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May 1985'
LiDS-P-1461
Tr-e
a WI
S
ANT.E£SZT WI:,=UP STZ:AT=C=S.
ATMATI-IctACTUATS
Petr-os Kapasouris and d.ticael Athans
Lboratory for Laformation and Dec-sion Systets
HassacnuseUts Institute of TecL--olo-y
Cambridge, .4assachuse'tS 02129
ex--ended i space then a relaation F is called
stable in tne extended 'L space if there exists
CT
ASSTXA
This paper contains preliminar-
yet proasi
q,
results for intoduc ing antiresec.windup MAW) p=per-ies in mulivariable feedback conr-ol -XYt
syses
wi-t multiple saturating actuator nonlinearit.es
and integrating ac.ion.
ie'ARW method introduces
simple nonlinear feedback around the integrators.
The multiloop cir-cle
criterion
is
Ca
I
I lrx '
;
vx:
where
used to derive
sufficient condit.ons for closed-Loop stability
that employ f-equency-domain singular value tests.
The imorovement in transient response due to the
ARW feedback is de.ons=.ated using a 2-input
2-output control system based upon the F0404 jet
encine dynaaics.
'1
such
Int~duc'0n
Reset-windups (11, [21 appear in feedbac
contr-ol
wthen a linear compensator with integrators is used
in a closed-Loop systrm'in the presence of 'sataing ac-uators.
When the ac--uaors are saturated
-.he error is continuously integrated:
this can leSd
to large overshoots on t=he response of the system.
The antireset windup (ARW) idea is to use a linear
compensator with a nonlinear feedback Loop wnich
te
-:=u-.s-Off' =%e integrating ac'-onwnenever
tsaturataon
-s.
Thenocrthe Large overs.oos
saturation oc
Shen the large overshoots ean
Ts.
be reduced and the performance of the system willr
.ogrove (small overshoot, smaler set -ing t-les).
ARW tecihniques have been used extensively for
single-input Single-output (SISO) systems ([1,1.
The SISO ARW techniques are based on engineering
intuition and .on extensive simulations. 'Litle
mathematical analysis has been done even in the
e
S5SO case and we are not aware of any ARw st=ate-.s
for .tTMO systems. In this paper, we shall use the
mzUtLL'variable version of the ci-cle cie-rion (see
Safonov and Athans [31)'to develop suf.'cien: candi-.ions for the global stability of .M--0 systems
wi-t saturation toqe=ther wit. ML=0 met.ods for ARW
strategies. The theoretical examples are illustrated by a sLmple 2-input 2-output LC/LTR
based design (41,(51 using a model of -e CZ '-404
2)
t
(
L
The nonli.neariz-es
3)
used in
t
used La this
paper are single
input single-output and
-invariant.
We will
say that a S3SO mapping 'f' is stric-.ly ins.de a
conic sec--or wit% radius r and center c if
f
2
f
for sxo
-cx
or so(4t:Ii. St-_%ement
of the ?.-oblem a4
?=oposed Solution
i i- Fi gure I shows a general M=MO closed loop system
with several S:SO satu=ratig aoni!nea.ities at the
inpun of t=e plan:. Except for the sat=a ing nion -Le sa
e n ---ies the system _s inear and -4ie- .va=
ant.
A reasonable assumption is that the SISO actuator
nonlinearities are memoryless (static) with upper
and lower bounds e and U' respectively. The
input/output cha-ac-eriz ation of the nonl.ineari-y
f is given by
H
u
-uS
-
u (t) U<u.;(t)<U(
_
L
. (
wit= itL,:.
jet engine.
p,
wnere for p--ontrol inputs
u (:;
22.
Mat=ematical PL-e iAa=r;.es
Assuming that all the signals belong in '-e
'Research supported by NASA Ames and Langley
Research Centers under grant NASA/NAG 2-297 and by
the General Electric Co.
u
U
1
'Li
(6)
AL
p
_
P
P
(~ ( f i (
-H((As|>
'
s
~
~ ~~ ~ ~
G(s-
~
u . (t )>
~ ~ ~~
~
H
U
I.(t)<U
L
wnere
and UL are defined in
u (t),
L: Closed 'oop
and
U?'(tr
j
.- #
Figure
(6)
(8)
u'(t)
.12C syste
For a class o =refe:ence signcals r(s) and initi.al
conditions of G(s), the co=aided conr-ol signal
L;mits defined 'Ji
u(t) can exceed t;e salraron
.hus. ',
e' error .siqn.als axe generated
and
The eLements of
and are intecrated con:inuously.
the contr-oL vecor u'(t does not necessarily c-.ange
sign wnenever those of the e__or vector e(t) do,
because the integrator states can have La-ge values
at that tie.
