* Co esponding au ho : [email p o ec ed]
Robus Con ol Sys em Design o a B ushed DC Mo o Using
LabVIEW Simula ion Loop
Wadee Khou y1,*, and Pé e Tamás Szemes2
1Mas e s S uden , Deb ecen Uni e si y, Mecha onics Depa men , 4028 Deb ecen, Hunga y
2Associa e P o esso , Deb ecen Uni e si y, Mecha onics Depa men , 4028 Deb ecen, Hunga y
Abs ac . This pape p esen s a obus con ol sys em design o a PMDC mo o using LabVIEW
so wa e. A i s , he ma hema ical equa ions o he mo o we e modelled and simula ed. Then, a
p opo ional in eg al con olle (PI) was used as a speed con olle . To gua an ee he obus ness o
he sys em agains he dis u bances, he PI pa ame e s we e uned by applying a sensi i i y
analysis. Finally, a p opo ional gain was used as a posi ion con olle o guide he sys em owa ds
he desi ed angle. The simula ion was achie ed by using LabVIEW Simula ion Loop, and he
sensi i i y analysis was ca ied using LabVIEW Ma hSc ip ool.
1 In oduc ion
The b ushed DC mo o s a e widely used in a ious
obo ic applica ions and indus ial a eas in ields anging
om oys o spacec a , due o hei lexibili y, high
eliabili y, and ela i ely low cos . One o he mos
common echniques used o con ol he posi ion and speed
o a DC mo o is he PID con olle , because o i s simple
s uc u e and comp ehensible con ol algo i hm.
Many pape s add ess he design o a DC mo o con ol
sys em using di e en me hods. Fo example, Yolchan e .
al. [1] compa ed PID and S a e Feedback me hods o
con ol eal- ime posi ion, ajec o y, and speed o a
b ushed DC mo o . Sahin e . al. [2] conduc ed a esea ch
on posi ion con ol o a DC mo o using A i icial Neu al
Ne wo k as he main con olle o hei sys em. Namazo
e . al. [3] used uzzy logic con ol. Mao e . al. [4] also
esea ched p ecision posi ioning o a DC mo o , bu in he
p esence o ae os a ic ic ion, and hey used a PID
con olle as well. Howe e , al hough many ad anced
con ol algo i hms we e used, PID con ol me hod has
p o ed i s e iciency, obus ness, and conside o be ully
e ec i e o DC Mo o applica ions.
2 DC Mo o Model
2.1 Ma hema ical Model
In a ma u e con ol o Pe manen Magne DC mo o s,
he ol age applied o he a ma u e o he mo o is
adjus ed wi hou changing he ol age applied o he ield.
Figu e 1 shows he Pe manen Magne DC Mo o
equi alen model.
Fig. 1. Pe manen magne DC mo o model [5].
Unde he assump ion o a homogeneous magne ic
ield, he di ec cu en (DC) mo o is modeled as a linea
ansduce om mo o cu en o elec ical o que. The
classical model o he DC mo o is composed o a coupled
elec ical and a mechanical subsys em.
The angula eloci y is con olled by he inpu ol age
𝑉
𝑎 wi h a cons an ol age d op a ibu ed o he b ush and
o o esis ance, and a back-elec omo i e o ce (back
em ) caused by he o a y a ma u e. The mo o induc ance
con ibu es p opo ionally o he change in he mo o
cu en . The mo o cu en couples he elec ical
componen wi h he mechanical one, as i gene a es he
d i ing o que. This o que is an agonized by he mo o
ine ia, s uc u e damping, ic ion, and he ex e nal load.
2.1.1 Elec ical Cha ac e is ics
A di e en ial equa ion o he equi alen ci cui o a
DC mo o , illus a ed in igu e 1, can be de i ed by using
Ki chho ’s ol age law a ound he elec ical loop.
Ki chho ’s ol age law s a es ha he sum o all ol ages
a ound a loop mus equal ze o, o
𝑉
𝑎− 𝑉𝑅𝑎 − 𝑉𝐿𝑎 − 𝑉
𝑐= 0 (1)
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Acco ding o Ohm’s law, he ol age ac oss he
a ma u e’s esis o can be ep esen ed as
𝑉𝑅𝑎 = 𝑖𝑎𝑅𝑎 (2)
Whe e 𝑖𝑎 [𝐴] is he a ma u e cu en . The ol age
ac oss he induc o is p opo ional o he change o cu en
h ough he coil wi h espec o ime as
𝑉𝐿𝑎 = 𝐿𝑎
𝑑
𝑑𝑡 𝑖𝑎 (3)
Whe e 𝐿𝑎 [𝐻] is he induc ance o he a ma u e coil.
