O iginal Pape
Measu emen and Con ol
2022, Vol. 55(1-2) 94–101
© The Au ho (s) 2022
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DOI: 10.1177/00202940221076961
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The PID and 2DOF con ol o he in eg al
sys em - influence o he 2DOF pa ame e s
and p ac ical implemen a ion
Radek Gu as
1
, Radek S ambe sky
1
and Mi osla Mahdal
1

Abs ac
The a icle deals wi h he issue o using he Two Deg ee o F eedom (2DOF) PID con olle o con ol an in eg al sys em and
in es iga es by he simula ion and expe imen al measu emen wha influence i has on he cou se o he con ol p ocess
compa ed o s anda d PID con olle . The con olled plan is ep esen ed by he DC elec ic mo o wi h wo m gea and i s
ou pu sha o a ion angle. The a icle s udies he e ec o he added pa ame e s o he 2DOF con olle on he dynamics o
he closed-loop con ol. The influence o hese pa ame e s is hen e alua ed using he quali y o eedback con ol c i e ia
ITAE. The pape s udies how he o e shoo o he con olled a iable du ing he se poin s ep is elimina ed using 2DOF
con ol heo y. The o e shoo is caused due o an agg essi e uning o he con olle o elimina e he dis u bance e ec on
he con olled a iable o he in eg al plan s wi h dead zones.
Keywo ds
in eg al sys em, 2DOF PID con olle , con ol sys em, DC mo o
Da e ecei ed: 9 Sep embe 2021; accep ed: 8 Janua y 2022
In oduc ion
In he indus ial en i onmen oday, he posi ion con ol o
a ious de ices is a c ucial ask. We a e inc easingly en-
coun e ing equi emen s o imp o e he accu acy and e fi-
ciency o p oduc ion p ocesses, whe e i is necessa y o
eplace he posi ioning o he limi swi ches wi h eedback
con ol o achie e op imal condi ions o he echnology. This
ask hus e y o en leads o in eg al sys em con ol, o en
wi h nonlinea p ope ies, as desc ibed in he pape .
1
The ypical app oach o he eedback con ol is using he
PD con olle .
2–4
The e is no pe manen con ol de ia ion o
linea in eg al sys ems such as pneuma ic, elec ic, o hy-
d aulic sys ems unless he e a e non-linea i ies like s a ic
ic ion. PID con ol can lead o emo ing he pe manen
con ol de ia ion in such sys ems.
5–7
Howe e , he PID
con ol may no always be ideal o his ask, leading o
o e shoo s and unwan ed slip and slide e ec s. The e o e, i
is o en beneficial o use o he algo i hms ha can handle he
p oblem be e . One such op ion is o use he PID con olle
algo i hm wi h wo deg ees o eedom (2DOF).
8–13
Two Deg ee o F eedom PID con olle s add se poin
weigh s o he p opo ional and de i a i e e m o he al-
go i hm o ensu e ha he e ec o he dis u bance is quickly
elimina ed while supp essing o e shoo when acking he
se poin .
8,9
Thedeg eeo eedomo acon olle isdefined as he
numbe o closed-loop ans e unc ions ha can be uned
independen ly,
14–18
which p o ides addi ional op ions o uning
he con olle ega ding change o se poin o dis u bance alue.
Ac i e dis u bance ejec ion con ol, which uses he ex-
ended s a e obse e
19–21
o he Dis u bance obse e ,
22,23
can
be used as an al e na i e. Fo bo h o hese me hods, we a e
equi ed o c ea e he in e se model o he plan . The simpli-
fica ion o he 2DOF con olle is ha he fil e is applied o he
se poin signal and depends on al eady known PID pa ame e s
only. Due o ha , he implemen a ion o he 2DOF is also
possible wi hou he need o de ailed knowledge abou he
plan . 2DOF con olle can be uned manually o by quan i a i e
me hods as
9,24
based on educing he c i e ia desc ibed in
Chap e 6. o his pape . Ano he op ion is also he combina ion
o he 2DOF con olle wi h he dis u bance obse e .
