Robus PID Con ol o he Quad o o
Helicop e ?
R.A. Ga c´ıa, F.R. Rubio and M.G. O ega
Dp o Ing Sis emas y Au om´a ica. Uni e sidad de se illa, Escuela
T´ecnica Supe io de Ingenie ´ıa, 41092, Se illa, Spain.
{ among , ubio, mo ega}@us.es
Abs ac : In his pape a obus PID con ol s a egy ia a ine pa ame iza ion is designed o
an mul i a iable nonlinea unmanned ae ial ehicle. The obus ness o he con olled sys em is
assu ed by using he H∞no m o he weigh ed complemen a y sensi i i y unc ion. Simula ion
esul s ca ied ou using a comple e nonlinea model a e shown, whe ein he pe o mance
achie ed wi h his con ol s a egy is shown.
Keywo ds: PID, Robus con ol, UAV, Simula ion, Model.
1. INTRODUCTION
Unmanned Ae ial Vehicles, UAVs, a e well ad anced in
ecen yea s, especially he quad o o equipmen s. They
a e helicop e s whose main ea u e is ha hey ha e ou
coplana mo o s. Fu he mo e, as helicop e , i enables
a ho e , allowing he es ing o con ol echniques o
s abiliza ion and heigh egula ion ega dless o he dis-
placemen in he XY plane.
This kind o ai ship also equi es con inuous con ol ac ion,
because he sys em is uns able. The sligh es change in an
uncon olled ac ion om he engines can cause he sys em
lea e he s able ope a ing poin .
A lo o esea ches ha e been ca ied ou in a wide ange
o ields in con ol and obo ics, such as compu e ision
eedback, usion senso , and linea and nonlinea con ol
me hodologies o imp o e pe o mance o his kind o
sys em.
Equa ions o mo emen s o his kind o helicop e can be
ound in (Cas illo e al (2007), Ra o e al (2008)). These
equa ions can be used o he con ol o he helicop e
by means o many con ol s a egies, such as MPC, H∞,
and PID, among o he s (Alexis e al (2011), Chen e al
(2003,AIAA), Chen e al (2003)). Feedback in o ma ion
can be p o ided by ine ial measu emen uni s, GPS, o
e en compu e ision sys ems (Al ug e al (2002)).
In his pape , a ixed-s uc u e obus PIDs a e used o
con ol bo h he a i ude and he heigh o a quad o o
helicop e . PIDs a e e y ex ended in indus y due o
he possibili y o expe imen al se ings and he insigh
o hei pa ame e s uning. Focusing on obus con ol,
his con ol heo y allows o conside unce ain ies which
a e no desc ibed in a nominal model, such as delays o
a ia ions in ce ain alues due o complex measu emen s.
In Saeki (2006) can be ound a wo k conce ning obus
PID design ia LMIs.
?The au ho s wan o hank he MICINN o unding his wo k
h ough p ojec DPI2010-19154.
In his wo k, he PID con olle s ha e been syn hesized
h ough a ine pa ame iza ion. This me hodology can
cancel he s able-known dynamics o he sys em. And
o he unknown dynamics and pa ame e unce ain y, a
obus design using H∞ heo y is implemen ed. By com-
bining obus and a ine app oaches, PID con olle s a e
ob ained which gua an ee unc ionali y unde unce ain
condi ions.
The emainde o he pape is o ganized as ollows. Sec ion
2 desc ibes he model o quad o o used in his wo k. In
Sec ion 3, he con olle s s uc u e and i s implemen a ion
a e explained. Simula ion esul s a e p esen ed in Sec ion
4, and inally he main conclusions a e d awn in Sec ion 5.
2. SYSTEM MODELING
A scheme o he sys em is p esen ed in Fig. 1, whe ein
di e en o ces and o ques ac ua ing on he quad o o can
be obse ed. Basically, hey a e he ollowings: o al o ce,
F, oll o que, τφ, pi ch oque, τθ, and yaw o que, τψ.
Howe e he ac ua o s a e ou o o s ep esen ed by hei
o ces, i. The e o e, he desi ed o al o ce and o ques
ha e o be achie ed by exe ing some combina ion o ias
ollows:
F= 1+ 2+ 3+ 4
τφ=l·( 2− 4), τθ=l·( 3− 1)
τψ=k
b·( 1+ 3− 2− 4),
(1)
whe e lis he dis ance om he cen e o o a ion o he
o o s, k is he d ag coe icien o he mo o s and bis he
h us coe icien o he o o s. The ollowing alues ha e
been conside ed o his wo k om a pa icula quad o o :
k = 1.06 ·10−8Nm2τψ=k ω2
b= 5.8·10−7Ns2 = bω2.
