- Email: [email protected]
A Robust Control of Fuel Injection Servo System
Copyright © IFAC Design Mcthod s of Control Systcms. Zurich. Switzerland. 1991
A ROBUST CONTROL OF FUEL INJECTION SERVO SYSTEM Y. Yanagita, T. Eisaka and R. Tagawa Department of E/eccrica/ Engineering. Hokkaido University. Sapporo. Japan
£\bst~~t:t,
In t h is P~f1"I, 1, ,, iJu s t :'I l,,
fll el in jecli"ll jllI11 11 ' "fdi""~l ' II' !l gillC car. T he p ~ I ';[ lll ell' r ~ uf thi s pl:llI t
V;H )'
u will g tu lh e chang e
vf temperature, lL is de~ir,d,j,; tll;I L. e\'ell i f pa r~ lI11 ete r ::; "f plant the l'h:lnge . the spil l pvs iti v n Lrncks Lil e rt.'fcrcnc e input r ' '1 ,idl:,' ~1Il( 1 c"lTe d iy, 1(;\1 ;\ 1 is n d es ig n lIl e th ud " f r e ali zin g Ill ud e l m atc hin g fr olll r efl'n 'l le'" inplll s t" C(, "l rtliie d v~ ri ;[ iJi es a n d a ssuri n g high l " h u stn css fi) r th e p:lr;lIneler unq; r t a inti ('~ i, v lh" ndlll s t. c U l1lpe n s ~lt.u r s illlui t'1I 1el'lls iy, Since Lh is pl'l nt is ~ n u n minimum p h " sl' 'lIld s i n g lv illPlll s in g le ll Ut.put Sy St.l' lll, the r"bu s t. 1·" nlp e ll S:l Lu r is d es ign e d a s f() l lo\Vs : (1) '1\) a,: iti ('v(' rtd)lI SI s\ <" :ldy ·s tat.e pro pe r ly. t h e r obu st Cl) /lll 'l' n ~rl t,,, r is d es ig n ed so as to minimi ze If :? n pl'ln (,f ". \I'" .; wit hi n the des ire d bpll nd c d fr cq ue n e'Y r~ n ge,
(2 )
T o main tain
s talJi li ty . l lt r.' r" b u n (' u n 1l' cJlsalu r is d e si g n e d s uc h L1 w t th e Lr:1I1 " fe r fu" eli,, " 11 \\' '''11 fr olll the COnl'cplu ,,j inpul t n Llt l' ", ," t nd \'"ri;l iJI l' h a s s u ita h le Ill~ r g ill \)rg~l i ll ,"](1 ph'lsl', Si mu laL io ll u nd expe rim e n L r esu l t" s lll'\\' ti1; ll, Lh(' Jl wtlwd pr" pused in thi s p; '1 )(; r is effc L'Li vl.' lh e l'os iLi l) n co nL r o l o f fu (' I i !ljfT!. i"n I'U IllP ,
1!'iTI W lJ l i e Tl ():\
S imulati lln n n d l'XpCr illl e n l. l'C.' sulLs s h ow that it is cffe l,t i ve fo r th e co n t ;'u l o f pl a n ts w i l h param e t e r
D iese I en gi n E'
c~
I' is ('I'(, n um il'" I il l l he u ~l' u i' fUI' I an d
v " riati o n s , S till, lh e fll ll ()w in g pl'u blem s remain:
us ed in th e wo rl d wi d e , In th e futu r (' . t ill ' l· xpans io n o f
( I ) lL is diffi c ul t L" show co n t r u l s pe cificati o ns
i ls a ppli ca Lio n s is eXj1f:l'led ,
ba se d nn a Lim e d ll maill, (2 )
Th e fuc l inj e,:li"n s (' I'\'" " l(','I" l lli ~ lil " C d ies ,, 1 pUIllP
It is diffi culL
( 11
adju s t d eli,.'nle ly the property
f)fc o n t r o l sy ",Le m ,
r eg ul ;ltcs t h e V()lllllll: " I' fl le l s (' n l. tn l'<.lmiJus li u n th e c ham be r of diesc'l e n f~ i n (', Til (' r" f" re II C0. i n p u L of Lh e
IU l l\ 1 tl\.. Tn gaw a 198 (i ) is fn vo r nh le fo r l renling these
se r vo sysLe m is d ec ided ba s ed .. " si g n al s o f vari o u s
pr o bl e m s, Thi s is a d es ig n JJl e th ud of r e alizing model
se n so rs so a s t li se nd til " d csi r :lbl e vo lulll e , Thi s
matchin g fr o m r e fer c n ce input s t o control l ed
s y s Le m is r(' quesled t" pos ses s rapid a nd co rr ec t
va r i ab les ,l n d
~ s 5 llrin g
hi gh r o bu s tne ss fo r the
r e fe r e n ce in put prnppdy , Sine'p th" 51' 1'\'1) II1 ech[l ni s Il1
param e le r un certa in t ies by th e r obu s t compe n sato r
is so ak e d in fu e l (,il . l.he P:1I':LIIl (' l" r s nf th e pl a n t " "ry
s imul t an e o u s ly , Th e r u bu s t
ow in g Lo l h e ch a n g e o f tll<: l' ,,(· f'lj c il'llt " f \' is cos i ly by
lll :ll\es t h e tra ns fe r fu n l,t i" n I{Wd;' from funcl a menta l
\Vid e vari [l l.i o n o f lC II1p e r :llll r e, Thus , i l. is im pO l' Lant
e quival e nL ci is l.url'l;[ nl' Cs to th e co n t roll e cl va riables
compe n s~to r
whic h
thaL, ev e if pa r;III1 Plpr s ch";l ,lY , th e c" ntro l systeIl1
r Inse to Ze n ) is ;l(ldcd to l h e m ndel n18 Lching contro lle r
has n o tr ac kin g e IT,) r n n d i ts r ci'el'l.' n ('e inpuL r esp')J1se
d es igne d by u su alm e l h ocis ,
hardl y chan g es, B y a p pl y in g Rn b ll s t ;\'l ode l ;Vl n tchin g to the min im um
11..> o ptim a l d es ig n o f fu e l injec Li c li r u m p ba sed o n
phn se planl. , in C[l s C o f cll nLinu o us -data syste m, low
mixed sen s iti v ity pro ble m h:l s be e n cl o n e Ol.Kur[lo ica
s e n s iti vity ~lnd rub u s t s ta b i lity can be g ua r anteed for
and other s . 1989) , In th is m el h od. lh e co n l l'oll er
an y bo u n d e d parame le r va riativn s unless the number
s ati s fy in g the foll o win g co ndil ion s is des igne d:
o f un s t a ble po le ehan g es ( y, Z h o n g and othe r s, 1989).
