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Con ol Design o Mixed Sensi i i y P oblem o
Educa ional Model o Helicop e
S epan OZANA , Pe VOJCINAK , Ma in PIES , Rado an HAJOVSKY
Depa men o Cybe ne ics and Biomedical Enginee ing, Facul y o Elec ical Enginee ing and Compu e
Science, VSB–Technical Uni e si y o Os a a, 17. lis opadu 15, 708 33 Os a a-Po uba, Czech Republic
[email p o ec ed], pe . o[email p o ec ed], [email p o ec ed], ado an.hajo[email p o ec ed]
Abs ac . The pape deals wi h he design o H-∞ o-
bus con olle , pa icula ly wi h mixed sensi i i y p ob-
lem o ele a ion con ol. I b ie ly in oduces basic
ma hema ical backg ound conce ning obus con ol ap-
p oach, which is hen applied o ypical example o
MIMO sys em, ha is a helicop e model. The ob ained
esul s a e e i ied on eal educa ional physical model
CE 150 by Humuso , l d.
Keywo ds
Algo i hms and so wa e, simula ion o dy-
namic sys ems, obus con ol o nonlinea sys-
ems.
1. In oduc ion
The objec i e o he obus con ol is o design a
dynamic con ol sys em ope a ing in a eal en i on-
men . The changes o he su ounding condi ions can
be caused by he ollowing ac o s acco ding o [15]:
•componen aging,
• empe a u e e ec ,
•e ec o he wo king en i onmen .
The con ol sys em mus no only be esis an o
he a o emen ioned ac o s bu i also mus elimina e
inaccu acy o he model, i.e. obus ness is he ele an
abili y o he con ol sys em o accep changes. The
equi ed ou pu alue will be eached e en when he
changes in he p ope ies o he con olled sys em a e
limi ed and cons an dis u bance signals a e ope a ing.
F om he ma hema ical poin o iew, he obus con-
olle is no only sui able o one pa icula sys em
bu o a se o sys ems [15].
In o he wo ds, obus ness plays a signi ican ole
in he design o con ol sys ems as eal sys ems a e
p one o ex e nal dis u bances and measu emen noise.
Mo eo e , he e a e o en di e ences be ween he p o-
posed ma hema ical models and ac ual eal sys ems.
A ypical example is he design o a con olle ha
will s abilize he sys em e en i i is o iginally uns a-
ble and accep a pa icula le el o pe o mance a he
p esence o dis u bance signals, noises, ha d- o-model
p ocess dynamic cha ac e is ics o p ocess pa ame e
a iables. Such asks a e bes sol ed by a eedback
con ol mechanism as hey b ing along a whole ange
o p oblems acco ding o [4]:
•high p ice (e.g. use o senso s),
•sys em complexi y (e.g. possibili ies o implemen-
a ion and eliabili y),
•sys em s abili y (e.g. equi emen o in e nal s a-
bili y and s abilizing con olle s).
The need and signi icance o obus ness as a pa
o con ol sys em designs ha e been de eloping since
1980s. Robus ness in he s anda d SISO con ol is
p o ided by a sui able gain ma gin and phase ma -
gin. When he i s design echniques o mul i- a iable
sys ems de eloped in 1960s, emphasis was laid on he
achie emen o good pe o mance, no obus ness. The
me hods ha use mul i a iables we e based on he
linea –quad a ic c i e ion and Gaussian dis u bances.
I was demons a ed ha hey can be success ully used
in a whole ange o a ia ion applica ions whe e i is
possible o se up p ecise ma hema ical models, in-
cluding he desc ip ions o ex e nal dis u bance signals
o noises. Howe e , he applica ion o hese me h-
ods, called LQG me hods (linea -quad a ic Gaussian
con ol), in o he indus ial b anches e iden ly showed
hei bad p ope ies om he poin o obus ness which
led o he e o o de elop a heo y ha would explic-
i ly deal wi h he issue o obus ness in he con ol
eedback design. The pionee ing wo k on he de el-
opmen o he heo y, oday known as he heo y o
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op imal con ol in H-∞, was in oduced a he begin-
ning o 1980s by Geo g Zames and B uce A. F ancis.
The H-∞app oach i s speci ied he model o sys em
unce ain y, i.e. addi i e pe u ba ion and/o ou pu
dis u bances. In mos cases, i is enough o ind a sui -
able con olle so ha he closed loop achie es some
obus s abili y. The pe o mance is also a pa o he
op imiza ion loss (objec i e) unc ion. The elegan o -
mula ions o he solu ion a e based on he solu ions o
Ricca i equa ions, e.g. in MATLAB [4].