Finally, he back em can be w i en as
𝑉
𝑐= 𝑘𝑣𝜔𝑎 (4)
Whe e 𝑘𝑣 [𝑉/𝑟𝑎𝑑/𝑠𝑒𝑐] is he eloci y cons an
de e mined by he lux densi y o he pe manen magne s,
he eluc ance o he i on co e o he a ma u e, and he
numbe o u ns o he a ma u e winding 𝜔𝑎 [𝑟𝑎𝑑/𝑠𝑒𝑐] is
he o a ional eloci y o he a ma u e.
Subs i u ing equa ions (2-4) in o equa ion (1) gi es he
ollowing di e en ial equa ion:
𝑉
𝑎− 𝑖𝑎𝑅𝑎− 𝐿𝑎
𝑑
𝑑𝑡 𝑖𝑎− 𝑘𝑣𝜔𝑎= 0 (5)
2.1.2 Mechanical Cha ac e is ics
By balancing he ene gy o he sys em, he sum o he
o que mus be equal o ze o. The e o e,
𝛵𝑒− 𝛵𝜔′− 𝛵𝜔− 𝛵𝐿= 0 (6)
Whe e 𝛵𝑒 [𝑁𝑚] is he elec omagne ic o que,
𝛵𝜔′ [𝑁𝑚] is he o que due o o a ional accele a ion o he
o o , 𝛵𝑤 [𝑁𝑚] is he o que p oduced om he eloci y
o he o o , and 𝛵𝐿 [𝑁𝑚] is he o que o he mechanical
load. The elec omagne ic o que 𝛵𝑒 is p opo ional o he
cu en h ough he a ma u e winding 𝑖𝑎 and can be
w i en as:
𝛵𝑒= 𝑘𝑡𝑖𝑎 (7)
Whe e 𝑘𝑡 [𝑁𝑚/𝐴] is he o que cons an and, like he
eloci y cons an , is dependen on he lux densi y o he
ixed magne s, he eluc ance o he i on co e, and he
numbe o u ns in he a ma u e windings. 𝛵𝜔′ can be
w i en as:
𝛵𝜔′ = 𝐽 𝑑
𝑑𝑡 𝜔𝑎 (8)
Whe e 𝐽 [𝑘𝑔𝑚2] is he ine ia o he o o and he
equi alen mechanical load. The o que associa ed wi h
he eloci y is w i en as:
𝛵𝜔= 𝐵𝜔𝑎 (9)
Whe e 𝐵 [𝑁/𝑟𝑎𝑑/𝑠𝑒𝑐] is he damping coe icien
associa ed wi h he machine mechanical o a ion sys em.
Subs i u ing equa ions (7-9) in o equa ion (6) gi es he
ollowing di e en ial equa ion:
𝑘𝑡𝑖𝑎− 𝐽 𝑑
𝑑𝑡 𝜔𝑎− 𝐵𝜔𝑎− 𝛵𝐿= 0 (10)
2.2 T ans e Func ion and Block Diag am
A block diag am o he sys em can be de eloped om
he di e en ial equa ions (5) and (10). Taking he Laplace
ans o m o each equa ion gi es
𝑠𝐼𝑎(𝑠)− 𝑖𝑎(0)= − 𝑅𝑎
𝐿𝑎
𝐼𝑎(𝑠)−𝑘𝑣
𝐿𝑎
𝜔𝑎(𝑠)+1
𝐿𝑎
𝑉
𝑎(𝑠) (11)
𝑠𝜔𝑎(𝑠)− 𝜔𝑎(0)=𝑘𝑡
𝐽𝐼𝑎(𝑠)−𝐵
𝐽𝜔𝑎(𝑠)−1
𝐽𝑇𝐿(𝑠) (12)
I pe mu a ions a ound some s eady-s a e alue a e
conside ed, he ini ial condi ions go o ze o and all he
a iables become some change a ound a e e ence s a e,
and he equa ions can be exp essed as ollows:
𝐼𝑎(𝑠)=−𝑘𝑣𝜔𝑎(𝑠)+ 𝑉
𝑎(𝑠)
𝐿𝑎𝑠 + 𝑅𝑎
(13)
𝜔𝑎(𝑠)=𝑘𝑡𝐼𝑎(𝑠)− 𝑇𝐿(𝑠)
𝐽𝑠 + 𝐵 (14)
The abo e equa ions can hen be easily pu in o a block
diag am ha ep esen he ma hema ical model o
pe manen magne DC (PMDC) Mo o , as shown below.