25
The
same echnique can also be used wi h o he ypes o eedback
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1
Depa men o Con ol Sys ems and Ins umen a ion, VSB –Technical
Uni e si y o Os a a, Os a a, Czech Republic
Co esponding au ho :
Radek Gu as, Depa men o Con ol Sys ems and Ins umen a ion,
VSB –Technical Uni e si y o Os a a, 17. lis opadu 15/2172, Os a a
708 00, Czech Republic.
Email: adek.gu as@ sb.cz
loop con ol as Fuzzy con ol, which o e us he op ion o
design he esponse p ope ies based on expe expe ience, S a e
con olle , which allow us o design he con ol sys em based on
he ull s a e eedback, o F ac ionalPIDcon olwhe e he
de i a ion and in eg a ion e ms o PID con olle a e exchanged
wi h he ac ional-o de in eg a ion and de i a ion e m which
leads o ackle he p oblem o dead zones nonlinea i ies in
in eg al sys ems.
26
Fo bo h me hods, he addi ional fil e can be
added o al e na e he se poin signal. Howe e , he quan i a i e
design o he fil e o non-PID con ol is hen based on Me a-
heu is ic op imiza ion algo i hms.
27
The implemen a ion o he
2DOF fil e was published be o e in e . 28, whe e adi ional
2DOF is modified o 2DOF wi h an in e nal model o ime-
delay p ocesses. The combina ion o he 2DOF and F ac ional
PID con olle design was hen published in e . 29.In hescope
o his pape , we will ocus on he adi ional 2DOF PID
Con olle , whe e he design, implemen a ion, and compa ison
be ween he nume ical simula ions and ac ual single-chip
compu e implemen a ion.
Desc ip ion o he sys em
The in eg al sys em (see Figu e 1) is a DC mo o wi h a wo m
gea , whe e he con olled a iable is he angle o o a ion o he
ou pu sha . A esis ance senso p o ides he measu emen in a
po en iome ic se up. The A duino Nano mic ocon olle was
used as a model con ol sys em o coope a e wi h an ex e nal
AD con e e and an adap ing elec onic boa d.
The ac ion a iable is ealized ia a DC mo o d i e
(Pololu Simple High-Powe Mo o Con olle 18 15, see
Figu e 2 and Figu e 3) using a PWM signal and H-b idge
di ec ion swi ching as a esponse o he con olle ou pu . The
MOSFET swi ching ca ie equency o he d i e is 25 kHz.
All he pa ame e s o he d i e a e configu able h ough he
USB po and dedica ed so wa e.
The ou pu signal is an analog alue in he ange o 0–5V
p ocessed by a 12-bi AD con e e . The ange o he angle o
o a ion o he ou pu sha is app oxima ely 0–230°.
By i s physical na u e, he sys em i sel exhibi s nonlin-
ea i ies, some o which a e compensa ed by he algo i hm,
such as he ini ial insensi i i y o he mo o o he inpu signal
caused by ic ion and nonlinea exci a ion o he mo o coils.
This beha io is ela i ely supp essed by he o se o he
inpu signal o p e iously expe imen ally measu ed alues.
O he nonlinea i ies o he sys em we e neglec ed o his
wo k and conside ed in he iden ifica ion o he sys em by
selec ing he mos sui able sys em pa ame e s.
A duino nano was chosen as a sui able mic ocon olle -
based con ol sys em o con ol his plan . I s main ad an-
ages a e he possibili y o as and simple p og amming, easy
po abili y o code o o he simila boa ds, he possibili y o
p og amming om he MATLAB/Simulink en i onmen while
main aining a low-cos concep , andeasyimplemen a ionin o
exis ing ha dwa e, unlike e.g., gene al PIC MCU.
30
The desc ibed block diag am o he con ol sys em can be
seen in Figu e 2.Figu e 3 shows he elec onics im-
plemen a ion o he expe imen .