I we design a con olle o a pa icula in e al o
pa ame e s, hen he con ol ci cui is obus ly s able.
Ano he impo an pa ame e o obus con olle s is
hei pe o mance, mee ing he equi emen s o pa-
ame e s acco ding o [15]:
•con ol,
•dis u bance,
•speed o esponse (se ling ime).
The p oblem o he design o he obus con olle
is based on he eedback ci cui (closed loop) which
enables wo king wi h he sensi i i y and elimina ion o
he dis u bance. On one side, he eedback o a non-
s able sys em s abilizes i ; on he o he side, i may
des abilize a s able sys em [15]. Conside he s anda d
con ol diag am acco ding o Fig. 1.
Fig. 1: Classical con ol scheme wi h he de ini ion o ci cui
signals.
Desc ip ion o he signals in he con ol ci cui :
•W(s): Laplace ans o m o he he e e ence sig-
nal,
•E(s): Laplace ans o m o he con ol e o sig-
nal,
•U(s): Laplace ans o m o he manipula ed alue
signal,
•V1(s): Laplace ans o m o he dis u bance: low-
equency known and unknown dis u bances; he
sys em mus elimina e hem,
•V2(s): Laplace ans o m o he dis u bance: sen-
so s o measu emen , high- equency cha ac e
wi h insigni ican e ec .
We use he ules o block algeb a o de ine he ma he-
ma ical ela ions wi hin he con ol ci cui , i.e. acco d-
ing o [15], [9]:
• o he open-loop ans e unc ion:
L(s) = K(s)G(s),(1)
• o he ans e unc ion o he con ol e o - he
sensi i i y unc ion:
GE(s) = E(s)
W(s)=1
1 + L(s)=S(s),(2)
• o he closed-loop ans e unc ion - he comple-
men a y sensi i i y unc ion:
GW(s) = Y(s)
W(s)=L(s)
1 + L(s)=T(s),(3)
•limi ing condi ion applies o he sum o he sen-
si i i y unc ion and complemen a y sensi i i y
unc ion:
S(s) + T(s) = 1
1 + L(s)+L(s)
1 + L(s)= 1.(4)
Conside he equi emen s o he sensi i i y unc-
ion and complemen a y unc ion acco ding o [15]:
•a 1( ) = 2( )=0, he e ec o he con ol ac-
ion p edomina es and so he sensi i i y unc ion
ˆ
GE(s) = S(s)mus be small and he complemen-
a y unc ion GW(s) = T(s)will be la ge,
• he en i e con ol ci cui ca ies ou he elimina-
ion o he low- equency noise 1( ), and so again,
he e ec o he con ol ac ion p edomina es: he
sensi i i y unc ion GE(s) = S(s)mus be small
and he complemen a y unc ion GW(s) = T(s)
will be la ge,
• he en i e con ol ci cui mus no a ec he elimi-
na ion o he high- equency noise 2( )and so we
elimina e he e ec s o he con ol ac ion and con-
ol e o , i.e. he sensi i i y unc ion GE(s) =
S(s)mus be small and he complemen a y sensi-
i i y unc ion GW(s) = T(s)will also be small.
As he a o emen ioned opposing equi emen s can-
no be me by one con olle , i is necessa y o ind a
comp omise be ween he sizes o he sensi i i y unc-
ion and he complemen a y sensi i i y unc ion [15],
see Fig. 2.
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Fig. 2: Loga i hmic ampli ude equency cha ac e is ics o he
sensi i i y unc ion (on he le ) and complemen a y
sensi i i y unc ion (on he igh ).
2. Robus Con olle Design
2.1. Robus Design Me hods
The me hods o he s a e space in he ime domain
allowed o a oid he p oblems wi h ans e unc ion
ma ices and also p o ided means o he analysis and
design o MIMO sys ems wi h mo e inpu s and ou -
pu s. App oxima ely a he same ime when he me h-
ods o op imal con ol we e being de eloped, esea ch
ocused on he ex ensions o means o MIMO sys em
s anda d con ol was conduc ed. The obus design is
based on he inding o such a con olle so ha he
esul ing sys em in he closed loop is also obus . Ro-
bus ness became he main s andpoin in he ield o
con ol, he e o e speci ica ions and me hods ollowed
sho ly, i.e. acco ding o [15], [5]:
•H∞me hod,
•H2me hod,
•LTR me hod (loop ans e eco e y),
•µ: syn hesis,
•QFT me hod (quan i a i e eedback heo y),
•Kha i ono heo em o he examina ion o obus
s abili y,
•speci ica ion o he small-gain heo em,
•speci ica ion o s uc u ed singula alues.