Trherefore one or aore elements of
t-e e--ro vec-or e(t} may no Sbe nu led-out for a
substantiaL time before the cor=ec component of
and
changes sign, and th.s can lead to overshoots aV
times in the trasLent respones.
large settling
Because the resec-windup problem is caused by
t'e
the integrators it makes sense to 'tus=n-off
Figure 2
integrators whenever saturation occurs.
shows a proposed nonLinear feedback system on the
compensator K(s) and the integrators that will
deactivate the integrators when such sacuratr on
occurs.
From figure 2 it can be easily seer. Lat when u(t)
imis ts e following is
exceeds theSat.ration
-
U(s) - s[
'K(s)e(s)
'.'ectiveLy the nonlinear feedback system replaces
he design
the
tegraors wi
a lag network.
onsr ate odf he
parameter o dewterines the ti
of '
closed-Loo
the
eS sai;
plarad
ita
shows the s..abiliy analysis
Sec-ion
system.
and how C can be decermined.
v.
Anavysis
Sraei:t-!
The closed-Loop system in Figure 2 can be rest-uc:=red :o an ecuival.n one, as shown in
Note the sepaati on of the Linea= 'me
Figure 3.
pa._ ':Cs) and the memoryless noninvariant (L)
Lnva-=ian pa.t- F.
linca. t=e
r
.-
aI _9q,,
-!
r
l
l
!
(
Fi:ure Z:
es.
CLosed-Loop system wi-: ARW prope-rt
- The input/output c.a.ractari=atzon o' the
'
l
non-
Linearity, wnich consists of SISO 'dead-:one'
type elements, is jiven by
'-- 'ure
(9)
t(s) -ets)
-
-_
3:
Me closed
pope:--res.
loop sys=em wit- ARW
I
It is impormant to point out that since C(s), and
/s are Linear systems t=ie r(t) can enter at the
u(t) signal as shown in 'igure 4 wviLout tcArngn
euie stability of the closed Lcop system
r-.
r--7--~---F
_
-Lr
SuCh modification 'is i por! az L ord.r to' apply
The mathe- athe multiloop circle criterion [3].
tical equivalnce of Fiqure 4 and ioure 3 can
on or by
be easily seen by block diagram nanipa
explicitly calculatinq the y(s)/U(s) =ansfe:
functions for both configurae.on f'-on Fires
3
or
()
3.
system equivalent to that of Fg.
Fiure 4:
_
Ct)i
tt,
-4
4.
~'-
,-L-e
u (t)
(1
(u(t))
-
(u'
u (t -,
%
S
~-'~~~~~~2. ~-
-
ts
,,
3)'
'
(t)
_al e
_u= ei
i.
conic setor
UnThe
2
(15)
Then the f,
<
UL
and f
Yt;
oC U.<:
Yi
(16)
noninearities.
can be
of the f. nonlinearit'es,
and a radus
C
: _L ltf
: wohee
L
C>O; C-10
-^
-
c
.he u(t signals (unsaturated controls) can ae
consie:lred bounded (the bounds can be very asrge)
of the f
iouF e 5b: Conic sec-r
defined by a center C
i u(t,
(-):Y
(17
iU.
WnTch imolies - at
L
(
U-
nonlineari:_es are strictly
inside some conic sector as snown in Figures SA
and Sa.
