he con ol p ocess, nominal pa ame e
alues need o be speci ied. The ini ial alue o he pa ame e
ec o p[see (16)] has been ob ained h ough an iden i ica ion
p ocedu e, i.e., i ing he model o expe imen al measu emen s
om one speci ic elay. No e, howe e , ha he con ol p o-
cess may also be ini ialized wi h a bi a ily chosen nominal
pa ame e s. Mo eo e , he desi ed ajec o y ime cons an s
ha e been de ined o closely ma ch he swi ching imes o a
s anda d uncon olled scena io (0 o 24 V squa e signal). The
selec ed imes a e c= 6.5 ms and = 8 ms. Rega ding he
PI con olle [see (20)], he con ol gains ha e been p ese
so ha he closed-loop esponse has a se ling ime (wi h a
5 % e o band) less han o equal o 2ms and a damping
coe icien g ea e han o equal o 1. Then, hey ha e been
expe imen ally ine- uned, esul ing in he pa ame e alues
Kp= 37500 Wb/V and Ki= 1.15 ×108Wb/(Vs).
In a eal scena io, only one i e a i e con ol p ocess would
be needed o con ol any gi en elay. None heless, due o he
high a iabili y be ween and wi hin elays, we ha e decided o
conduc se e al ials in o de o show he a e age pe o mance
and he a iabili y o he con ol p ocess. Speci ically, he
p oposed con ol algo i hm has been applied o 10 di e en
elays and, o each one, 10 epe i ions o he con ol p ocess
ha e been execu ed. Thus, 100 di e en ials ha e been
pe o med, each one s a ing om he nominal pa ame e
se . In addi ion o ha , no e also ha in a eal si ua ion
he con olle would be wo king inde ini ely, adap ing i s
pa ame e s each ime he elay is swi ched. Howe e , in he
g aphs in his sec ion we only show he esul s ob ained
o he i s 300 swi ching ope a ions, which is a su icien
numbe o show he con e gence o he me hod. In o de o
demons a e he obus ness o esis ance a ia ions, a 150 Ω
esis o has been added in se ies o he coil du ing he las
50 ope a ions. This emula es a sudden empe a u e inc ease
o 25 ◦C, which would imply a 10% inc ease in he coppe
elec ical esis i i y. Addi ionally, o compa ison pu poses,
i e s anda d swi ching ope a ions wi h a cons an 24 V ol age
signal ha e been ca ied ou wi h no addi ional esis ance, and
(a)
(b)
Fig. 8: Dis ibu ion o alues o he esis ance (a) and no malized cos s
(b) o he 100 expe imen al ials as a unc ion o he numbe o
ope a ions. Bo h g aphs include he median alue (p50), he in e qua ile
ange ([p25, p75]) and he 10 h o 90 h pe cen ile in e al ([p10, p90]) o
he co esponding dis ibu ions. The cos s g aph (b) shows he alues
ob ained wi h con ol and wi h s anda d ope a ions.
ano he i e s anda d ope a ions wi h he 150 Ω se ies esis o .
No e ha , in a p ac ical scena io, en i onmen al noise can
nega i ely a ec he con ol pe o mance. Thus, o a ai e
expe imen al alida ion, he es s ha e been ca ied ou wi h
no sound isola ion. I is also wo h ema king ha he elays
a e enclosed and he e o e a isual inspec ion o hei mo ing
pa s is no possible. Thus, in o de o main ain he in eg i y
o he elays, he angula posi ion has no been measu ed.
In he es s, he esis ance alue has been es ima ed o each
o he 300 ope a ions o he 100 pe o med expe imen al ials.