(2)
The aim is o con ol he heigh and he s abiliza ion o
he UAV simul aneously. Nex , he nonlinea equa ions o
Fig. 1. Model scheme: Main o ces and o ques
mo ion o he heigh , zand o hese h ee angles a e
shown, whe ein he oll, pi ch and yaw Eule angles a e
no ed as φ,θand ψ espec i ely (Ra o e al (2010)), (Ra o
e al (2008)).
•Equa ion o he heigh exp essed wi h espec o an
ine ial ame W:
¨z=−g+1
m(cos θcos φ)F+Az
m,(3)
being Aza dis u bance ac ing on he Zaxis. ¸c
•Equa ions o he h ee Eule angles:
I(η)¨η+C(η, ˙η) ˙η=τη+τηdis .(4)
Being I(η) inne ma ix, ηEule angles, CCo iolis
ma ix, τη o ques applied and τηdis dis u bance
o ques.
τη="τφ
τθ
τψ#(5)
I(η) = "i11 i12 i13
i21 i22 i23
i31 i32 i33 #(6)
i11 =Ixx, i13 =−Ixx sin θ
i12 =i21 = 0
i22 =Iyy cos2φ+Izz sin2φ
i23 = (Iyy −Izz) cos φsin φcos θ
i31 =−Ixx sin θ, i32 = (Iyy −Izz) cos φsin φcos θ
i33 =Ixx sin2θ+Iyy sin2φcos2θ+Izz cos2φcos2θ
(7)
C(η, ˙η) = "c11 c12 c13
c21 c22 c23
c31 c32 c33 #(8)
c11 = 0
c12 = (Iyy −Izz )( ˙
θcos φsin φ+˙
ψsin2φcos θ)
+(Izz −Iyy )˙
ψcos2φcos θ−Ixx ˙
ψcos θ
c13 = (Izz −Iyy )˙
ψcos φsin φcos2θ
c21 = (Izz −Iyy )( ˙
θcos φsin φ+˙
ψsin2φcos θ)
+(Iyy −Izz )˙
ψcos2φcos θ+Ixx ˙
ψcos θ
c22 = (Izz −Iyy )˙
φcos φsin φ
c23 =−Ixx ˙
ψsin θcos θ+Iyy ˙
ψsin2φcos θsin θ
+Izz ˙
ψcos2φsin θcos θ
c31 = (Iyy −Izz )˙
ψcos2θsin φcos φ−Izz ˙
θcos θ
c32 = (Izz −Iyy )( ˙
θcos φsin φsin θ+˙
φsin2φcos θ)
+(Iyy −Izz )˙
φcos2φcos θ+Ixx ˙
ψsin φcos θ
−Iyy ˙
ψsin2φsin θcos θ−Izz ˙
ψcos2φsin θcos θ
c33 = (Iyy −Izz )˙
φcos φsin φcos2θ−Ixx ˙
θsin2φcos θsin θ
−Izz ˙
θcos2φcos θsin θ+Ixx ˙
θcos θsin θ
(9)
These equa ions desc ibe he sys em beha io in all i s
wo kspace. Howe e , in his pape a ho e ing ligh is
conside ed, and he e o e, he desi ed equilib ium poin
consis s in keeping null alues o he Eule angles, a leas
o he pi ch and oll angles. Keeping in mind his ac ,
he o iginal model can be simpli ied, yielding he ollowing
linea exp essions:
¨z=F
m,¨
φ=τφ
Ixx
,¨
θ=τθ
Iyy
,¨
ψ=τψ
Izz
(10)
The alues o he mass, he leng h om he cen e o
he mass o he o o and he ine ias conside ed can be
obse ed below, hey would be ob ained expe imen ally:
m= 2.24 kg, l = 0.332 m, Ixx = 0.038425 kgm2
Iyy = 0.038421 kgm2, Izz = 0.061547 kgm2(11)
Then he plan wi hou conside ing he o o s dynamics is
gi en by:
Gi(s) = Ki
s2,{i=z, φ, θ, ψ}.(12)
These ans e unc ions ep esen a simpli ied linea
model ob ained om mo emen equa ions. Ne e heless
in his model, nei he he delays om communica ion
(o abou 20 ms) due o he mo o s d i e no mo o s
dynamics a e conside ed. Besides, some coupling is no
conside ed ei he , which appea s when he helicop e is
wo king a om he desi ed equilib ium poin s. All hese
ac s will be deal as a sys em unce ain ies.