DC se r vo m o tor, its efficacy is
( 1) internall y sta bili ty o f cl os ed·l ollp s y s te lll,
In l he a pp licatio n La a
(2 ) lo w se n s iti v ity .
co nfirm e d with com p liLe r simulati on5 and labo r ato ry
(3) hi g h l'I)b us l s t.nbili ty,
exp e rimenls ('1'. Ei s a k a a nd ot h e r s, J 989 ),
185
The fuel injection servo mechanism is non-minimum pha~e
plant. If this syste m was designed in the same
way as minimum phase plant, it becomes unstable system. In this paper. we propose IUvL'vl meth od applicable to the non -minimum ph;1se and single input single output (8 180) plant. (1) To achieve l o w sensitivity. the robust
co mpe nsat.or is desi gne d so
~s
to minimize J[2
n orm of I! IF, i.\" . To
(2)
lllainL~in
sLability,
the
Fi g .l gnin I'r lll'p rt y "fLhe plnnLs
r obusL
cO lllpen satl) r is desi g ned such thaL Lhe Lrans["er ["uncLi o n 1;lV l . ,,, fr om conceptual
Assullle L1wt when the' par:1lllelers of plnnt vary from
input to cl1nLrol variable has suitable margin
A. 13.
of gai n and phase.
U' I-6U'
e'
and
to 1\ -I "" 1\. 13 16 1]. C' -16.C" ancl
f)'
r e~pectively.
the sU ILes , co ntrol variab les
and measuremenL vnrinbles of Lhe and y'" to
pl~lI1t
chnl1ge from
:111'[ y' respecl.ively under the
Computer simulatio ns and experiment shows that
x"'
RI\1I\1 is effective fur pan1meler variaLions.
same n'fercnce input and the srlIlle zer o initial the
iL "
.1'.
IJ
states of the "<.InLI'lII sysLem. At this time, the state equation is described as fuliows.
IDENTIFICATION
i:
= A .x -/
/3 u I d I
(4)
'
y'· =C"·x-/-J)'·x /- «("
Ollr fuel injection pump is different from Lhe one
(5)
where el l = 6/\ -,r-l o.iJ ·u,
Rura oka and others (1989) used, buL h as the same
lG)
d? = o.C'x/-6/Yu.
sLructure. This pinnL was identified as same way. The transfer fUllction of the pump at 25"C is as follows.
(7)
ell and ,12 are defined as l.he equivalent disturbances corresponding to the p:uallleters vnr ia tions 6A. 6D,
/'
12:'1= ",'
o.C,6D*
S:I -I· 11I89·,'/+8 .J:'9x IO: 1·5-1 l .159x lO"
197 5' (S - I%)IS+SG!';j 15 + 17.0 7 H S -H5.9 :t.lGS.45)
(1)
The plant can be desl'l'ib e d in terms of transfer Similarly, transfer fUllctions at O"C and GO"C are
funcLion as
follows. )"=1'"y. u+['",·" .
__ 1117 2.<;2 _ ,1% 1 X 10 3 " - 7 :J GGx IOs I'
(0)=
,n
5'12(i:'7 ..<;2; Z.9ZIXI0'S-t-JO(i2X l o 5
l'"y, (Si (2)
C~
(9)
l C' ·( sI -Ai -1.D -1- D*]
1\ly,(S;=lC(sI-J\) - I , lJ .
- 1.'l .:':l.';2_ 4 .:'!iZX 10:1.,';-1.'19 1 x IOs
I'
(8)
wh ere rL=ld;l' . cl/V, ,
(l0) (11)
(fiO)= - -- - - - - - - - - - - - - - -
",.
s:I''!!i .97S2 + 8.452X 103. 5
+ 1.4 48x IU 5
Let left fractoriznti on
(3)
( T~ gn wa
1991) of chnracLerisLic
transfer funcLion 1'"" ns follows:
Fig.l shows g;)in property of plants.
1'"),, =I'dy,·N I(
(12)
OUTLINE OF RMlVl
Pd y ' is the lefL element and
NI1 IS the right element.
Pdy' satisfies realizotion condition and has no zeros.
In RM1VI, disturbances are defined to represent the
I'd J ' is also give n by lllulLiplying Pdy' by real valued
influence of parameter var iations. Let x denote the
matrix
state, u the control variable, y the controlled variable,
n fr om right side. Jlenc e, (1 = n·el. Therefore, d
is defined as fund8l11enUll e'lu ivnlent disturbance.
s> the measuremen t variable of plant withou t y and y* all output (equals to [yT, yT]T) .
l~ ubust
Equivalent disturbance
compensator
StrucLure of RMM syslem for SISO plant is shown in Fig.2. CI", is lhe left element of left fractorization of
186
[C r ", CoY·,,] , It is th e input to C'IIl' In the case of S1S0
h as zeros of I'l;lnt as eigenvalues. Th er efore, a noth e r
system, C'IU is equal to 1 1 D r.;.
m eth od is used to make the gain of 11 Wciy sma ll.
Model
Hi'vI:V1 FOE i\ON-l\1IN IM UM PIlASE r
PL ANT
y
In thi s p:lper. wc use foll owing sy mbo ls . N ominal plant; I'" = N" " 1 /)" " . 1'l anLwi l h I'ar:lm e tcr varied; /'=iV,, / IJI' . IVr\"= L ,y / D.
Moclc lt ra ns f"rr fun dion;
l'vIl/d e l m:l k hing co ntroll er;
,l _____ __ ___ _ ___ _ __ ___ ____ ___ ___ l,,
[ 1' //" .1'.'",,1::=1 -NI'" , iJ" "I . Hubus tlilLc r ; F =jf {J,,"
llobust Compensator
11: = 1" ,; 1" -1-
Wh e re N I'" . {J/ ,,, . !.f r ". M y" . D c and DF a r e po lYll ulIlinl s .
d from u and y ' and is a polynomial
matrix. Eq .(8) leads to
d =Pci.l' - I -it -Pci.l' -1·p"y' U , Th e refore,
r =[-Pci y -
I
Puy, Pciy - /J.