The ollowing p esen a ion will only ocus on he H∞
me hod.
2.2. H∞Me hod
A con ol sys em is obus i i s ays s able and mee s
pa icula beha io al c i e ia a he p esence o possi-
ble unce ain ies. The H∞op imiza ion me hod, de-
eloped since 1980s, has p o ed o be a e y e icien
and po en design me hod o obus con ol in he ield
o linea , ime-in a ian con ol sys ems [15], [11], [3].
H∞con olle s ha e hei own e minology, no a ion
and concep ion. This me hod leads o a se o sui -
able s able ans e unc ions ha a e physically iable.
Simila ly o LQR and LQG con olle s, we expec op-
imiza ion o he objec i e unc ion ha will compa e
di e en ans e unc ions and selec he mos sui able
one om he se . The equi emen s o he closed loop
a e he ollowing, i.e. acco ding o [15]:
•Physical iabili y: The o de o he ans e unc-
ion denomina o mus be highe o equal o he
o de o he ans e nume a o .
•S abili y: The ans e poles mus lie in he le
hal plane o he Gaussian plane o in he a ea o
he Laplace ans o m con e gence (p o ided ha
he con ol s aigh line and imagina y axis a e
iden ical).
The basic p e equisi e o he H∞me hod is he
knowledge o he ans e unc ion o he gi en sys em,
e alua ing ∞-no m acco ding Fig. 5 acco ding o [15],
[1]:
kGk∞=supω{|G(jω)|} .(5)
The no m can be g aphically ep esen ed as he max-
imum o he Bode diag am p o ided ha he ans e
unc ion is de ini e and has no imagina y poles, while
i s objec i e is o minimize i in he ∞-no m. I de-
c eases he apex o he Bode diag am, which inc eases
he obus s abili y ma gin [15].
2.3. Mixed Sensi i i y P oblem
Usually, p ac ical indus ial applica ions do no only
use one objec i e unc ion bu a combina ion o se e al
unc ions like ha , e.g. accomplishmen o he good
pe o mance o acking he e e ence signal a limi ed
ene gy o he e e ence signal. Then we sol e he mixed
sensi i i y ask, o he ’S o e KS’ p oblem de ined by
a gene al ela ion ( o he SISO sys em) acco ding o
[8], [12]:
minKs 



S(s)
K(s)S(s)


∞
=
= minKs 



[1 + L(s)]−1
K(s) [1 + L(s)]−1


∞
.(6)
The Eq. 6 can also be exp essed by he equi emen s
o he design conce ning he addi i e pe u ba ion, e.g.
nominal beha io , good pe o mance o acking he
e e ence signal o he elimina ion o dis u bance sig-
nals and obus s abili y [4].
Figu e 3 shows he s anda d block diag am o he
H∞con igu a ion using he linea ac al ans o ma-
ion (LFT) wi h speci ica ion o he indi idual ex e nal
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Fig. 3: S anda d block diag am a he H∞con igu a ion.
inpu s, ex e nal ou pu s, inpu s in o he con olle and
i s ou pu s, see [1] (p. 438). The con ol ci cui con-
ains a obus con olle wi h ans e unc ion K(s)
and a pe u bed (also ex ended o gene alized) sys em
wi h ans e unc ion P(s) ha has wo inpu s and
wo ou pu s acco ding o [15]:
•~w ( )inpu e e ence signal ec o ; ex e nal inpu
signals,
•~ ( )ou pu manipula ed signal ec o ; ou pu
con ol signals om he con olle .
The main di e ence be ween he ec o s is ha he
con olle does no a ec he inpu s. The inpu e e -
ence signal ec o ~w ( )includes an ex e nal noise, noise
om he senso s and acking ( e e ence) signals. To
he con a y, he ou pu s om he sys em a e di ided
in o wo g oups acco ding o [15]:
•~y ( )ou pu signal ec o ; measu ed ou pu s,
•~z ( )con olled ou pu s; minimized o penalized
ou pu s.