3. CONTROLLER DESIGN
Fou con olle s a e needed, one o he heigh and one
o each angle. A PID based on a ine pa ame iza ion is
going o be p esen ed. This classic con olle , C(s), has
he ollowing s uc u e:
CP ID(s) = KP
TITDs2+TIs+ 1
TIs(τDs+ 1) ,(13)
Based on an a ine pa ame iza ion, he con olle can be
exp essed in he ollowing o m (Goodwin e al (2000):
C(s) = Q(s)
1−Q(s)G(s),(14)
whe e he ans e unc ion Q(s) ac s as pa ame e which
may cancel he s able dynamics o he plan . The e o e,
Q(s) can be exp essed as ollows:
Q(s) = FQ(s)G−1(s),(15)
whe e FQ(s) is chosen o sa is y some cons ain s (Good-
win e al (2000)).
In he case unde conside a ion,
G−1(s) = s2
K.(16)
I is known ha Q(s) mus be p ope , which yields FQ(s)
needs o ha e ela i e deg ee 2. In addi ion, since a speci ic
PID s uc u e o he con olle is desi ed, and aking in o
accoun ha inpu sensi i i y ans e unc ion
Sio(s) = [1 −Q(s)G0(s)]G0(s) (17)
mus ha e a ze o a he o igin in o de o be able o ejec
sus ained dis u bances, he ollowing gene al exp ession
o FQ(s) is p oposed:
FQ(s) = α2s2+α1s+ 1
α4s4+α3s3+α2s2+α1s+ 1 (18)
By subs i u ing his exp ession o FQ(s) in o (15) and (14),
and a e some manipula ions, he ollowing con olle is
ob ained:
C(s) = α2s2+α1s+ 1
α3Ks(α4
α3s+ 1) (19)
By compa ing he p eceding exp ession wi h Eq. (13), i
is easy o ob ain he ollowing co ela ions:
KP=α1
α3K, TI=α1, TD=α2
α1
, τD=α4
α3
(20)
In his wo k, he ollowing condi ions ha e been aken in o
accoun o he con olle s design. On he one hand, in
o de o op imize he dis u bances ejec ion, he ollowing
cons ain s has been imposed:
TI= 4TD→α2=α2
1
4(21)
On he o he hand, he ime cons an o he high equency
pole is chosen as ollows:
τD=TD
10 →α4=
α2
α1
10 α3(22)
Unde hese cons ain s, he con olle s ha e wo pa ame-
e s o adjus , α1and α3. Thei adjus men will be ca ied
ou in o de o p o ided obus ness o he con olled sys-
em.
Fo he design o a obus con olle , i s he ollowing
unce ain plan is conside ed:
G∗
i(s) = K∗
i
s2(τms+ 1)e−Ls (23)
In his unce ain plan , s uc u al and pa ame ic unce -
ain ies wi h espec o he nominal model a e included.
τm ep esen s he mo o ime cons an , whose alue has
been es ima ed abou 0.1 seconds, and Lis he delay due
o he ESC (Elec onic Speed Con olle ) d i e , whose
Fig. 2. Unce ain ies in Z coo dina e and es ima ed bound
Fig. 3. Unce ain ies in Roll coo dina e and es ima ed
bound
alue is a ound 20 ms. In addi ion, a gain a ia ion o a
±25% is also conside ed.
Once he unce ain plan s ha e been selec ed, he mul i-
plica i e unce ain y wi h espec he nominal model can
be es ima ed as ollows:
|Emi(jω)|=|G∗
i(jω)−Gi(jω)|
|Gi(jω)|,(24)
The modulus o he ou pu mul iplica i e unce ain y can
be uppe bounded by he modulus o a s able weigh ing
unc ion Wi(s). Bo h he es ima ed mul iplica i e unce -
ain y and he designed weigh ing unc ion a e depic ed in
Figs. 2 and 3 o he case o he heigh and he oll angle
espec i ely. The ones o he emaining Eule angles a e
e y simila o he oll angle case. I should be highligh ed
he ac ha he modulus o he weigh ing unc ion g ows
a high equencies, which is due o he ac ha he
unce ain models do no ep esen he eal plan a hese
equencies.