-!- I", .!-I + 1"11.
whcl'(' r; I i = 0, I , '" . k ) :1re fr ee p[lrnmcters .
Fig.2 Th e s tructu r e of RMM syste m l' estimates
[C , ,, . C mJ =[Mr ".
A1.'" 11 I / lJ('
( 13)
III Fi g .2. tran s fcr functi un is wrillcn a s
( 14 ) (1 8)
The ou tp ut of R is used as the inp u t to C'lII n o t
l Lle;ld s Lo
throughout r obust fi lter P, then, the tra n sfer functi on /{ Wdy fr om fundam ental equ iva lent dist ur bance to
U
11-'-
r/y
~
-
J
-
/ ) . /) I-'
If)
C
/i
V
· /1l)
IN I'll
( 19)
controlled variable o f th e system with r o bu st compensator added is as follows:
In th e case o f n on -minimum phase plnnt, we deLermined fre e param e ters of robust co mp ensato r so
( 15)
where MWJ y is t h e t ra nsfer function from
as to r eal ize r o bust s Leady-state pro pe r ty, robust
d
tt'ansicnt pt'ope rty , and robu s Lslability.
to y before we add rob ust compensato r.
l~o bll s t
If R mak es 11 W,iy zero, the re i s no inn uence of
parameter variations . This R is g ive n by Eq.(lG).
s tead y pro perty
We let 1J '1I be t h e chnracteri s lic po lyn omial of o ute r mode l in put to the co ntrol sy s tem, a nd apply the
(lG)
folluwing the orem (T[lgaw[I 1991). In Fig.2, if C",,·]? T is proper, r obus t compe n sa tor can realize strictly zeroing. When C",,·]?T is not proper,
Theorem
us ing robus t filter, we nH1k e C,",. F ·R ·r proper.
For the pLlJ1t with pnramete r variat ions, t h e ncce ss ary :1I1cl s llfri cienlly condit ions fo r r o bu st
After]? is given as Eq.(1G), Eq. (l5) lefl ds to
steacly -s Ln Le properLy (17)
( 1)
:HP
Lhe numeraLo r o f transfe r function /lWril' from ft lll(l:llll ental di s turbnnces to controlled
Therefore , by choosing a low-p ass filter, with a
vnriab lcs ha s DH! [IS fador,
suitable relative orde r , s u ch that its frequency
(2) robust s Lability is' sa ti sfi ed in rflnge so f
property is close to " 1 "in t h e frequen cy range from 0
specifi ed parameter var i atio n s o f the
to Wc, the II Wdy can be m ade close to ze ro wi thi n such
controll ed objec t.
frequency range the num erator of /{Wd y is (DDF-I-N /',,· R ). lIen ce, the This method is not avai lable for non -minimum phase
free paramett'r of robu st compensntor (Ire ch ose n so as
plant, this is because the contro l system a fLer RMM
to lTI fl k e (D·J)F -/- N/,,,·]?)
h ave 0," as f"clor, thus
robust sten dy-staLe property is ac hi eved.
187
(22)
Robust transient property we regard the plant with parameter variations as To achieve the robusl transient properly, in stead of
PIl + 6. Closed-loo p ~ystelll is described in Fig.3.
zcroing, we make the gain of IiWdy small based on
In Fig.3, /lW U '1I is wrillen in terms of the transfer
some evaluation. In this paper, after robust steady-
functi on of numinal plnnt, m odellllalching plnnt and
s lale properly is rea lized, we minimize lhe H2 norm
r obust compensatur.
of 11 Wd y using lhe rcmainin g free pa rameter. I
(~ 211
i~
" I
I
"
i 11' - i<:'l'lrOJ)i .1O)
(J
/) /'"
,11' • = - - (,1/
It
U
rly
2
I'
IJ · /) F
11
(23)
.+ 11 , /1)
. f) .\11
/.
"1
(20)
.
APPLICATION OF rUvlM '1'0 FUEL
I NJECTION l'UivlP
lnthe next equation ''le decide sampling time as 3 Illsec based o n ca lculating time uf nllllpute r . The discrete tran sfe r
(2 1)
funcLion of the of 2!i"C plnnL is ns follows. minimizes lhe 112 n orm of /(Wd y . /'
(:u,)
".\
llobusl stabili ly
'1.1)1111 X 10 .. 1.;//_ 1 1) ,1.1.,< 11 1- 1·1,+!i(;G7X 1 0 -~ = - - - - - - -- -- - - - - - - I /'-:11;:'(;'/' ·1· :1 JRU·I,-7 .213x IU -
In some cases, th e sys tem with this robust
'1.1111'<10-'·(1,_1177) ' (/'+ 1.8'1)
compensator is llns tnble fo r a plant with parameter
(1.
-I-
(24)
II%UHI, -11.8;'3 1:.1(178)
va rin t io ns.ln the se cnse, bccn use the purpose of HMM is to mnke the gnin of /( \Vd y small, we decide some free
In the same way, the plnllL ;lL WC and GO"C are given
parnmelers so ns to achieve robll st stability and such
as follows.
t h at the frequ ency property of /lWd)' with 1I2 norm
_ 1 98" X 111 - ' . I,~ -I 7r, ,1Gx 10 - ~.I,
minimized doe s not chn n ge largely.
I'
(11)=
-I
I .9lH X I U - 2
.<;:1_Z ,!:I:1-i' ,I.. 1 8K:JI,-'i.50 GX 10 - 1
fly
(25) -:'.'181 /)
(GO)=
x IO - '·1.~+G:,
IO - 2 1. - I .
2
1:" - 2 G821,'+21:IG·I,-7498XJ()-1
Ily
(26)
We ch oose Illodel transfer function such that se llling tim e is within 0.1 second. :1.I;:,xln -· :I ·II, ·, 2 . 12)(1, IO.IG)(1 -077)(I, + ) 8'i)
IV ')'
(I, -117 ,) /(1, - a.7ol(1, - U.8'li
rOIllJ)(·" .o:;nlo r ~
- - - - - - --- - - - - -- - -- --- - - - - - - ------ - - - - - --- -
--
(27) The model I1HlLching controller is deLermined ns Eq.(28) and Eq.(2f)) . -9 . 12!i X I O-'·I,'-2081 x IO - ll,-l.onx IIl - '
c:, "
i:: 2 _
0.:, ,I I 4';: -
O. I 7 G2
(28) G;,r;GI,~ - I I :)2.1, , 4 fl88 1,2 _ fI .!i 4 I'll, - O.17GZ
(29)
RobusL compensaLor is gi ve n by, I'll'/=
-4.000XIO- 2·Z'+1.044X10 - I ·Z -5.GG7XIO - 2 ,
(30)
ryq = ZI- 2.G56·Z 2 + 2.380·Z + 7.213 X 10 - 1, (31) Fig.3 Closed loop system
R=r2Z2+r/ ·Z+ ro,
[) =
Parameter chnnges are equivalent to additional
F
variation 6,
188
J
(I,_OG8):l
.