The ask is hen de ined so ha he in e nally s a-
bilizing con olle K(s)is sea ched o in he con ol
ci cui o he obus con ol o he gi en gene alized
sys em P(s) ha minimizes o penalizes he con olled
ou pu ec o ~z ( ). In o he wo ds, we minimize he
maximal no m o he ans e unc ion be ween ~w ( )
and ~z ( )by he gi en ela ion acco ding o [4]:
~z =P11 (s) + P12 (s)K(s)I−1
P K P21 (s)~w, (7)
whe e:
IP K =I−P22 (s)K(s),(8)
we ge a linea ac al ans o ma ion a e he adjus -
men :
~z =Fl[P(s), K (s)] ~w. (9)
Then, he H∞op imiza ion p oblem can be ex-
p essed by a ela ion, i.e. acco ding o [4]:
min
Ks
kFl[P(s), K (s)]k∞.(10)
Fig. 4: S anda d block diag am o he mixed sensi i i y p ob-
lem: con olle and pe u bed sys em con aining nomi-
nal sys em, con ol e o and manipula ed alue weigh -
ing il e s.
Figu e 4 shows he s anda d block diag am o he
mixed sensi i i y p oblem and i is basically a mo e
de ailed illus a ion o Fig. 3, whe e i is easy o deduce
he ollowing ela ions, i.e.:
• o ex e nal inpu signals:
~w ( ) = ( ),(11)
• o ou pu con ol signals om he con olle :
~u ( ) = u( ),(12)
• o measu ed ou pu s:
~y ( ) = e( ),(13)
• o minimized o penalized ou pu s:
~z = [z1( ), z2( )]T=
= [W1(s)e( ), W2(s)u( )]T.(14)
The ollowing applies o he gene alized sys em con-
aining weigh ing il e s acco ding o [4]:
P(s) = P11 (s)P12 (s)
P21 (s)P22 (s),(15)
while:
P11 (s) = W1(s) [I, 0]T= [W1(s),0]T,(16)
P12 (s)=[−W1(s)G(s), W2(s)I]T,(17)
P21 (s) = [I]T=I, (18)
P22 (s) = [−G(s)]T=−G(s).(19)
The weigh ing il e s W1(s)and W2(s)a e com-
monly used in p ac ice; in such case, he Eg. 6 can
be o mally adjus ed in o he o m desc ibing a objec-
i e unc ion, i.e. ( o he SISO sys em) acco ding o
[4]:
min
Ks 



W1(s)S(s)
W2(s)K(s)S(s)


∞
.(20)
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3. Con olle Design o
Helicop e Ele a ion
On he basis o he a o emen ioned heo e ical indings
om he ield o he H∞ obus con ol, we can design
a obus con olle acco ding o he selec ed concep ion
when we deal wi h he ques ion o he au onomy o he
con ol and he p oblem o mixed sensi i i y and also
conside he inpu signals. De i ed ma hema ical mod-
els in ele a ion and azimu h a e c ucial o he design
o con olle s, when we design he con olle o bo h
he ma hema ical model and eal model. Howe e , his
pape only ocuses on ele a ion, due o he ex en o
he ask. Finally, bo h esponses a e compa ed o one
ano he .
The ma hema ical desc ip ion o helicop e model is
aken om o icial manual, see [6]. The design o ele a-
ion con olle pa icula ly comes ou om Figs. 2.1,
2.2 s a ed on pages 8, 15 o his manual.
3.1. Au onomy Requi emen
The main p oblem o his MIMO sys em namely in-
cludes he elimina ion o he ele a ion-azimu h cou-
pling. All in all, we na u ally wan o elimina e bo h
ela ions bu we know ha he azimu h-ele a ion ela-
ion is no as signi ican . The concep ion is hus based
on he assump ion o designing wo independen con-
olle s, i.e. an ele a ion con olle and an azimu h
con olle .
The de ini ion o au onomy says ha he e e ence
signal ~w ( )mus only a ec jus one co esponding
ou pu signal ~y ( ), [7].
Au onomy also equi es ha he ans e ma ix o
he open loop is diagonal, he elemen s o he ma ix
a e only loca ed on he main diagonal. A p io i, diago-
nali y is equi ed in he con ol ma ix. Wi h espec o
he explici ly w i en sign, he ans e unc ion o he
co ec ion e m is gi en by he gene al ela ion based
on he heo y o ma ix de e minan s (Laplace comple-
men ) acco ding o [2]:
Ri,j (s)=(−1)j+1Rj,j (s)G∗
j,i (s)
G∗
j,j (s)
,(21)
whe e iis ma ix line index, jis ma ix column in-
dex, (−1)j+1 is Laplace (algeb aic) complemen , G∗
j,i
is ma ix de e minan G∗
j,i (s)and G∗
j,jis ma ix de-
e minan G∗
j,j (s).
The ela ion Eq. (21) a he has a heo e ical cha -
ac e . Di ec ly de i ed condi ions o au onomy pay
o a a small amoun o egula ed signals (in ou case
he e a e wo signals) a he han ha ing o exac ly
emembe i s con en and a oid making a mis ake in
he algeb aic complemen . The si ua ion is depic ed in
Fig. 5, di ec ly modi ied o he helicop e model.
mech.pa
(ele a ion)
G (s)
1M (s)
1(emp.)