Theo y o he 2DOF PID con olle
Ou con ol sys em aims o calcula e he co ec con ol signal
o he inpu o he plan desc ibed abo e. Fo he con ol, we
a e going o use he linea eedback con olle . All he signals
will be p esen ed in he complex domain using he Laplace
ans o m, whose defini ion is:
XðsÞ¼L xð Þg ¼ Z
∞
0
xð Þes d (1)
Whe e he ime ( ) domain signal x( ) is ans e ed in o he
complex (s) domain X(s).
Figu e 1. DC mo o as in eg al plan .
Figu e 2. Block diag am o he con ol sys em.
Gu as e al. 95
The p ima y me hod o con olling he plan s a e Y(s) is
he P opo ional-In eg al-De i a i e (PID) (based on P o-
po ional, In eg al, and De i a i e e ms) con olle . The
con ol loop diag am can be seen in Figu e 4. This me hod
uses he eedback om he con ol signal Y(s), which is hen
compa ed wi h he wan ed (se poin ) signal W(s). The e o
signal can be calcula ed as:
EðsÞ¼WðsÞYðsÞ(2)
Using he con olle desc ibed by he ans e unc ion:
GRðsÞ¼UðsÞ
EðsÞ(3)
whe e U(s) is he con ol ac ion signal which is he inpu o
he con olled plan . The con ol plan ans e unc ion de-
sc ibes he dynamics o he plan :
GSðsÞ¼YðsÞ
UðsÞVðsÞ¼0
¼YðsÞ
VðsÞUðsÞ¼0
(4)
whe e he V(s) is he dis u bance.
Usually, du ing he design phase o he con olle design,
we se he expec a ion o he closed con ol loop o be:
The ans e unc ion o he se poin o he ou pu is equal
o one.
GwyðsÞ¼YðsÞ
WðsÞVðsÞ¼0
→1 (5)
The ans e unc ion o he dis u bance o he ou pu is
ze o.
G yðsÞ¼YðsÞ
VðsÞWðsÞ¼0
→0 (6)
Due o he physical p ope y o he mechanical sys em, his
expec a ion canno be implemen ed as e e y s a e con ain some
dynamics, and he sys em canno c ea e he co esponding
con ol signal agains he u u e unknown dis u bance V(s).
The di e ence be ween he s anda d PID con olle and
2DOF PID con olle will be p esen ed on he plan , which
was desc ibed abo e, and he expe imen al iden ifica ion
(desc ibed in he nex chap e ) lead o he ans e unc ion:
GsðsÞ¼ k1
sðT1sþ1Þ(7)
whe e k
1
is he plan gain, and T
1
is he ime cons an due o
ine ia.
Fi s , le us desc ibe he p ope y o he PID con olle
whose ans e unc ion is:
GRðsÞ¼kp1þ1
TIsþTDs(8)
whe e k
P
is he con olle gain, and i is equal o he p opo ional
(posi ion) eedback, T
I
is he in eg al cons an o he con olle , and
he k
P
/T
I
is equal o he in eg a ion o he con ol e o whose
unc ion is o ob ain Y(s) = W(s) in he s eady s a e. The las e m is
he de i a i e e m con aining he con olle ’s de i a i e cons an
T
D
,whe ek
P
T
D
is equal o he de i a ion ( eloci y) eedback.
The 2DOF con olle has a simila s uc u e. The only
di e ence is he fil e wi h he wo cons an s band c. Thanks
o hese pa ame e s, we can con ol he influence o he
se poin a iable compa ed o he ou pu a iable. The 2DOF
con olle can be desc ibed wi h he equa ion:
UðsÞ¼KP8
>
<
>
:
bWðsÞYðsÞþ 1
TIs½WðsÞYðsÞ…
…þTDs½cWðsÞYðsÞ
9
>
=
>
;
(9)
I he b=1andc= 1, hen he 2DOF con olle is equi alen o
he common PID con olle . I he b=0andc=0 hen hefil e is
ully ac i e. We can di ide his con olle in o wo blocks- he
common PID con olle and a sepa a e fil e . A final diag am can
be seen in Figu e 5. Then he ans e unc ion o he fil e is:
GFðsÞ¼cTITDs2þbTIsþ1
TITDs2þTIsþ1(10)
The ans e unc ion o he se poin o he ou pu is equal
o:
Figu e 4. Closed loop con ol sys em diag am.