(32) (33)
(2) minimi7.e o r make small the Il2 norm of Trans[er [unction /lIVely is s uch that robust steady-
uIVJy , (3) increa se the stability margin of HIV u '".
state property is ach ie ved [or s tep reference inpuL. '-0 is described as a funcLion of I"] and 1'2, '-0=-1'2-1'/+2.703.
J\CI{NOWLI~DGEMEN TS
(34)
f(II'Jy is Cri kui::Ited rlS fol lows.
'I'he a uth ors exp r ess grntitud e t o H.Kura okil and
11 IV cl\" = - G.G88r/ -21.90·'-/·'-2+ 144.1·,-/
-2U)()·'-/+327.3·'-2-2529
N.Ohba, and thilnk Il&D De pt., N ipjlo nclenso Co., (3G)
Ltd., for suppo r ting this experiment.
Next, '-2 which minimizes Eq.(3G), is calculated as llEFEI{ENCES
foll ows . '-2= - O.!)·r/ 1'/
(36)
+7 .'171.
E isaka,T. and others. (1989). Evaluation of rrubu s t Mode l-MaLching for the Control oC a DC Servo Motur.
which minimiz es Eq.(3G), is calculated·as follows.
Int ernatio/tal Jou/"Iwl o("Colllro/. GO,479-493.
,-/ = -86 .S3 .
Kuraoka,Il. "nd ot hers. (1989). J\pplica Lion of 1[ ' " J\t this time, ro=3iUiG , '-2=GO.H9 (C :1 se 1). In this case, in Fig.4 , Nyl{uist
opti mal design 1.0 ilutcJtllOtive fuclcont rol. Proc. ACe,
plot of -c.(Z)·IIIV,, "II(Z) is
1%7 -19G2.
s hvwn. IL s hows that Lllis system becomes unstable
'I'a gawa, rr. (198G).
whpn parrlm ete rs \'aried. Th eref"re, wc set 1'/ = 0 so
Control. Computer and J\pplicat i on's Mook.
il'lod el i\latching and Robus t
t h at the Illrlrg in of sta bility is increasing. Under this
COlI!jJl.Ltrol, 13, G3 -G8, Corona, (in .Jrlpanese).
conditiun, set r () = -4. 76:l to achieved robust steady-
'l':1gaw:1,lC ( 1991). Dual Model Matching. Tu nppenr
state property (lnd 1"2=7.471 so as to minimize
[J2
at I s l1fAC symposium on Design Methods ofCont rvl
norm o[ /! IV ,iy(C,lse 2) .
System.
III these two cases, Fig.G shows the frequency prope rty
Model Matching Desig-n lVlethod Which Ensures
of IIW,iy(Z), rind Yig.G shows luc us of vecto r. Fig.7
Stabi li ty. Elec/ronics und COII IIllI.Lf!ica/i olls ill
s hows Nyq uist plot of - c.(Z)·u IVU"II(!.:) in Case 2 .
.III.l'IIN, l'arl III Fl.Lnciwncll/al Electronic Science, 72,
Fig.7 shows that this system is robust stable for the
%-104, (SC [U P'l'J\
Zhong,Y.
'°
plant which par,llllctcr vnr ied. Step re spo n ses nre
:1~
s hown in Fig.S. 1'~XPElniV1ENT I{I~SU LTS
4
2
'l'.I~ i saltil, 'l'.
'
=: t
("I)n tr o l of fuel in ject ion pump, simu lati on and
INC.).
. --.~- ----'-j
""'" \~' 1 \, i c'
,
\
/' \
./
:
11
1
""""'",------------------------///J'J
_'oL ____ ._,.______ ,_ __ ~ ___ _
ex periment t·e s ults s how that it is effedive whe n the
-10
-~
0
!!5
pbnt has pilnltn ete r \':1riations. Fig.4 Nyquist plot in case 1 (l)atO"C (2)ilt6()"C
Sti ll, it is not clea t· th;l t thc order and pole ass ignmen t of r obust compensator are m ost su itable. In this paper , to get r obust stilb ility, we set the free parameter dra wing the i\yquist pl ot of I! IV U"II a nd the frequency property of I! \\' d y . But, RMM is avo ilable for discreted-d:tta system and non-minimum phase plant, we ob tain r ob ust control ~yste m
j
///-----------------~---""""',
r /' r (
_4 ~ r{ubust Model i\]a tch in g is rlpplied to sp ill position
'['lII~C ll NICJ\,
-----.,------
J _] (2\
CONCLlJSlON
and Tagawa,lC (1989). llobu s t
by determining suitably the free parameters
of the robu st compe nsator so as to (1) satisfy the n ecessary and efficiently
condition for realizing robust steady -state property,
189
10
~~~C'
)
:~"~ 200
1 I
r----
I
o~
1-
200
~""'~:' )
:
t -· o~ ~
\
"
_o.L
'( 2 )
=::: ~r~_4-- '- ~ 10- 3
....... UH .. _ _
10-:11
~
_ . .
. ...... .
L.~'.
_---t- -" . . ........ t! • . _
10 -'
•
.
lOO
-,,_,-,-,-J
o
10'
,,. .. "' ..... "0>' (rod )
Fig.S frequency properly uf 11 Well' in Lh e (Gse 1 (2) ill Lhe ca se 2
1 I
lOO ..
I
Fig.G veclor locus of 11 W"',, in Lhe case 1 (2) in Lhe case 2
Fi g .7 Nyqu isL plot in cuse 2 (l)aLO"C
( • • 0)
Fig.S Step responses (1) al25"C (2) at onc (3) at 60 n C
---'-'
(1)
0. 4
t In"> .