U1(mo )
mech.pa
(azimu h)
R (s)
21
G (s)
11
G (s)
21
G (s)
22
uM
uT+++++
G (s)
M (s)
(emp.)
U1( eak)
G (s)
2M (s)
2(emp.)
U2(mo )
Fig. 5: Block diag am o he elimina ion o he ele a ion-
azimu h coupling; he azimu h-ele a ion coupling is ne-
glec ed.
I he manipula ed alue exp essed by ol age UM
induces an undesi able esponse in he ou pu o he az-
imu h mechanical pa desc ibed by he Laplace ans-
o m G21 (s)UM(s), hen i can be comple ely com-
pensa ed by he co ec ion e m R21 (s)unde he con-
di ion acco ding o [2]:
G21 (s)UM(s) + G22 (s)R21 (s)UM(s)=0
⇒R21 (s) = −G21(s)
G22(s)
.(22)
The ans e unc ion o he second co ec ion e m
can be ei he w i en di ec ly wi h he use o he p in-
ciple o cyclical subs i u ion o indexes o a condi ional
equa ion can be se up again Eq. (23), i.e. acco ding
o [2]:
G12 (s)UT(s) + G11 (s)R12 (s)UT(s)=0
⇒R12 (s) = −G12(s)
ˆ
G11(s)
.(23)
The ela ion Eq. (23) desc ibes he azimu h-
ele a ion coupling ha , howe e , is no signi ican , and
hus can be w i en acco ding o [2]:
R12 (s) = G12 (s) = 0.(24)
The diag am in Fig. 5 can also be in e p e ed in he
ollowing way: he in e nal physical coupling be ween
he ele a ion and azimu h in he o m o ans e unc-
ion G21 (s)canno be elimina ed wi hou he basic
(cons uc i e) in e e ence in o he sys em. Howe e ,
we can qui e easily implemen an ex e nal connec ion
be ween he inpu s o he helicop e se by he co -
ec ion e m R21 (s) ha will ensu e he same as he
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un easible elimina ion o he c oss-coupling inside he
MIMO sys em ans e ma ix, decomposi ion o he
model in o wo models ha seemingly do no in luence
one ano he . Thei con ol can hen be ensu ed by wo
con ol loops independen o one ano he [2].
I we subs i u e in he ela ion Eq. (23), we will e-
cei e ans e unc ion o he co ec ion e m, Eq. (25):
R21 (s) = −T 2s2+T 1s+ 1
B(s)
a UMs+b
a2UTs+b2
A(s),(25)
whe e A(s) = T2s2+ 12,B(s) = T1s2+ 12. The
ela ion (Eq. 25) clea ly shows ha he ans e unc-
ion in his o m does no mee he condi ion o phys-
ical iabili y as he nume a o o de is highe han
he denomina o o de . Thus we de ine wo ine ias
wi h condi ions ξ1T 1and ξ2T 2. The phys-
ically easible co ec ion ans e unc ion elimina ing
he ele a ion-azimu h coupling is gi en by he ela ion
Eq. (26):
R21 (s) = −T 2s2+T 1s+ 1
ξ2s2+ξ1s+ 1
a 0.55s+b
a20.2s+b2
A(s)
B(s).(26)
3.2. Con olle Concep ion
The concep ion o he helicop e model obus con-
ol design was pa ially explained in he equi emen s
o he au onomy o he ele a ion and azimu h con-
ol. The second pa conce ns he H∞ obus con ol,
namely he modi ica ion o he mixed sensi i i y p ob-
lem (MSP) whe e we also penalize he g oup o ex e nal
ou pu signals in addi ion o he con ol e o ~e ( )and
manipula ed alue ~u ( ):
•~
d1( ) = ( ) = ~w ( ) e e ence o ex e nal con ol
signal,
•~
d2( )low- equency signal (dis u bance),
•~
d3( )high- equency dis u bance signal (noise).
We use weigh ing ans e unc ions o also called
weigh ing il e s o penalize signals incoming and ou -
going om he ex ended sys em in he ele a ion o az-
imu h. The whole block diag am o he con ol ci cui
wi h ex ended sys em is shown in Fig. 6.