9
Figu e 3. Pic u e o he con ol sys em.
Figu e 5. Closed loop con ol sys em wi h inpu fil e .
3
96 Measu emen and Con ol 55(1-2)
GWY ðsÞ¼GSðsÞGRðsÞGFðsÞ
1þGSðsÞGRðsÞ(11)
GWY ðsÞ¼ k1kpðcTDTIs2þbTIsþ1Þ
T1TIs3þTDTIk1kpþTIs2þT1k1kpsþk1kp
(12)
F om his equa ion, we can see ha using he con en ional
PID con olle , we ob ain wo complex conjuga ed ze os.
Thei e ec can be canceled using he 2DOF con olle when
b¼0 and c¼0. I we se band c o di e en non-ze o o
non-one, he ze o can be al e ed.
I we use he subs i u ion:
GPðsÞ¼ k1kp
T1TIs3þTDTIk1kpþTIs2þT1k1kpsþk1kp
(13)
which is he ans e unc ion o he p opo ional sys em, we
ob ain he ans e unc ion om he se poin o he ou pu :
GWY ðsÞ¼GPðsÞþGPðsÞbTIsþGPðsÞcTDTIs2(14)
The fi s e m does no depend on any o he bo cpa-
ame e s, he second e m is he fi s de i a ion o he fi s e m,
and he influence can be con olled by he pa ame e b,and he
hi d e m is he second de i a ion o he fi s e m and can be
con olled by he pa ame e c.I bo cis nega i e, i can lead o
he unde shoo . bpa ame e can be used o make he sys em
esponse o he s ep se poin change mo e agg essi e. Pa ame e
cwill ha e only a minimal influence on he dynamics o he
sys em esponse o he s ep se poin change. Visible di e ences
will be no iceable igh a e he s ep se poin change.
The ans e unc ion o he dis u bance o he ou pu is
equal o:
G yðsÞ¼ GSðsÞ
1þGSðsÞGRðsÞ(15)
G yðsÞ¼ TIk1s
T1TIs3þTDTIk1kpþTIs2þT1k1kpsþk1kp
(16)
As you can see, in he s eady s a e G yðsÞ¼0. Also, his
ans e unc ion is no a ec ed by he 2DOF pa ame e s b
and c. The 2DOF PID con olle used o he expe imen al
e ifica ion in his pape is done by single chip compu e
implemen a ion.
31
Expe imen al iden ifica ion o he
in eg al plan
By di ec iden ifica ion wi h he s ep inpu signal, da a de-
sc ibing he p ope ies o he con olled sys em we e
ob ained.
The measu emen was pe o med by applying an inpu
signal o 50% o he maximum ange o he sys em inpu , and
he s ep esponse o he sys em o his signal was eco ded
(see Figu e 6). In he figu e, he alue o he ou pu a iable is
no malized o he maximal angle o he o a ion ange. The
same no maliza ion is also used in he ollowing pa ag aphs.
By p ocessing he inpu da a and no malizing he signals,
he sys em’s ans e unc ion was ob ained in he o m o an
in eg a ion sys em wi h fi s -o de ine ia.
GSðsÞ¼ 1
0:3039sð0:0603sþ1Þ¼3:2906
sð0:0603sþ1Þ(17)
Closed loop con ol
A con ol sys em was designed o he plan (sys em) de-
sc ibed by (17) based on he p e ious heo y. The PID
con olle pa ame e s we e uned using he nume ic op i-
miza ion me hod, and hey a e lis ed in Table 1.
The expe imen al measu emen was pe o med as a
esponse o a s ep change o se poin om he beginning o
he measu emen . A ime = 15 s, he e was a s ep change
o dis u bance in on o he plan . The exac p ocess was
nume ically simula ed in he MATLAB /Simulink en i-
onmen , whe e he iden ifiedsys emwascon olledbya
2DOF con olle wi h ac ion a iable limi a ion (see
Figu e 7). The pa ame e s (weigh s) o he inpu fil e we e
selec ed om he whole ange 0–1 and measu ed and
simula ed wi h he s ep o 0.2. The dis u bance is simula ed
by he so wa e d op o ac ion a iable ol age on he inpu
o he plan . The esul s o he measu emen s and simu-
la ions a e plo ed in Figu e 8.