I is well-known (Sigu d (2005)) ha o achie e a obus
H∞con olle , i is necessa y he in e se o he mul iplica-
i e unce ain y bound has o be an uppe bound o he
complemen a y sensi i i y ans e unc ion Ti(s)
Fig. 4. Complemen a y sensi i i y unc ion o Z coo di-
na e and i s uppe bound
Ti(s) = Gi(s)Ci(s)
1 + Gi(s)Ci(s).(25)
o all equencies. Consequen ly, he wo pa ame e s, α1
and α3, a e designed in such a way ha complemen a y
sensi i i y unc ion be unde he in e se mul iplica i e
unce ain y bound and o sa is y ha bandwid h be as
highe as possible.
By subs i u ing he exp essions (12) and (19) in o (25),
he ollowing exp ession o he complemen a y sensi i i y
unc ion is ob ained:
Ti(s) = 10α2
i1s2+ 40αi1s+ 40
αi1αi3s4+ 40αi3s3+ 10α2
i1s2+ 40αi1s+ 40 (26)
The e o e, he coe icien s α1iand α3i,i=z, φ, θ, ψ, mus
be compu ed in o de o make he co esponding Tibe
unde i s uppe bound. The alues o hese coe icien s
designed o his applica ion a e he ollowing:
αz1= 8 , αz3= 11.2
αi1= 5.8, αi3= 1.12 i=φ, θ, ψ
(27)
Figu es 4 and 5 show he complemen a y sensi i i y unc-
ions ob ained whi hese alues o he case o he heigh
and he oll angle, as example o he iple o angles,
espec i ely.
4. SIMULATION RESULTS
In his sec ion, simula ion esul s a e p esen ed whe ein
se e al s ep signals a e applied as a e e ence o he
iple o angles, as well as o he heigh . These kind
o e e ences will o ce he helicop e o mo e on he
XY plane wi hou a p ede ined ajec o y, al hough in
any case, hese mo emen s will no be conside ed in his
wo k. The e o e, i can be supposed ha he quad o o is
a ached o a elescopic a m o p e en such mo emen s.
Besides, a mo e ealis ic model han he one p esen ed
in Sec ion 2 has been used o he simula ion. This model
conside , among o he ac o s, a misalignmen be ween he
mass cen e and he geome ic cen e o he helicop e .
Fig. 5. Complemen a y sensi i i y unc ion o Roll coo -
dina e and i s uppe bound
Fig. 6. Time e olu ion o he Z coo dina e
The con olle has been implemen ed in disc e e ime, wi h
a sample ime o 20 ms, which simula es he ime equi ed
by ESC.
The beha io o he heigh and he iple o angles, oll,
pi ch and yaw, a e shown in Fig. 6, 7, 8 and 9. The coupling
e ec s o he sys em can be seen in hese igu es. Fo
example, i can be seen a la ge a ia ion in he oll angle
a =120 s due o a big e e ence change o he yaw angle.
Ne e heless, he slope o he s ep p oduces an o e shoo .
To mi iga e ha e ec ,a i s -o de il e o he e e ences
has been designed, whose ime cons an has been selec ed
o coun e ac s he con olle ze os. Figu es 10, 11, 12, 13
show how he beha io imp o es wi h he il e . In can
be seen ha he bene i s om he il e addi ion a e
double. On he one hand, he o e shoo s o he s ep
esponse disappea bu he ime esponse ge s slowe . On
he o he hand, his las ac makes ha a s ep change in
e e ence o one coo dina e a ec s in a smoo he way on
he o he s coo dina es, which imp o es he decoupling o
he con olle sys em.