(1) '20
._._ _ ~_____,.__ ~J
(2)at60"C
190
O .D
A ROBUST CONTROL OF FUEL INJECTION SERVO SYSTEM Y. Yanagita, T. Eisaka and R. Tagawa Department of E/eccrica/ Engineering. Hokkaido University. Sapporo. Japan
£\bst~~t:t,
In t h is P~f1"I, 1, ,, iJu s t :'I l,,
fll el in jecli"ll jllI11 11 ' "fdi""~l ' II' !l gillC car. T he p ~ I ';[ lll ell' r ~ uf thi s pl:llI t
V;H )'
u will g tu lh e chang e
vf temperature, lL is de~ir,d,j,; tll;I L. e\'ell i f pa r~ lI11 ete r ::; "f plant the l'h:lnge . the spil l pvs iti v n Lrncks Lil e rt.'fcrcnc e input r ' '1 ,idl:,' ~1Il( 1 c"lTe d iy, 1(;\1 ;\ 1 is n d es ig n lIl e th ud " f r e ali zin g Ill ud e l m atc hin g fr olll r efl'n 'l le'" inplll s t" C(, "l rtliie d v~ ri ;[ iJi es a n d a ssuri n g high l " h u stn css fi) r th e p:lr;lIneler unq; r t a inti ('~ i, v lh" ndlll s t. c U l1lpe n s ~lt.u r s illlui t'1I 1el'lls iy, Since Lh is pl'l nt is ~ n u n minimum p h " sl' 'lIld s i n g lv illPlll s in g le ll Ut.put Sy St.l' lll, the r"bu s t. 1·" nlp e ll S:l Lu r is d es ign e d a s f() l lo\Vs : (1) '1\) a,: iti ('v(' rtd)lI SI s\ <" :ldy ·s tat.e pro pe r ly. t h e r obu st Cl) /lll 'l' n ~rl t,,, r is d es ig n ed so as to minimi ze If :? n pl'ln (,f ". \I'" .; wit hi n the des ire d bpll nd c d fr cq ue n e'Y r~ n ge,
(2 )
T o main tain
s talJi li ty . l lt r.' r" b u n (' u n 1l' cJlsalu r is d e si g n e d s uc h L1 w t th e Lr:1I1 " fe r fu" eli,, " 11 \\' '''11 fr olll the COnl'cplu ,,j inpul t n Llt l' ", ," t nd \'"ri;l iJI l' h a s s u ita h le Ill~ r g ill \)rg~l i ll ,"](1 ph'lsl', Si mu laL io ll u nd expe rim e n L r esu l t" s lll'\\' ti1; ll, Lh(' Jl wtlwd pr" pused in thi s p; '1 )(; r is effc L'Li vl.' lh e l'os iLi l) n co nL r o l o f fu (' I i !ljfT!. i"n I'U IllP ,
1!'iTI W lJ l i e Tl ():\
S imulati lln n n d l'XpCr illl e n l. l'C.' sulLs s h ow that it is cffe l,t i ve fo r th e co n t ;'u l o f pl a n ts w i l h param e t e r
D iese I en gi n E'
c~
I' is ('I'(, n um il'" I il l l he u ~l' u i' fUI' I an d
v " riati o n s , S till, lh e fll ll ()w in g pl'u blem s remain:
us ed in th e wo rl d wi d e , In th e futu r (' . t ill ' l· xpans io n o f
( I ) lL is diffi c ul t L" show co n t r u l s pe cificati o ns
i ls a ppli ca Lio n s is eXj1f:l'led ,
ba se d nn a Lim e d ll maill, (2 )
Th e fuc l inj e,:li"n s (' I'\'" " l(','I" l lli ~ lil " C d ies ,, 1 pUIllP
It is diffi culL
( 11
adju s t d eli,.'nle ly the property
f)fc o n t r o l sy ",Le m ,
r eg ul ;ltcs t h e V()lllllll: " I' fl le l s (' n l. tn l'<.lmiJus li u n th e c ham be r of diesc'l e n f~ i n (', Til (' r" f" re II C0. i n p u L of Lh e
IU l l\ 1 tl\.. Tn gaw a 198 (i ) is fn vo r nh le fo r l renling these
se r vo sysLe m is d ec ided ba s ed .. " si g n al s o f vari o u s
pr o bl e m s, Thi s is a d es ig n JJl e th ud of r e alizing model
se n so rs so a s t li se nd til " d csi r :lbl e vo lulll e , Thi s
matchin g fr o m r e fer c n ce input s t o control l ed
s y s Le m is r(' quesled t" pos ses s rapid a nd co rr ec t
va r i ab les ,l n d
~ s 5 llrin g
hi gh r o bu s tne ss fo r the
r e fe r e n ce in put prnppdy , Sine'p th" 51' 1'\'1) II1 ech[l ni s Il1
param e le r un certa in t ies by th e r obu s t compe n sato r
is so ak e d in fu e l (,il . l.he P:1I':LIIl (' l" r s nf th e pl a n t " "ry
s imul t an e o u s ly , Th e r u bu s t
ow in g Lo l h e ch a n g e o f tll<: l' ,,(· f'lj c il'llt " f \' is cos i ly by
lll :ll\es t h e tra ns fe r fu n l,t i" n I{Wd;' from funcl a menta l
\Vid e vari [l l.i o n o f lC II1p e r :llll r e, Thus , i l. is im pO l' Lant
e quival e nL ci is l.url'l;[ nl' Cs to th e co n t roll e cl va riables
compe n s~to r
whic h
thaL, ev e if pa r;III1 Plpr s ch";l ,lY , th e c" ntro l systeIl1
r Inse to Ze n ) is ;l(ldcd to l h e m ndel n18 Lching contro lle r
has n o tr ac kin g e IT,) r n n d i ts r ci'el'l.' n ('e inpuL r esp')J1se
d es igne d by u su alm e l h ocis ,
hardl y chan g es, B y a p pl y in g Rn b ll s t ;\'l ode l ;Vl n tchin g to the min im um
11..> o ptim a l d es ig n o f fu e l injec Li c li r u m p ba sed o n
phn se planl. , in C[l s C o f cll nLinu o us -data syste m, low
mixed sen s iti v ity pro ble m h:l s be e n cl o n e Ol.Kur[lo ica
s e n s iti vity ~lnd rub u s t s ta b i lity can be g ua r anteed for
and other s . 1989) , In th is m el h od. lh e co n l l'oll er
an y bo u n d e d parame le r va riativn s unless the number
s ati s fy in g the foll o win g co ndil ion s is des igne d:
o f un s t a ble po le ehan g es ( y, Z h o n g and othe r s, 1989).