The ob ained dis ibu ion o alues is ep esen ed in Fig. 8a,
by means o he 10 h, 25 h, 50 h, 75 h and 90 h pe cen iles
(p10,p25,p50,p75 and p90, espec i ely), as a unc ion o he
numbe o ope a ions, n∈[1 . . 300]. I is shown ha he e
is some a iabili y due o empe a u e luc ua ions du ing he
expe imen al es ing, and due o di e ences be ween he coils
o he es ed elays. None heless, he majo esis ance changes
occu be ween n= 250 and n= 251, when he se ies esis o
is connec ed.
Then, Fig. 8b summa izes he con ol esul s. Fo a clea
compa ison, all he compu ed cos s ha e been no malized
wi h espec o he median cos o he s anda d swi ching
ope a ions a oom empe a u e (i.e., wi hou he se ies esis-
o ). Equi alen ly o he esis ance plo , his g aph shows he
dis ibu ion o alues o he no malized cos , Jno m, ob ained
o he 100 expe imen al ials pe o med, as a unc ion o
he numbe o con ol ope a ions. Fo compa a i e pu poses,
(a) (b) (c)
(d) (e) ( )
(g) (h) (i)
Fig. 9: E olu ion o he con ol pe o mance o a single elay. Vol age (a), lux linkage (d) and audio (g) in a s anda d ope a ion. Vol age (b), lux
linkage (e) and audio (h) in he las con ol i e a ion wi h low esis ance (n= 250). Vol age (c), lux linkage ( ) and audio (i) in he i s con ol i e a ion
wi h inc eased esis ance (n= 251).
i also displays he pe cen iles o he con en ional swi ching
ope a ions wi hou (n≤250) and wi h (n≥251) he se ies
esis o . The igu e shows he high a iabili y be ween he
di e en es ed de ices, e en hough hey a e all o he same
model. This a iabili y is mo e e iden a he beginning o
he con ol p ocess. No e ha he cos s a e la ge han in
he uncon olled scena ios, e en hough he con ol has been
ini ialized using he es ima ed pa ame e s om a simila elay.
This shows he impo ance o he i e a i e adap a ion o he
model pa ame e s, as explained in Sec ion III-D. Then, he
con ol esul s imp o e g ea ly as he numbe o i e a ions
inc eases. I is shown ha he con ol equi es abou 50
i e a ions o imp o e he median wi h espec o s anda d
ope a ions. In a simila way, he con ol needs a ound 100
i e a ions o imp o ing he 90 h pe cen ile. The esul s also
show ha he con ol is obus agains esis ance a ia ions,
as he sudden esis ance change does no seem o a ec he
no malized cos s. In con as , he s anda d ope a ions a e qui e
sensi i e o empe a u e. In his ega d, i can be seen ha when
he esis ance inc eases, he median cos dec eases and he 10
o 90 h pe cen ile in e al widens.
As a mo e in ui i e demons a ion o he con ol pe o -
mance and i s obus ness agains esis ance a ia ions, a com-
pa ison o ol age, lux linkage and audio signals is p esen ed
in Fig. 9. These esul s co espond o he ial wi h he la ges
audio cos s in he s anda d uncon olled scena io. Figs. 9a, 9d
and 9g ep esen he ol age applied in a s anda d ope a ion,
he es ima ed lux linkage and he esul ing audio signal,
espec i ely. Then, Figs. 9b, 9e and 9h show he same signals
o he 250 h con ol i e a ion. As can be seen in Fig. 9e,
he lux- acking con ol wo ks co ec ly. The acking only
ails a he end o he con ol in e al, i.e., close o 8ms,
bu his is due o ol age sa u a ion (see Fig. 9b). Despi e he
ol age signal limi a ions o he implemen a ion, i manages o
educe conside ably he audio in ensi y, which indica es ha
he swi ching impac s a e also dec eased. Then, Figs. 9c, 9
and 9i co espond o he 251 h con ol i e a ion, in which he
esis ance inc eases by 150 Ω. Due o his change, he esul ing
ol age signal is modi ied (see Fig. 9c). Howe e , as expec ed,
he desi ed lux is una ec ed by his change (see Fig. 9 ).