Fig. 2. Block diag am o a PMDC mo o sys em.
F om he block diag am, and assuming ha ΤL=
0 [Nm], we can ind he equi alen ans e unc ion:
𝜔𝑎(𝑠)
𝑉
𝑎(𝑠)=𝑘𝑡
𝐿𝑎𝐽𝑠2+(𝑅𝑎𝐽 + 𝐿𝑎𝐵)𝑠 + 𝑅𝑎𝐵 + 𝑘𝑡𝑘𝑣
(15)
We ake he AXEM DC se omo o , F9M2, which has
he ollowing cha ac e is ics [6]:
Table 1. Mo o Cha ac e is ics.
Pa ame e [uni ]
Value
To al mo o esis ance 𝑹𝒂 [Ω]
0.98
To que cons an 𝒌𝒕 [𝑵𝒎/𝑨]
0.0274
Back em cons an 𝒌𝒗 [𝑽/𝒓𝒂𝒅/𝒔𝒆𝒄]
0.0297
Damping cons an 𝑩 [𝑵/𝒓𝒂𝒅/𝒔𝒆𝒄]
7.2 × 10−5
To al sys em ine ia 𝑱 [𝒌𝒈𝒎𝟐]
3.2 × 10−5
Induc ance 𝑳𝒂 [𝑯]
25 ×10−6
Ra ed o que [𝑵𝒎]
0.282
Ra ed speed [𝑹𝑷𝑴]
3000
Ra ed ol [𝑽]
14
Acco ding o he selec ed mo o , he ans e unc ion
o he speed [ωa(s)/Va(s)], assuming 𝛵𝐿= 0, is:
𝐺(𝑠)=0.0274
8×10−10𝑠2+ 3.136 ×10−5𝑠 + 0.00088434 (16)
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3 Robus sensi i i y analysis
In eali y, 𝛵𝐿 is no ze o, and i s alue is conside ed as
a dis u bance o he sys em. The e o e, o ge he bes
esul s and make he sys em obus owa ds he
dis u bances and noise signals ha can a ec he
esponse, as shown in igu e 3, we a e going o analyze
he sensi i i y o he open loop sys em o he angula
speed in he p esence o he PI con olle .
Fig. 3. Closed loop sys em in he p esence o dis u bances and
noise signals [7].
Gene ally, mos o he dis u bances occu a low
equencies, which can a ec he esponse, and mos o
he noise signals occu a high equencies, which can
a ec he measu emen . I is ound ha he sensi i i y
unc ion, 𝑆(𝑠), is p opo ional o he dis u bances, and he
complemen a y sensi i i y unc ion, 𝑇(𝑠)= 1 − 𝑆(𝑠), is
p opo ional o he noise signals. The e o e, he sensi i i y
unc ion mus be o low magni ude a low equencies, so
he e ec s o he dis u bances a will be minimized, and i
mus be o high magni ude a high equencies so he
e ec s o he noise signals will be minimized as well. As
a esul , we ob ain a obus sys em ha is capable o
wi hs anding un-modelled dynamics, dis u bances, noises
…e c.
Since he open-loop unc ion, 𝐿(𝑠) = 𝐺(𝑠)𝐺𝐶(𝑠), is
in e sely p opo ional o 𝑆(𝑠), hen i mus be o high
magni ude a low equencies, and o low magni ude a
high equencies. In o he wo ds, he loop unc ion should
beha e as an in eg a o , and o achie e his, we mus
design ou con olle o gi e us such beha iou .
Fo his pu pose, we choose a pa allel PI con olle :
𝐺𝐶(𝑠)= 𝑘𝑝+𝑘𝑖
𝑠=10 +20
𝑠 (17)
The loop ans e unc ion is gi en as:
𝐿(𝑠)= 𝐺(𝑠) 𝐺𝐶(𝑠) (18)
𝐿(𝑠)=0.274𝑠 + 0.548
8 × 10−10𝑠3+ 3.136 ×10−5𝑠2+ 0.00088434𝑠 (19)
The sensi i i y unc ion o he open loop sys em is:
𝑆(𝑠)=1
1 + 𝐿(𝑠) (20)
𝑆(𝑠)=𝑠3+ 39202.2𝑠2+ 1.10543 × 106𝑠
𝑠3+ 39202.2𝑠2+ 3.436 ×108𝑠 + 6.85 ×108 (21)
To illus a e his analysis, we will d aw he Bode plo
o he open loop sys em, as well as o he sensi i i y
unc ion using LabVIEW Ma hSc ip ool, which will
esul in he ollowing plo s.