DC se r vo m o tor, its efficacy is
( 1) internall y sta bili ty o f cl os ed·l ollp s y s te lll,
In l he a pp licatio n La a
(2 ) lo w se n s iti v ity .
co nfirm e d with com p liLe r simulati on5 and labo r ato ry
(3) hi g h l'I)b us l s t.nbili ty,
exp e rimenls ('1'. Ei s a k a a nd ot h e r s, J 989 ),
185
The fuel injection servo mechanism is non-minimum pha~e
plant. If this syste m was designed in the same
way as minimum phase plant, it becomes unstable system. In this paper. we propose IUvL'vl meth od applicable to the non -minimum ph;1se and single input single output (8 180) plant. (1) To achieve l o w sensitivity. the robust
co mpe nsat.or is desi gne d so
~s
to minimize J[2
n orm of I! IF, i.\" . To
(2)
lllainL~in
sLability,
the
Fi g .l gnin I'r lll'p rt y "fLhe plnnLs
r obusL
cO lllpen satl) r is desi g ned such thaL Lhe Lrans["er ["uncLi o n 1;lV l . ,,, fr om conceptual
Assullle L1wt when the' par:1lllelers of plnnt vary from
input to cl1nLrol variable has suitable margin
A. 13.
of gai n and phase.
U' I-6U'
e'
and
to 1\ -I "" 1\. 13 16 1]. C' -16.C" ancl
f)'
r e~pectively.
the sU ILes , co ntrol variab les
and measuremenL vnrinbles of Lhe and y'" to
pl~lI1t
chnl1ge from
:111'[ y' respecl.ively under the
Computer simulatio ns and experiment shows that
x"'
RI\1I\1 is effective fur pan1meler variaLions.
same n'fercnce input and the srlIlle zer o initial the
iL "
.1'.
IJ
states of the "<.InLI'lII sysLem. At this time, the state equation is described as fuliows.
IDENTIFICATION
i:
= A .x -/
/3 u I d I
(4)
'
y'· =C"·x-/-J)'·x /- «("
Ollr fuel injection pump is different from Lhe one
(5)
where el l = 6/\ -,r-l o.iJ ·u,
Rura oka and others (1989) used, buL h as the same
lG)
d? = o.C'x/-6/Yu.
sLructure. This pinnL was identified as same way. The transfer fUllction of the pump at 25"C is as follows.
(7)
ell and ,12 are defined as l.he equivalent disturbances corresponding to the p:uallleters vnr ia tions 6A. 6D,
/'
12:'1= ",'
o.C,6D*
S:I -I· 11I89·,'/+8 .J:'9x IO: 1·5-1 l .159x lO"
197 5' (S - I%)IS+SG!';j 15 + 17.0 7 H S -H5.9 :t.lGS.45)
(1)
The plant can be desl'l'ib e d in terms of transfer Similarly, transfer fUllctions at O"C and GO"C are
funcLion as
follows. )"=1'"y. u+['",·" .
__ 1117 2.<;2 _ ,1% 1 X 10 3 " - 7 :J GGx IOs I'
(0)=
,n
5'12(i:'7 ..<;2; Z.9ZIXI0'S-t-JO(i2X l o 5
l'"y, (Si (2)
C~
(9)
l C' ·( sI -Ai -1.D -1- D*]
1\ly,(S;=lC(sI-J\) - I , lJ .
- 1.'l .:':l.';2_ 4 .:'!iZX 10:1.,';-1.'19 1 x IOs
I'
(8)
wh ere rL=ld;l' . cl/V, ,
(l0) (11)
(fiO)= - -- - - - - - - - - - - - - - -
",.
s:I''!!i .97S2 + 8.452X 103. 5
+ 1.4 48x IU 5
Let left fractoriznti on
(3)
( T~ gn wa
1991) of chnracLerisLic
transfer funcLion 1'"" ns follows:
Fig.l shows g;)in property of plants.
1'"),, =I'dy,·N I(
(12)
OUTLINE OF RMlVl
Pd y ' is the lefL element and
NI1 IS the right element.
Pdy' satisfies realizotion condition and has no zeros.
In RM1VI, disturbances are defined to represent the
I'd J ' is also give n by lllulLiplying Pdy' by real valued
influence of parameter var iations. Let x denote the
matrix
state, u the control variable, y the controlled variable,
n fr om right side. Jlenc e, (1 = n·el. Therefore, d
is defined as fund8l11enUll e'lu ivnlent disturbance.
s> the measuremen t variable of plant withou t y and y* all output (equals to [yT, yT]T) .
l~ ubust
Equivalent disturbance
compensator
StrucLure of RMM syslem for SISO plant is shown in Fig.2. CI", is lhe left element of left fractorization of
186
[C r ", CoY·,,] , It is th e input to C'IIl' In the case of S1S0
h as zeros of I'l;lnt as eigenvalues. Th er efore, a noth e r
system, C'IU is equal to 1 1 D r.;.
m eth od is used to make the gain of 11 Wciy sma ll.
Model
Hi'vI:V1 FOE i\ON-l\1IN IM UM PIlASE r
PL ANT
y
In thi s p:lper. wc use foll owing sy mbo ls . N ominal plant; I'" = N" " 1 /)" " . 1'l anLwi l h I'ar:lm e tcr varied; /'=iV,, / IJI' . IVr\"= L ,y / D.
Moclc lt ra ns f"rr fun dion;
l'vIl/d e l m:l k hing co ntroll er;
,l _____ __ ___ _ ___ _ __ ___ ____ ___ ___ l,,
[ 1' //" .1'.'",,1::=1 -NI'" , iJ" "I . Hubus tlilLc r ; F =jf {J,,"
llobust Compensator
11: = 1" ,; 1" -1-
Wh e re N I'" . {J/ ,,, . !.f r ". M y" . D c and DF a r e po lYll ulIlinl s .
d from u and y ' and is a polynomial
matrix. Eq .(8) leads to
d =Pci.l' - I -it -Pci.l' -1·p"y' U , Th e refore,
r =[-Pci y -
I
Puy, Pciy - /J.