--------------
C. (
i
l--~-c-
2C
./"-
(20)
~
-1--------~-------`---
zn the zulticannot be Meet quaranting stabi!ity.
loor circle c:itoro
xn for awO the AwR Cirit
is
noe used and the system is uneffected by th:e nonARW feedback.
where
-
1
ULH
.
|lU'
U1
.
s
U-.Linear
U'^analysis
UXM
_ P__p.sufficien.:
represents the high sa:turati:on ';-;
The U.
each input c.annel and i: is
a physica.L cs
y a
will be parame-ers in t-e sa'i
The U
linearities can be defined by a center Ci
r2
radius r. as
E~>;
ncessy boud for U (t)
sjsli:zo
ynoa
and they represent t=e worst bound on the .sa.r='I will be defined in the
ated signals U. (t.,
stabi
:y test' to guarantee closed loop s-'y-Similarly a conic sector of eac: of the ' non-
:2x'tt)
- S: '- C/:
i
'
A very imoranrn f'naal point on the sa.ility
Is tha: the s-abiil;i:y test is oniv a
czndi on whch means tat
if = .e c nd.nons
axe saJsfied
h
-en
- he closed loop syse m
s
e.
''
e
s uane
o
Lare not saisfiec
:~en 'the closed loop system
may
4
or may not e closed loop stable, or the U is
V.
.
A linearined
_
odel of t-e GE T404 jez engine
has
been c.sen o = .espord.c to rated -th=us con.ditin
a
ude.
e example is used here
purpose onLy.
And the sta:e-space
representation of the scaled F-404 model is given
by
a
'
35,000 ft
:for
academic
-
(23)
-0
..
32
.175
IL
define
-:a:i
d
]24)
C
l
_-~
1^r~2
°
Q
C-
0
CIC'O
C
e
(t)
3
(30)
(20)
_
vector
r
0C
,. RIOr
2. ~ 0 L
components
ul(t):
scaled fuel
u
scaled 3OZne area with limits
-3ui2
<
(t)<-3
<+3
(tI:
226)
~-2<u
Z
-t
are
-3-<u
(t) <
flow with limits
The conic sectors than
The closed loop system has to be stable of the
nonlinearities are replaced by their centers
of their sectors C.
a')
2)
The following has to be t-ue for all
cuence.es
(C
.027
-L.24
rJL
1)
where
v'(t)L -
.708
.472
O
--e ulti'oop circle criterion [31 can now be
applied and sufficient conditions for stabi-ity
are as follows:
T(jL,
!R
1
x(t)
_-.
-he components of the state
vector X (t) are scaled
and represent
low pressure speed, high pressure
The control
speed, and thermocouple dynamics.
Stabilit4y ResuLt
aax
0
0
-2.23
-.39
0
CCG(s)- C(sZ-A)
125)
0[a , _2
_0
(r2
Ca
-r 0
Nv.eriCcal Exa:mple
T(ja)
2. I
T(s) has been defined by equation
(7
(1).
The only swo parameters are a and U
where
Notice tCan
is the feedback Loop gain parameter.
different a's can be used for different direc-ions
on the feedback Loop.
f for a-. the Ue given by the sa-biliry test is
not satisfactory, the linear compensator desian
has to cnanqe or the specificai-ons of the proolem-
the input nonlinear
ies
are strictly inside are given in fiqure 6. The
conic sectors that the autowindup feedback nonlinearities
are st=ic-tiy inside axe the same as
with
always to thce ones given in Secti4on ZV.
-le given definition of the conic sectors the C
and R matrices are
fe-
'v
Level
:-t
(d/)
/2 (1+E-3 (Ut)
0
using the :LG/LT.-R met=odology
integrator
-
O
[4,51 i2cluding an
zcNssca
-X
'h
H
-aumter
roop
on the
over ftequency region was set beween aspproxiately
values of
..
e :esuI.tan singlax4 and 1O rad/sec,
are shown in Figure 7.
the loop trans=e= 3ar.-x
was designed t e folovin
After the com.ensaor
ma:-,-.x was forme
.C(s(s)
F
Z
I/
:
(rc-/sec)
log W
-
32)
s
.gCs)G(s)
- (Z
I
a
S
Figcre
7
'
: SinguJa-a values of the G(s)x(s)
S
L
IIuE
S
(/3
V
-;f
U
(')
/Sec)
elog
of
Fi ure 6a: Conic secor for the sat30nio
...
fue.l inpu of -. e .F;C4
Four.e 8:
RecaU tha
,
Si:
'.
vaue s
ity
-es:
(eq.3).
for closed loop s-ability
the Zes
was
c-
t _jw)
E]C
aC_ _R
a
'
~I S
Zt was found tZ.a: for a-L.