The e a e many di e en ways o he ex ension o
he nominal sys em. Howe e , he mo e ex e nal in-
pu s and penalized (e o ) ou pu s he e a e, he mo e
di icul i is o selec he weigh ing il e s. The weigh -
ing il e s a e gene ally s able ans e unc ions (no
necessa ily p ope a ional unc ions) o a pa icula
o de . Thus, he mo e we add, he highe he o de
he esul ing sys em will ha e. Such an in e connec ed
sys em can be hen used o exp ess he s a e desc ip-
ion, o he ans e unc ion o he H∞op imal, o
E(s) U(s)
K(s)
W (s)
e
W (s)
u
G (s)
0
z ( )
1
z ( )
2
W (s)
d
W (s)
cmd
W
noise
d ( )
1
d ( )
2
d ( )
3
Y(s)
d( ) +y( )
2
d d( ) + ( )+y( )
2 3
-
+++
+
+
(s)
d( )
1
d( )
2
d( )
3
e( )
u( )
Fig. 6: Block diag am o he connec ion o he H∞ obus con-
olle , nominal sys em and weigh ing il e s o he he-
licop e model.
subop imal obus con olle K(s)wi h one deg ee o
eedom (1DOF con igu a ion).
The meaning o he indi idual blocks is as ollows
acco ding o [10]:
•Wcmd (s): his weigh ing ans e unc ion akes
ca e o he e e ence acking. A no malized signal
appea s a he inpu and he signal a he ou pu
is in ele an physical uni s,
•Wd(s): his weigh ing ans e unc ion adjus s
he equency and ampli ude cha ac e is ics o ex-
e nal low- equency dis u bance signals a ec ing
he nominal sys em,
•Wnoise (s): his weigh ing ans e unc ion ep-
esen s he models o noises o senso s in he e-
quency domain. I ies o de ec a pa icula piece
o in o ma ion in he con ol de i ed om labo-
a o y expe imen s o p oduc ion measu emen s.
Na u ally, he noise shows a high- equency cha -
ac e ,
•We(s): his weigh ing ans e unc ion penalizes
he con ol signal om he obus con olle and
hus he con ol e o . I de e mines he in e ed
alue o he expec ed o m o he ou pu signal.
The signal ha appea s a he inpu o he il e
is in ele an physical uni s and i is no malized
a he ou pu ,
•Wu(s): his weigh ing ans e unc ion penalizes
he con ol signal om he obus con olle and
hus he manipula ed alue signal. I de e mines
he in e ed alue o he expec ed o m o he ou -
pu signal. The signal ha appea s a he inpu
o he il e is in ele an physical uni s and i is
no malized a he ou pu ,
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The comple e calcula ion o bo h he ans e unc-
ion o he H∞ obus con olle and he key H∞no m
is execu ed in MATLAB. The calcula ion algo i hm
is based on he co ec in e connec ion o he nomi-
nal sys em wi h he weigh ing il e s in co espondence
wi h Fig. 6. The p og am solu ion in he M- ile appea s
as ollows, i.e.:
sys emnames = ’G Wcmd Wd Wnoise We Wu’;
inpu a = ’[d1; d2; d3; u]’;
ou pu a = ’[We; Wu; Wcmd-G-Wnoise]’;
inpu _ o_Wcmd = ’[d1]’;
inpu _ o_Wd = ’[d2]’;
inpu _ o_Wnoise = ’[d3]’;
inpu _ o_G = ’[u]’;
inpu _ o_We = ’[Wcmd-G-Wnoise]’;
inpu _ o_Wu = ’[u]’;
cleanupsysic = ’yes’;
P = sysic
NCon ol = 1;
NMeasu e = 1;
= [NCon ol NMeasu e];
[K,CL,gop ] = hin syn(P,NMeasu e,NCon ol);
Fi s o all, we de ine wha sys ems we will in-
e connec , he a iable sys emnames. Then we de-
ine he inpu signal ec o (ex e nal con ol signals
and con ol signal om he con olle ), he a iable
inpu a . The ou pu is ep esen ed by he a i-
able ou pu a con aining he penaliza ion o he con-
ol e o (We), manipula ed alue (Wu) and he mea-
su ed ou pu (Wcmd-G-Wnoise). Subsequen ly, we con-
nec all he inpu s o he weigh ing il e s. By he
cleanupsysic command wi h he a ibu e alue se
o yes we con i m ha we wan o emo e he a i-
ables sys emnames,inpu a and ou pu a om he
MATLAB wo k en i onmen (Wo kspace) immedia ely
a e he c ea ion o he sys em in e connec ion.
The a iable P ep esen s he ex ended sys em o sys-
em in e connec ion (sysic, Sys em In e connec ion).