Fu he mo e, he lux- acking con ol pe o ms e y simila ly
despi e he sudden change in i s dynamics and, consequen ly,
he esul ing noise educ ion is almos iden ical (see Fig. 9i).
C. Fu he esul s: lux-based s. ol age-based con ol
As explained in Sec ion III, he p oposed lux-based eed o -
wa d con olle is ad an ageous o e he cu en - and ol age-
based coun e pa s because i depends on ewe pa ame e s.
To complemen he heo e ical explana ion, his subsec ion
includes an expe imen al compa ison o ou p oposal wi h an
al e na i e ol age-based con olle . This con olle is based
on he same p inciples as he o he , bu he eed o wa d block
calcula es di ec ly he inpu ol age om he desi ed ajec o y
as desc ibed in (18). The lux- acking con olle and es ima o
om Sec ion III-C a e hus no used in his case. None heless,
no e ha he eed o wa d con olle s ill uses he esis ance
es ima ion in o de o calcula e he ol age signal.
Fig. 10: Median alues o he no malized cos s o he 100 expe imen al
ials as a unc ion o he numbe o ope a ions. Compa ison be ween
he lux-based con olle and he al e na i e ol age-based con olle .
The al e na i e con olle has been expe imen ally es ed
in he same manne and using he same en elays as in
he p e ious subsec ion o he i s 250 i e a ions (i.e., wi h
no se ies esis o ). The ob ained esul s a e compa ed in
Fig. 10 wi h he ones om he lux-based con olle . Fo cla i y
easons, he g aphic depic s only he median cos s Jno m o
he con olle s ins ead o he comple e dis ibu ions o alues.
The e is an ini ial phase in which he un- o- un adap a ion
law pe o ms a b oad sea ch in he pa ame e space. This can
be seen in he ac ha he cos s ob ained wi h any o he
wo con olle s a y g ea ly in he i s 20 ope a ions. Fo
he ollowing ope a ions, bo h con olle s s a o con e ge,
esul ing in dec easing cos s. No e ha , al hough he esul s a e
ini ially simila , he e is a la ge a iabili y be ween consecu-
i e ope a ions o he ol age-based con olle . Then, o he
las ope a ions (n≥150), he mos no iceable aspec is ha he
a e age cos s om he lux-based eed o wa d con olle a e
consis en ly smalle , which con i ms ha he p oposed design
pe o ms be e han he ol age-based e sion.
V. CONCLUSION
In his pape we ha e p oposed an audio-based con ol
s a egy o educe he nega i e e ec s o impac s du ing elay
swi ching ope a ions, accoun ing o he nonlinea dynamics
and high a iabili y o hese de ices. The expe imen al esul s
show ha his con ol p oposal is able o educe no only he
a e age impac noise, bu also i s a iabili y.
The p oposal includes lux-based eed o wa d and eedback
con ol blocks, which ha e se e al ad an ages o e ol age- o
cu en -based ones. Mos impo an ly, hey make he con ol
mo e obus agains modeling e o s and empe a u e a ia-
ions. Addi ionally, an adap i e lea ning- ype un- o- un block
is inco po a ed o ci cum en modeling and o he e o s and o
imp o e he con ol pe o mance. This is specially use ul o
de ices wi h high a iabili y be ween ope a ions o be ween
uni s, as he s udied case. This p oposed un- o- un adap a ion
law elies on mic ophone measu emen s, bu , in a mo e
gene al case, hey can be easily eplaced o complemen ed
wi h auxilia y measu emen s, e.g. he con ac ol ages o
de ec ing bounces. Mo eo e , he modula con ol s uc u e is
e y e sa ile, pe mi ing he eplacemen o modi ica ion o
any o he con ol blocks.