Finally, i can be no iced by compa ing Figs. 4 and 5 ha
he con ol bandwid h o he heigh is no as high as
he one o he angles. Howe e , al hough he bandwid h
Fig. 7. Time e olu ion o he Roll angle
Fig. 8. Time e olu ion o he Pi ch angle
Fig. 9. ime e olu ion o he Yaw angle
con ol o he z coo dina e may be inc eased, his ac
may cause undesi able e ec s due o he coupling. As o
example, Figs. 14 and 15 show he beha io o he Z and
oll coo dina es o he case o he same con ol bandwid h
o all coo dina es, which is achie ed by selec ing he
same alues o αz1and αz3 han he ones o αφ1and
αφ3 espec i ely. Ob iously, he con ol o he heigh ge s
Fig. 10. Time e olu ion o he Z coo dina e wi h il e ed
e e ence
Fig. 11. Time e olu ion o he Roll angle wi h il e ed
e e ence
Fig. 12. Time e olu ion o he Pi ch angle wi h il e ed
e e ence

Fig. 13. Time e olu ion o he Yaw angle wi h il e ed
e e ence
Fig. 14. Time e olu ion o he Z coo dina e wi h high
heigh -con ol bandwid h
Fig. 15. Time e olu ion o he Roll angle wi h high heigh -
con ol bandwid h
be e , bu a e e ence change in Z a ec s in excess on
he pe o mance o he angles. This is clea ly undesi able
o u u e wo ks whe e he displacemen s in he XY plane
ha e o be conside ed.
5. CONCLUSION
In his pape a obus PID con ol s a egy, uned ia
a ine pa ame iza ion, has been p esen ed. A i s , a
decoupled sys em has been assumed, bu in igu es o
simula ion esul s i is obse ed ha as long as a e e ence
angle changes, he o he ones can be modi ied. This
is because he equa ions used in he simula ion model
a e coupled, whe eas he equa ions used o con ol a e
decoupled.
I can be seen how he beha io o he sys em is qui e good
e en wi h he simpli ica ions made. So a nonlinea sys em
can be con olled wi h a decoupled linea con olle . This
con olle is a PID uning by a ine pa ame iza ion and
obus hanks o H∞ heo y.
A ine pa ame iza ion allows o cancel some dynamics in
o de o ob ain a desi ed dynamics. In addi ion, H∞ heo y
allows o conside e unce ain ies such as he plan wi h
di e en gain and he ac ua o , mo o s in his case.
Howe e i could be obse ed an o e shoo due o he
applied e e ence. To sol e ha a il e in he e e ence
was used and he esul s ob ained we e be e and slowe .
REFERENCES
K. Alexis, G. Nikolakopoulos and A. Tzes Model P e-
dic i e Con ol Scheme o he Au onomous Fligh o
an Unmanned Quad o o Indus ial Elec onics (ISIE),
2011 IEEE In e na ional Symposium on. 2243–2248,
2011
E. Al ug, J.P. Os owski, R. Mahony Con ol o a Quad o-
o Helicop e Using Visual Feedback Robo ics and
Au oma ion, 2002. P oceedings. ICRA ’02. IEEE In e -
na ional Con e ence on Vol.1 72 – 77, 2002
P. Cas illo, P. Ga cia, R. Lozano, P. Albe os. Modelado
y Es abilizaci´on de un Helic´op e o con cua o o o es.
Re is a Ibe oame icana de Au om´a ica e In o m´a ica
Indus ial RIAI, ISSN: 1697-7912 4:41–57, 2007.
M. Chen, M. Huzmezan A Combined MBPC/2 DOF H∞
Con olle o a Quad Ro o UAV P oc. AIAA Guid-
ance, Na iga ion, and Con ol Con e ence and Exhibi
Texas, USA, 2003.
M. Chen, M. Huzmezan A Simula ion Model and H∞
Loop shaping Con ol o a Quad Ro o Unmanned
Ai Vehicle P oceedings o Modelling, Simula ion, and
Op imiza ion, 320–325, 2003
G. C. Goodwin, S. F. G aebe, M. E. Salgado. Con ol
Sys em Design. Valpa ´ıso, 2000 .
M. Saeki Fixed s uc u e PID con olle design o s anda
H∞con ol p oblem. Au oma ica, 42:93–100, 2006.
G. V. Ra o, M. G. O ega, F. R. Rubio An in eg al p e-
dic i e/nonlinea H∞con ol s uc u e o a quad o o
helicop e Au oma ica, 46:29–39, 2010.
G. V. Ra o, M. G. O ega, F. R. Rubio Robus Back-
s epping/Nonlinea H∞con ol o pa h acking o a
quad o o unmanned ae ial ehicle. P oceedings Ame -
ican Con ol Con e ence (Acc’08) Sea le. Wa. Usa,
2008.
S. Skoges ad and I. Pos le hwai e Mul i a iable Feedback
Con ol Analysis and Design. John Wiley and Sons L d,
2005