To comple e he enume a ion o pa ame e s o he cal-
cula ion, we ha e o de ine he numbe o con ol ou -
pu s om he con ol (NCon ol) and he numbe o
measu ed ou pu s (NMeasu e). We will ob ain he cal-
cula ion o he H∞con olle (K), closed loop ans e
(CL) and maximum closed loop ans e no m (gop )
by ac i a ing he hin syn unc ion wi h he ollowing
pa ame e s: P,NMeasu e and NCon ol.
The ac i a ion o he hin syn unc ion can also be
ex ended by mo e inpu and ou pu pa ame e s; in his
ac ual case acco ding o [10]:
• wo algeb aic Ricca i equa ions a e sol ed,
•γ∈(0,+∞),
• he closed-loop ans e unc ions is calcula ed
wi h he use o he linea ac al ans o ma ion
CL =F{P(s), K (s)},
•γ0=kCLk∞=kF{P(s), K (s)}k∞.
MATLAB, namely he Robus Con ol Toolbox, con-
ains o he unc ions ha can be used o sol e he issue
o he design o a con inuous o disc e e H∞ obus con-
olle . Fo comple eness, we only gi e he p o o ype o
he unc ion ocused on he s anda d mixed sensi i i y
p oblem:
[K, CL, gop , INFO] = mixsyn(G, W1, W2, W3)
The p oblem o he mixsyn unc ion is he numbe
o he weigh ing il e s and hei cha ac e as hey only
penalize he con ol e o (W1), manipula ed alue (W2)
and measu ed ou pu (W3). Wi h ega d o he selec ed
design concep ion, i would no be possible o penalize
inpu s wi h his unc ion.
3.3. Ele a ion Con olle o
Ma hema ical Model
The ans e unc ion o he dynamics o he ma he-
ma ical model in ele a ion is gi en by he ela ion ac-
co ding o he ela ion Eq. (27):
Gψ(s) = Ψ (s)
UM(s)=
=7.3315s+ 1.1883
3s4+23s3+116s2+519s+ 1000.(27)
The ampli ude and phase equency cha ac e is ics
a e shown in Fig. 7 which also clea ly show ha i con-
ains he highes alue unde he ollowing condi ions:
• equency: ωMAX = 4.9448  ad ·s−1,
• ans e unc ion module:
|Gψ(jω)|MAX =
=−16.05 [dB]∼
=0.1576 [−].(28)
The sys em is o he ou h o de and con ains ou
s able poles, ou o which wo a e complex conjuga e
and one is a double pole:
p1=−0.2105 + j4.9448,(29)
p2=p1=−0.2105 −j4.9448,(30)
p3=p4=−4.(31)
The maximum no m o he gi en sys em is (in acco -
dance wi h he maximum alue o he ans e unc ion
module):
kGψ(s)k∞= 0.1577 [−].(32)
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-200
-150
-100
-50
0
10
-2 10
-1 10
010
110
210
3
-270
-180
-90
0
90
F equency ( ad/s)
Magni ude (dB)
Phase (deg)
Fig. 7: Ampli ude equency cha ac e is ic and phase equency
cha ac e is ic o he gi en ans e unc ion o he ma h-
ema ical model in ele a ion.
The o ms o he weigh ing il e s o he gi en sys-
em a e as ollows, i.e.:
•weigh ing ans e unc ion o e e ence signal:
Wcmd (s) = 1
0.25s+ 1,(33)
•weigh ing ans e unc ion o low- equency dis-
u bance signal:
Wd(s) = 0.5
0.1s+ 1,(34)
•weigh ing ans e unc ion o high- equency dis-
u bance signal (noise):
Wnoise (s) = 0.01s+ 1
s+ 1 ,(35)
•weigh ing ans e unc ion o con ol e o signal:
We(s) = Ke
1
Mes+ωbe
s+ωbeεe
= 0.001 s+ 0.5
s+ 0.0005,(36)
•weigh ing ans e unc ion o manipula ed alue
signal:
ˆ
Wu(s) = Ku
s+ωbu
Mu
εus+ωbu
= 10−7s+ 1
0.01s+ 2.(37)
The weigh ing il e s o he con ol e o and manip-
ula ed signal ha e a p esc ibed ans e unc ion o m
acco ding o [14], he ans e always con ains he same
nume a o and denomina o o de as o ensu e he s a-
bili y o he in e ed ans e unc ions. The il e o
he con ol e o is low-pass and o he manipula ed
alue i is high-pass. The g aphic dependences o he
sensi i i y unc ions and in e ed ans e unc ions o
-60
-50
-40
-30
-20
-10
0
10
-5 10
-4 10
-3 10
-2 10
-1 10
010
1
-90
-45
0
F equency ( ad/s)
-150
-140
-130
-120
-110
-100
10
-2 10
-1 10
010
110
210
310
4
0
45
90
F equency ( ad/s)
Magni ude (dB)
Phase (deg)
Magni ude (dB)
Phase (deg)
Fig. 8: Ampli ude and equency cha ac e is ics o he low-
pass weigh ing il e We(s)(uppe pa ) and high-
pass weigh ing il e Wu(s)(lowe pa ): ma hema ical
model in ele a ion.