6,7
Acco ding o he esul s, he influence o he bpa ame e is
significan ly g ea e han he cpa ame e when acking he
se poin in he o m o he s ep. P o ed also by he simula ion,
he bpa ame e supp esses he e ec o p opo ional e m and
elimina es he o e shoo o he con ol p ocess while he
esponse o a dis u bance s ep emains unchanged.
Figu e 6. DC mo o ou pu sha angle o he o a ion s ep
esponse.
Table 1. 2DOF PID Con olle unable pa ame e s.
k
P
T
I
T
D
bc
0.2012 2.577 0.056 0–10–1
Gu as e al. 97
The e ec o he indi idual e ms acco ding o (13) and
(14) a e shown in Figu e 9. We spli he h ee e ms in o
sepa a e cou ses ha gi e he o e all con ol sys em esponse
using he same basic PID unable pa ame e s and b= 0.4 and
c= 0.8.
Quali y o eedback con ol
Mul iple me hods can be used o e alua e he quali y o he
eedback con ol. As i was desc ibed ea lie , one o he
expec a ions o he closed-loop con ol is ha he ans e
unc ion o he se poin o he ou pu is equal o one. Tha also
means ha he e o should be ze o. Fo physical sys ems, we
wan he sys em e o o minimize as as as possible.
As an example, we can choose one o hese c i e ia o
e alua ing he se poin s ep change esponse cou se:
·In eg al o Absolu e E o (IAE)
IIAEðsÞ¼Z
∞
0
jeð Þjd (18)
·In eg al o Squa ed E o (ISE)
IISEðsÞ¼Z
∞
0
e2ð Þd (19)
·In eg al o Time mul iplied by Absolu e E o (ITAE)
IITAEðsÞ¼Z
∞
0
jeð Þjd (20)
whe e a lowe numbe means a as e con ol p ocess. All
hese me hods can be used o bo h non-oscilla ing and
oscilla ing p ocesses. The disad an age o IAE and ITAE is
ha i canno be calcula ed as a de i a i e o a ze o-c ossing
o absolu e alue is no defined. The e o e, hese me hods
can be calcula ed only nume ically. The ISE is defined, bu
he oscilla ing p ocesses ha e a lowe e o han non-
oscilla ing p ocesses. These me hods a e usually used o
he quan i a i e me hods o calcula ing he con ol a iables
as published in
9,24
. We calcula ed he ITAE in his a icle.
Nume ical simula ion is hen compa ed o he measu emen s.
The calcula ed esul s o di e en pa ame e s band ccan
be seen in ables: measu emen s in Table 2 (ITAE) and
Figu e 7. MATLAB/Simulink nume ical simula ion schema ic.
Figu e 8. Closed loop con ol ( ed –simula ed, g ay - measu ed).
Figu e 9. Compa ison o he influence o he 2DOF e ms (b= 0.4,
c= 0.8).
Table 2. Quali y o eedback con ol acco ding o ITAE c i e ia.
b/c 0 0.2 0.4 0.6 0.8 1
0 11.46 11.54 11.58 11.36 11.41 11.35
0.2 10.37 10.36 10.52 10.26 10.48 10.45
0.4 9.35 9.37 9.35 9.45 9.47 9.31
0.6 8.36 8.41 8.39 8.19 8.40 8.45
0.8 7.33 7.36 7.28 7.33 7.44 7.31
1 7.09 7.07 7.12 7.16 7.08 7.40
98 Measu emen and Con ol 55(1-2)

nume ical simula ions in Table 3 (ITAE). Bo h he ables a e
plo ed in he su ace g aph in Figu e 10 and Figu e 11
(simula ed).