Fig. 4. Magni ude diag am o he sensi i i y unc ion.
As shown in igu e 4, he e a e no peaks in he
magni ude o he sensi i i y, which means ha he e ec s
o he dis u bances and he noise signals on he sys em a e
well minimized a all equencies.
Fig. 5. Bode plo o he loop ans e unc ion L(s).
As illus a ed by he magni ude plo in igu e 5, he
loop unc ion beha es like an in eg a o , which, in his
case, will inc ease he obus ness o he sys em depending
on ou ea lie discussion. Addi ionally, he phase ma gin
is abou 80 deg ees, and he gain ma gin is in ini y. This
means ha he sys em is s able a e e y alue o he gain,
and he phase ma gin is la ge enough o ensu e ha he
sys em is no close o -1 in he Nyquis plo . Howe e , he
sensi i i y analysis is good enough o ensu e ha some
amoun o unce ain ies migh no ha e e ec s on ou
sys em, which, in u n, ensu es ha he sys em is obus
unde some bounda ies. Ne e heless, since we do no
ha e any igh hal ze oes o ime delays in ou sys em,
he sys em does no su e om he “lack o obus ness”,
bu i emains o he unknown unce ain ies o de e mine
how much ou sys em is ulne able, o obus .
4 LabVIEW implemen a ion
To design he sys em wi hin LabVIEW Simula ion
loop, se e al simula ion Sub-VIs we e de eloped, and
wo con olle s we e used, one o he speed (PI), and
ano he o he posi ion (P). The implemen a ion p ocess
is de ailed shown in igu es 6-8.
Fig. 6. Posi ion loop subsys em a chi ec u e.
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Fig. 7. Speed loop subsys em a chi ec u e.
Finally, he mo o dynamics subsys em, ha ing he
same a chi ec u e in igu e 2, is shown below:
Fig. 8. Mo o dynamics subsys em a chi ec u e.
5 Resul s
Acco ding o he sensi i i y analysis and he sys em
equi emen , he inal alues o he con olle s we e ound
as 𝑘𝑝=10 & 𝑘𝑖=20 o he PI speed con olle , and a
p opo ional gain o 40 o he posi ion con olle .
To es he e ec i eness o he sys em, we a e going
o apply a s ep esponse o 90 deg ees, and when he
sys em is s abilized, we will add a load o 0.282 [𝑁. 𝑚] a
0.4 𝑠𝑒𝑐 in o de o es he obus ness o he sys em. The
esul s a e shown below.
Fig. 9. Angula posi ion esponse, wi h applying 𝑇𝐿 a 0.4 sec.
The sys em eaches he desi ed posi ion in 0.2 𝑠𝑒𝑐
wi h no o e shoo o oscila ions, and he sys em is well
s able when applying he o que load.
Fig. 10. Speed esponse, wi h applying 𝑇𝐿 a 0.4 sec.
Fig. 11. Vol age esponse, wi h applying 𝑇𝐿 a 0.4 sec.
We can see ha all he pa ame e s o he mo o a e
wi hin he maximum limi s, his is achie ed by adding he
nonlinea unc ion, Sa u a ion, in he con ol sys em.
Addi ionally, he obus ness o he sys em is ully able o
o e come he unce ain ies (caused by he o que load)
and nonlinea i ies (caused by he sa u a ion), and he inal
esponse o he posi ion is dynamically well beha ed.
Howe e , he posi ion esponse could be achie ed in
a sho e ime by choosing di e en pa ame e s o he
PID, bu i would cause oscilla ions in he magni ude o
sensi i i y, which in u n will dec ease he obus ness.
6 Conclusion
LabVIEW Simula ion Loop was used in building a
simula ion sys em o a PMDC Mo o . The sensi i i y
analysis allowed us o apply he bes alues o a obus
con olle wi h no oscilla ions o o e shoo in he inal
esponse, e en when he maximum load applied.
Acknowledgmen
The wo k/publica ion is suppo ed by he EFOP-3.6.1-
16-2016-00022 p ojec . The p ojec is co- inanced by he
Eu opean Union and he Eu opean Social Fund.
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DC mo o used in sola panels wi h a i icial neu al
ne wo k, 2, 487-492 (2012).
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con olle s wi h di e en de uzzi ica ion me hods,
(2010).
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