-!- I", .!-I + 1"11.
whcl'(' r; I i = 0, I , '" . k ) :1re fr ee p[lrnmcters .
Fig.2 Th e s tructu r e of RMM syste m l' estimates
[C , ,, . C mJ =[Mr ".
A1.'" 11 I / lJ('
( 13)
III Fi g .2. tran s fcr functi un is wrillcn a s
( 14 ) (1 8)
The ou tp ut of R is used as the inp u t to C'lII n o t
l Lle;ld s Lo
throughout r obust fi lter P, then, the tra n sfer functi on /{ Wdy fr om fundam ental equ iva lent dist ur bance to
U
11-'-
r/y
~
-
J
-
/ ) . /) I-'
If)
C
/i
V
· /1l)
IN I'll
( 19)
controlled variable o f th e system with r o bu st compensator added is as follows:
In th e case o f n on -minimum phase plnnt, we deLermined fre e param e ters of robust co mp ensato r so
( 15)
where MWJ y is t h e t ra nsfer function from
as to r eal ize r o bust s Leady-state pro pe r ty, robust
d
tt'ansicnt pt'ope rty , and robu s Lslability.
to y before we add rob ust compensato r.
l~o bll s t
If R mak es 11 W,iy zero, the re i s no inn uence of
parameter variations . This R is g ive n by Eq.(lG).
s tead y pro perty
We let 1J '1I be t h e chnracteri s lic po lyn omial of o ute r mode l in put to the co ntrol sy s tem, a nd apply the
(lG)
folluwing the orem (T[lgaw[I 1991). In Fig.2, if C",,·]? T is proper, r obus t compe n sa tor can realize strictly zeroing. When C",,·]?T is not proper,
Theorem
us ing robus t filter, we nH1k e C,",. F ·R ·r proper.
For the pLlJ1t with pnramete r variat ions, t h e ncce ss ary :1I1cl s llfri cienlly condit ions fo r r o bu st
After]? is given as Eq.(1G), Eq. (l5) lefl ds to
steacly -s Ln Le properLy (17)
( 1)
:HP
Lhe numeraLo r o f transfe r function /lWril' from ft lll(l:llll ental di s turbnnces to controlled
Therefore , by choosing a low-p ass filter, with a
vnriab lcs ha s DH! [IS fador,
suitable relative orde r , s u ch that its frequency
(2) robust s Lability is' sa ti sfi ed in rflnge so f
property is close to " 1 "in t h e frequen cy range from 0
specifi ed parameter var i atio n s o f the
to Wc, the II Wdy can be m ade close to ze ro wi thi n such
controll ed objec t.
frequency range the num erator of /{Wd y is (DDF-I-N /',,· R ). lIen ce, the This method is not avai lable for non -minimum phase
free paramett'r of robu st compensntor (Ire ch ose n so as
plant, this is because the contro l system a fLer RMM
to lTI fl k e (D·J)F -/- N/,,,·]?)
h ave 0," as f"clor, thus
robust sten dy-staLe property is ac hi eved.
187
(22)
Robust transient property we regard the plant with parameter variations as To achieve the robusl transient properly, in stead of
PIl + 6. Closed-loo p ~ystelll is described in Fig.3.
zcroing, we make the gain of IiWdy small based on
In Fig.3, /lW U '1I is wrillen in terms of the transfer
some evaluation. In this paper, after robust steady-
functi on of numinal plnnt, m odellllalching plnnt and
s lale properly is rea lized, we minimize lhe H2 norm
r obust compensatur.
of 11 Wd y using lhe rcmainin g free pa rameter. I
(~ 211
i~
" I
I
"
i 11' - i<:'l'lrOJ)i .1O)
(J
/) /'"
,11' • = - - (,1/
It
U
rly
2
I'
IJ · /) F
11
(23)
.+ 11 , /1)
. f) .\11
/.
"1
(20)
.
APPLICATION OF rUvlM '1'0 FUEL
I NJECTION l'UivlP
lnthe next equation ''le decide sampling time as 3 Illsec based o n ca lculating time uf nllllpute r . The discrete tran sfe r
(2 1)
funcLion of the of 2!i"C plnnL is ns follows. minimizes lhe 112 n orm of /(Wd y . /'
(:u,)
".\
llobusl stabili ly
'1.1)1111 X 10 .. 1.;//_ 1 1) ,1.1.,< 11 1- 1·1,+!i(;G7X 1 0 -~ = - - - - - - -- -- - - - - - - I /'-:11;:'(;'/' ·1· :1 JRU·I,-7 .213x IU -
In some cases, th e sys tem with this robust
'1.1111'<10-'·(1,_1177) ' (/'+ 1.8'1)
compensator is llns tnble fo r a plant with parameter
(1.
-I-
(24)
II%UHI, -11.8;'3 1:.1(178)
va rin t io ns.ln the se cnse, bccn use the purpose of HMM is to mnke the gnin of /( \Vd y small, we decide some free
In the same way, the plnllL ;lL WC and GO"C are given
parnmelers so ns to achieve robll st stability and such
as follows.
t h at the frequ ency property of /lWd)' with 1I2 norm
_ 1 98" X 111 - ' . I,~ -I 7r, ,1Gx 10 - ~.I,
minimized doe s not chn n ge largely.
I'
(11)=
-I
I .9lH X I U - 2
.<;:1_Z ,!:I:1-i' ,I.. 1 8K:JI,-'i.50 GX 10 - 1
fly
(25) -:'.'181 /)
(GO)=
x IO - '·1.~+G:,
IO - 2 1. - I .
2
1:" - 2 G821,'+21:IG·I,-7498XJ()-1
Ily
(26)
We ch oose Illodel transfer function such that se llling tim e is within 0.1 second. :1.I;:,xln -· :I ·II, ·, 2 . 12)(1, IO.IG)(1 -077)(I, + ) 8'i)
IV ')'
(I, -117 ,) /(1, - a.7ol(1, - U.8'li
rOIllJ)(·" .o:;nlo r ~
- - - - - - --- - - - - -- - -- --- - - - - - - ------ - - - - - --- -
--
(27) The model I1HlLching controller is deLermined ns Eq.(28) and Eq.(2f)) . -9 . 12!i X I O-'·I,'-2081 x IO - ll,-l.onx IIl - '
c:, "
i:: 2 _
0.:, ,I I 4';: -
O. I 7 G2
(28) G;,r;GI,~ - I I :)2.1, , 4 fl88 1,2 _ fI .!i 4 I'll, - O.17GZ
(29)
RobusL compensaLor is gi ve n by, I'll'/=
-4.000XIO- 2·Z'+1.044X10 - I ·Z -5.GG7XIO - 2 ,
(30)
ryq = ZI- 2.G56·Z 2 + 2.380·Z + 7.213 X 10 - 1, (31) Fig.3 Closed loop system
R=r2Z2+r/ ·Z+ ro,
[) =
Parameter chnnges are equivalent to additional
F
variation 6,
188
J
(I,_OG8):l
.