As we can see om he esul s, he di e ences be ween he
simula ion and measu emen s con aining small nonlinea i ies
a e neglec able. The pa ame e chas only a iny influence on he
quali y, while lowe ing he bpa ame e slows down he e-
sponse. The esponse o he dis u bance is no a ec ed by he
2DOF con olle pa ame e s as s a ed abo e.
Using an inpu 2DOF fil e , supp ession o he o e shoo
o he con ol p ocess cou se is achie ed, while wi hou i , he
con ol p ocess cou se shows an o e shoo o app oxima ely
7% a a gi en se ing.
Conclusion
The pape shows ha using he 2DOF con olle , we ob ain
mo e flexibili y in designing he final desi ed sys em e-
sponse. The heo y was implemen ed in o he A duino-
based con ol sys em expe imen ally iden ified and con-
olled by he 2DOF PID con olle .
The ans e unc ion o se poin acking and he e-
ac ion o he ou pu o he dis u bance was calcula ed o
he in eg al sys em wi h he fi s -o de ine ia. The closed-
loop con ol using he ypical PID con olle con ains wo
ze os, which o he in eg al sys em makes he esponse
mo e agg essi e om he beginning a e he se poin s ep
change. Tha leads o as elimina ion o he dis u bance
change bu causes an o e shoo . This beha io can be
supp essed using he 2DOF PID con olle , whe e hese
ze os can be canceled by se ing he band cpa ame e s o
lowe alues. Using he bo cpa ame e s, we can also
design he dynamics o se poin acking wi hou a ec ing
he beha io o he closed-loop con ol sys em o he
dis u bances.
The measu emen and simula ion p o ed ha he
o e all influence o he bpa ame e is conside ably mo e
dominan when acking he se poin alue. Acco ding o
he c i e ia used o e alua e he quali y o he con ol
p ocess, you can conclude ha he se ling ime is no
significan ly ex ended (i we conside a su ficien ly na ow
ole ance band). Howe e , he se poin s ep unc ion e-
sponse o e shoo is supp essed, which can be e y im-
po an o some dynamic sys ems. Depending on he
se ings o he 2DOF fil e pa ame e s, we can hen achie e
an o e shoo - ee cou se while main aining he same
se ling ime. In he case o he pa icula sys em used, his
se ing is a ound he alue 0.8–0.9 o pa ame e b.
Decla a ion o conflic ing in e es s
The au ho (s) decla ed no po en ial conflic s o in e es wi h espec
o he esea ch, au ho ship, and/o publica ion o his a icle.
Funding
The au ho (s) disclosed eceip o he ollowing financial suppo
o he esea ch, au ho ship, and/o publica ion o his a icle:
This wo k was suppo ed by he Eu opean Regional De elopmen
Fund in he Resea ch Cen e o Ad anced Mecha onic Sys ems
p ojec , CZ.02.1.01/0.0/0.0/16_019/0000867 wi hin he Ope a-
ional P og amme Resea ch, De elopmen and Educa ion and he
p ojec SP2022/60 Applied Resea ch in he A ea o Machines and
P ocess Con ol suppo ed by he Minis y o Educa ion, You h
and Spo s.
ORCID iDs
Radek Gu as h ps://o cid.o g/0000-0002-8704-1395
Mi osla Mahdal h ps://o cid.o g/0000-0002-9720-2201
Table 3. Quali y o eedback con ol acco ding o ITAE c i e ia
(simula ed).
b/c 0 0.2 0.4 0.6 0.8 1
0 11.09 11.10 11.10 11.10 11.10 11.10
0.2 10.11 10.12 10.12 10.12 10.12 10.12
0.4 9.13 9.14 9.14 9.14 9.14 9.14
0.6 8.15 8.16 8.16 8.16 8.16 8.16
0.8 7.17 7.18 7.18 7.18 7.18 7.18
1 7.09 7.08 7.08 7.08 7.08 7.08
Figu e 10. Resul s o he ITAE c i e ion o di e en band c
pa ame e s.
Figu e 11. Resul s o he ITAE c i e ion o di e en band c
pa ame e s (simula ed).
Gu as e al. 99
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