APPENDIX
A. S abili y p oo o he lux- acking con olle
As s a ed, he lux linkage dynamics (19) is con olled using
a PI con olle in pa allel o m. Le he con olle equa ion (20)
be e o mula ed as
u( ) = Kpλd( )−ˆ
λ( )+Kiσ( ),(22)
˙σ( ) = λd( )−ˆ
λ( ),(23)
whe e σis he in eg al o he e o . Then, using (19), (22) and
(23), i is possible o desc ibe he closed-loop dynamics as
˙
λ( )
˙σ( )=−a( )−KpKi
−1 0 λ( )
σ( )
+KpKp
1 1 λd( )
˜
λ( ),(24)
whe e ˜
λ=λ−ˆ
λis he es ima ion e o . The eigen alues o
he closed-loop sys em ma ix a e gi en by
eig −a( )−KpKi
−1 0
=−a( )−Kp±q(a( ) + Kp)2−4Ki
2.(25)
Since a( )>0 o all , he eigen alues o he sys em will
ha e nega i e eal pa o any posi i e alue o Kpand Ki.
Thus, i can be concluded ha he closed-loop sys em is s able
i Kp>0and Ki>0. The s eady s a e egime is gi en by
λ( )
σ( )=1 1
a( )/Kia( )/Ki λd( )
˜
λ( ),(26)
which shows ha λcon e ges o he desi ed alue λdp o ided
ha he es ima ion e o ˜
λcon e ges o ze o.
REFERENCES
[1] M. Naidu, S. Gopalak ishnan, and T. W. Nehl, “Faul - ole an pe manen
magne mo o d i e opologies o au omo i e X-by-wi e sys ems,” IEEE
T ans. Ind. Appl., ol. 46, no. 2, pp. 841–848, Ma . 2010.
[2] J. Ace o, J. M. Bu dio, L. A. Ba agan, D. Na a o, R. Alonso, J. R.
Ramon, F. Mon e de, P. He nandez, S. Llo en e, and I. Ga de, “Domes ic
induc ion appliances,” IEEE Ind. Appl. Mag., ol. 16, no. 2, pp. 39–47,
Ma . 2010.
[3] S. Haghbin, S. Lundma k, M. Alakula, and O. Ca lson, “G id-connec ed
in eg a ed ba e y cha ge s in ehicle applica ions: Re iew and new
solu ion,” IEEE T ans. Ind. Elec on., ol. 60, no. 2, pp. 459–473, Feb.
2013.
[4] T. C. Beh, M. Ka o, T. Imu a, S. Oh, and Y. Ho i, “Au oma ed Impedance
Ma ching Sys em o Robus Wi eless Powe T ans e ia Magne ic
Resonance Coupling,” IEEE T ans. Ind. Elec on., ol. 60, no. 9, pp.
3689–3698, Sep. 2013.
[5] P. Ba kan, “A S udy o he Con ac Bounce Phenomenon,” IEEE T ans.
Powe App. Sys ., ol. PAS-86, no. 2, pp. 231–240, 1967.
[6] J. McB ide, “An expe imen al in es iga ion o con ac bounce in medium
du y con ac s,” IEEE T ans. Compon., Hyb ids, Manu . Technol., ol. 14,
no. 2, pp. 319–326, Jun. 1991.
[7] T. S. Da ies, H. Nou i, and F. W. B i on, “Towa ds he con ol o
con ac bounce,” IEEE T ans. Compon., Packag., Manu . Technol. A,
ol. 19, no. 3, pp. 353–359, 1996.
[8] P. M. d. S. D. de Mo aes and A. J. Pe in, “An Elec onic Con ol Uni o
Reducing Con ac Bounce in Elec omagne ic Con ac o s,” IEEE T ans.
Ind. Elec on., ol. 55, no. 2, pp. 861–870, 2008.
[9] E. Rami ez-Labo eo, C. Sagues, and S. Llo en e, “A New Run- o-Run
App oach o Reducing Con ac Bounce in Elec omagne ic Swi ches,”
IEEE T ans. Ind. Elec on., ol. 64, no. 1, pp. 535–543, Jan. 2017.