150
100
50
0
-50
-100
-150
-200
10 10 10 10 10
-4 -2 024
F equency( ad/s)
S
T
1/W
u
e
1/W
Singula Values (dB)
Fig. 9: Ampli ude equency cha ac e is ic o he sensi i -
i y unc ion S(s), complemen a y sensi i i y unc-
ion T(s), in e ed ans e unc ions 1/We(s)and
1/Wu(s): ma hema ical model in ele a ion.
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he ma hema ical model in ele a ion a e as s a ed in
Fig. 8.
One o he gene al equi emen s o he ampli ude
equency cha ac e is ics o he sensi i i y unc ion
S(s)and he in e ed ans e unc ion 1/We(s)de-
ines obus beha io :
∀ω∈R:|S(jω)|≤|1/We(jω)|=
= 1/|We(jω)|⇔kWe(s)S(s)k∞≤1.(38)
Figu e 9 implies ha he ela ion Eq. (38) is ully
me . Simila ly, i is also possible o de ine he ampli-
ude equency cha ac e is ics o he in e ed ans e
unc ion 1/Wu(s)and he p oduc o he con olle
ans e and he sensi i i y unc ion K(s)S(s):
∀ω∈R:|K(jω)S(jω)| ≤ |1/Wu(jω)| ⇔
⇔ kWu(s)K(s)S(s)k∞≤1.(39)
150
100
50
0
-50
-100
10 10 10 10 10
-4 -2 024
F equency( ad/s)
KS
S
1/W
u
e
1/W
Magni ude (dB)
Phase (deg)
Fig. 10: Ampli ude equency cha ac e is ics o he sensi-
i i y unc ion S(s), p oduc o ans e unc ions
K(s)S(s), in e ed ans e unc ions 1/We(s)and
1/Wu(s): ma hema ical model in ele a ion.
Figu e 10 implies ha he ela ion Eq. (39) is ully
me . Fo comple eness, we gi e he alues o he key
H∞no ms, i.e.:
•op imal H∞no m:
γ= 5.752010−4,(40)
•closed loop H∞no m:
kF{P(s), K (s)}k∞= 4.783710−4< γ, (41)
•sensi i i y unc ion H∞no m:
kS(s)k∞= 1.5357,(42)
•complemen a y sensi i i y unc ion H∞no m:
kT(s)k∞= 0.9997.(43)
The o de o he designed H∞con olle o he
ma hema ical ele a ion model co esponds wi h he o-
al o he o de s o he indi idual elemen s o he ex-
ended sys em, he nominal sys em is o he ou h o -
de a he mos and all i e weigh ing sys ems a e in
he i s o de a he mos . The a o emen ioned implies
ha he con olle will be a sys em o he nin h o de
a he mos . I s equency cha ac e is ics a e shown in
Fig. 11.
150
100
50
0
10 10 10 10 10
-4 -2 024
F equency( ad/s)
180
90
0
-90
-180
Magni ude (dB)
Phase (deg)
Fig. 11: Ampli ude equency cha ac e is ic and phase e-
quency cha ac e is ic o he H∞con olle o ma he-
ma ical model in ele a ion.
Acco ding o Fig. 6, a model o he H∞con olle
and he ex ended sys em was c ea ed in Simulink. The
con igu a ion o he g oup o ex e nal inpu signals is
as ollows:
• e e ence:
d1( = 0) = ( = 0) = −0.25 [−]
o ∈(0; 20i,(44)
d1( ) = ( = 20) = −0.05 [−]
o ∈(20; 100i,(45)
•LF dis u bance: no included in he model,
•HF dis u bance: band-limi ed whi e noise wi h
powe o 0.00001 [W] .
The esponse o he modeled ele a ion sys em o e -
e ence ~
d1( )is as shown in Fig. 12. Thanks o he
balanced a io o he sensi i i y unc ion and he com-
plemen a y sensi i i y unc ion, he elimina ion o he
noise and dis u bance will be e ec i e.
3.4. Ele a ion Con olle o Real
Model
We will use he ans e unc ion om he ma hema -
ical model o he design o he con olle o a eal
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