(32) (33)
(2) minimi7.e o r make small the Il2 norm of Trans[er [unction /lIVely is s uch that robust steady-
uIVJy , (3) increa se the stability margin of HIV u '".
state property is ach ie ved [or s tep reference inpuL. '-0 is described as a funcLion of I"] and 1'2, '-0=-1'2-1'/+2.703.
J\CI{NOWLI~DGEMEN TS
(34)
f(II'Jy is Cri kui::Ited rlS fol lows.
'I'he a uth ors exp r ess grntitud e t o H.Kura okil and
11 IV cl\" = - G.G88r/ -21.90·'-/·'-2+ 144.1·,-/
-2U)()·'-/+327.3·'-2-2529
N.Ohba, and thilnk Il&D De pt., N ipjlo nclenso Co., (3G)
Ltd., for suppo r ting this experiment.
Next, '-2 which minimizes Eq.(3G), is calculated as llEFEI{ENCES
foll ows . '-2= - O.!)·r/ 1'/
(36)
+7 .'171.
E isaka,T. and others. (1989). Evaluation of rrubu s t Mode l-MaLching for the Control oC a DC Servo Motur.
which minimiz es Eq.(3G), is calculated·as follows.
Int ernatio/tal Jou/"Iwl o("Colllro/. GO,479-493.
,-/ = -86 .S3 .
Kuraoka,Il. "nd ot hers. (1989). J\pplica Lion of 1[ ' " J\t this time, ro=3iUiG , '-2=GO.H9 (C :1 se 1). In this case, in Fig.4 , Nyl{uist
opti mal design 1.0 ilutcJtllOtive fuclcont rol. Proc. ACe,
plot of -c.(Z)·IIIV,, "II(Z) is
1%7 -19G2.
s hvwn. IL s hows that Lllis system becomes unstable
'I'a gawa, rr. (198G).
whpn parrlm ete rs \'aried. Th eref"re, wc set 1'/ = 0 so
Control. Computer and J\pplicat i on's Mook.
il'lod el i\latching and Robus t
t h at the Illrlrg in of sta bility is increasing. Under this
COlI!jJl.Ltrol, 13, G3 -G8, Corona, (in .Jrlpanese).
conditiun, set r () = -4. 76:l to achieved robust steady-
'l':1gaw:1,lC ( 1991). Dual Model Matching. Tu nppenr
state property (lnd 1"2=7.471 so as to minimize
[J2
at I s l1fAC symposium on Design Methods ofCont rvl
norm o[ /! IV ,iy(C,lse 2) .
System.
III these two cases, Fig.G shows the frequency prope rty
Model Matching Desig-n lVlethod Which Ensures
of IIW,iy(Z), rind Yig.G shows luc us of vecto r. Fig.7
Stabi li ty. Elec/ronics und COII IIllI.Lf!ica/i olls ill
s hows Nyq uist plot of - c.(Z)·u IVU"II(!.:) in Case 2 .
.III.l'IIN, l'arl III Fl.Lnciwncll/al Electronic Science, 72,
Fig.7 shows that this system is robust stable for the
%-104, (SC [U P'l'J\
Zhong,Y.
'°
plant which par,llllctcr vnr ied. Step re spo n ses nre
:1~
s hown in Fig.S. 1'~XPElniV1ENT I{I~SU LTS
4
2
'l'.I~ i saltil, 'l'.
'
=: t
("I)n tr o l of fuel in ject ion pump, simu lati on and
INC.).
. --.~- ----'-j
""'" \~' 1 \, i c'
,
\
/' \
./
:
11
1
""""'",------------------------///J'J
_'oL ____ ._,.______ ,_ __ ~ ___ _
ex periment t·e s ults s how that it is effedive whe n the
-10
-~
0
!!5
pbnt has pilnltn ete r \':1riations. Fig.4 Nyquist plot in case 1 (l)atO"C (2)ilt6()"C
Sti ll, it is not clea t· th;l t thc order and pole ass ignmen t of r obust compensator are m ost su itable. In this paper , to get r obust stilb ility, we set the free parameter dra wing the i\yquist pl ot of I! IV U"II a nd the frequency property of I! \\' d y . But, RMM is avo ilable for discreted-d:tta system and non-minimum phase plant, we ob tain r ob ust control ~yste m
j
///-----------------~---""""',
r /' r (
_4 ~ r{ubust Model i\]a tch in g is rlpplied to sp ill position
'['lII~C ll NICJ\,
-----.,------
J _] (2\
CONCLlJSlON
and Tagawa,lC (1989). llobu s t
by determining suitably the free parameters
of the robu st compe nsator so as to (1) satisfy the n ecessary and efficiently
condition for realizing robust steady -state property,
189
10
~~~C'
)
:~"~ 200
1 I
r----
I
o~
1-
200
~""'~:' )
:
t -· o~ ~
\
"
_o.L
'( 2 )
=::: ~r~_4-- '- ~ 10- 3
....... UH .. _ _
10-:11
~
_ . .
. ...... .
L.~'.
_---t- -" . . ........ t! • . _
10 -'
•
.
lOO
-,,_,-,-,-J
o
10'
,,. .. "' ..... "0>' (rod )
Fig.S frequency properly uf 11 Well' in Lh e (Gse 1 (2) ill Lhe ca se 2
1 I
lOO ..
I
Fig.G veclor locus of 11 W"',, in Lhe case 1 (2) in Lhe case 2
Fi g .7 Nyqu isL plot in cuse 2 (l)aLO"C
( • • 0)
Fig.S Step responses (1) al25"C (2) at onc (3) at 60 n C
---'-'
(1)
0. 4
t In"> .
(1) '20
._._ _ ~_____,.__ ~J
(2)at60"C
190
O .D