[10] L. Tang, Z. Han, and Z. Xu, “A Sequen ial Adap i e Con ol S a egy
o he Con ac Colliding Speed o Con ac o s Based on Fuzzy Con ol,”
IEEE T ans. Ind. Elec on., ol. 68, no. 7, pp. 6064–6074, Jul. 2021.
[11] E. Moya-Lashe as, E. Rami ez-Labo eo, and C. Sagues, “P obabili y-
Based Op imal Con ol Design o So Landing o Sho -S oke Ac ua-
o s,” IEEE T ans. Con ol Sys . Technol., ol. 28, no. 5, pp. 1956–1963,
Sep. 2020.
[12] B. Ca se, N. La sen, H. Nou i, and T. Da ies, “An app oach o he
educ ion o con ac bounce using uzzy con ol,” in P oc. IEEE In .
Symp. Ind. Elec on., ol. 3, pp. 1025–1029. Bled, Slo enia: IEEE,
1999.
[13] X. Wang, H. Lin, S. L. Ho, S. Fang, and P. Jin, “Analysis o Dy-
namic Cha ac e is ics o Pe manen Magne Con ac o Wi h Senso less
Displacemen P o ile Con ol,” IEEE T ans. Magn., ol. 46, no. 6, pp.
1633–1636, Jun. 2010.
[14] A. G. Espinosa, J. R. R. Ruiz, and X. A. Mo e a, “A senso less me hod
o con olling he closu e o a con ac o ,” IEEE T ans. Magn., ol. 43,
no. 10, pp. 3896–3903, Oc . 2007.
[15] H. Lin, X. Wang, S. Fang, P. Jin, and S. L. Ho, “Design, Op imiza ion,
and In elligen Con ol o Pe manen -Magne Con ac o ,” IEEE T ans.
Ind. Elec on., ol. 60, no. 11, pp. 5148–5159, No . 2013.
[16] L. Tang, H. Qu, and Z. Xu, “Resea ch on Double Closed-Loop Con ol
S a egy o Con ac o s Based on Flux Linkage Obse e s,” IEEE T ans.
Ind. Elec on., ol. 69, no. 3, pp. 2769–2779, Ma . 2022.
[17] E. Moya-Lashe as, C. Sagues, and S. Llo en e, “An e icien dynamical
model o eluc ance ac ua o s wi h lux inging and magne ic hys e e-
sis,” Mecha onics, ol. 74, p. 102500, Ap . 2021.
[18] E. Rami ez-Labo eo, C. Sagues, and S. Llo en e, “A New Model o
Elec omechanical Relays o P edic ing he Mo ion and Elec omag-
ne ic Dynamics,” IEEE T ans. Ind. Appl., ol. 52, no. 3, pp. 2545–2553,
May. 2016.
[19] J. L´
e ine, “On necessa y and su icien condi ions o di e en ial
la ness,” Appl. Algeb a Eng., Commun. Compu ., ol. 22, no. 1, pp.
47–90, 2011.
[20] E. Rami ez-Labo eo, E. Moya-Lashe as, and C. Sagues, “Real- ime
elec omagne ic es ima ion o eluc ance ac ua o s,” IEEE T ans. Ind.
Elec on., ol. 66, no. 3, pp. 1952–1961, 2019.
[21] J. A. Nelde and R. Mead, “A Simplex Me hod o Func ion Minimiza-
ion,” Compu . J., ol. 7, no. 4, pp. 308–313, 1965.
[22] T. G. Kolda, R. M. Lewis, and V. To czon, “Op imiza ion by di ec
sea ch: New pe spec i es on some classical and mode n me hods,” SIAM
Re iew, ol. 45, no. 3, pp. 385–482, 2003.
[23] E. Moya-Lashe as and C. Sagues, “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, ol. 25, no. 6, pp. 2645–2656, Dec.
2020.