C/2-.000
(2 )
VW
1j
)]
5
and *- - Z tre
was sazisfied for al' ftequencies
(;')
ll:y
3. S.nce the eznpie LS only
as shown in r!-;qe
for
.oe naqniL;de bound uo
for academ.c&: pnSurses
.equa
u,U ()
IL'
.<-e~~
2
._ST~
the u
S'e
-
signals w-:.'.
_o~tLC
-5:O:
be
Su=_
t:en. Ze CompersazJr .Cs) :nus:
L .Te
r.ciure
~
su fic.en..
ne no_ even for CL-0
be redesined.
was to replace the
second Zest for s-oilI'
so fuor
heiu centees C.
es uWI
nonlnea:
h
l
6b: Conic sec or for ine sa._a- on of y
.
noz=le inepu of he F404.Z3q
~
~
~
~
~~~_~~~~~~~~~~~~_
..si.de:e-
U
~
-
~ller:::e
¢e-.ar--s
w-~:.~:.%ei.-----
C,
SO
.~ori----
j, ·
.
rr
FI
!:
C '-
.{.24
S5 J
relac-ng :e no~nlneari- ies vi-th C -e
system was f'nd to be stable.
Since stai';-.ty is
No.
r
2. Apr
*--·,¥~'
*
-"--'-o
e
-
1.81.
L9
,. Dovye and C. Stein. "'.ultivariable Feed1([41..
Conceots for a Classicai/modern
back desian:
'
'=.-- .Trans. on Auto. Control,
s<;vnthes-s.
_.c_. _~.o.
C5
. Ste;n and .'.Atans, The LQG/LtR procedure
f'osrul.va.able feedback con-tol. design,"
(2;5
}
'
· ·. 1
.5
_0
:=-=J.
'
Vol. AC-26,
'fS-P-1384. , .XZ, May 1984.
4loc
closed
guaranteed a: least for
A si=ulation was perfor-ed with in
5- .
U
referenrce signal r-(t)
condi:.ons x(O), cons n.t
d-)
(R
e
and conszanc output di(turnaw-
ut
aL
A
-.5
-[-3j] _
[w]
1
[ , ] 2viithout AWR
wit-h ARW
/
(3S)
Figure 9 shows the time responses of the closed
One
loop system wi-t and without ARW prooeries.
can eadily see the porfor=ance imorovement of the
system with tne ARW feedback.
2.
n
.
o.
,e3 (sec)
Figure 10 shows the t:ansient response of the
sat=uated cont--o input u (t). Notice that the
t sces ac-tion
system with the ARW properties
faster than the original closed loop systes.
V.
-icure 9: The output for both the orqi-g.al system
and the one wi-th AWR feedback. The top figure shows
scaled low pressure speed transient and the bot-om
Conclusions
fic.ue shows scaled temperature transient.
his paper we have presented a promising
In -tec.hnicue for introducing ant.reset windup (ARW)
proper-tes in a Linear TflO cont-ol system with
integrators which are followed by saturating non-.
lineari-tes. The design of the ARW feedback loops
must be done i such a way so as to guaran.ee t he
stability of the aoniLnear con=tol system. We
have explored the use of the aul-ilo-o circle to
guide us in the ARW design.
Ite results in this paper as proising
but cer.ainly not complete. Much more research is
needed to a=rive at a generic design methodology
to introduce ARW properties in a MIMO control system with multipLe saturation nonlinearities. in
particular, we plan to exa-ine the degree of conservatis=m nduced by the sufficiency conditions
inherent in -teuse of the multiloop circle critenion. and means by wnich the conservatism can
be decreased.
--
Z.3
/
...
[21
[31
References
l;acrf sder and W.ScaufeLberger, 'SaSility analysis o.f single loop contoL. sys tems w--n satu=raton and ant'reset-wLncup
Trans. on Auto. Control,
clrcui-s,' ;__
Decemoer l983.
Vol. AC-28, No. il.
R. Hanns, "A new tecnni.ue for preventing
Conf. on Auto.
wvndup nuisances," Pr-oc. :':?
for Sae-tv in Shiooinc and Of-snore Petrol.
Ooerat::ons, 1980, pp. 221-24.
M. Sagonov and M. Athans, 'A .ult:iloop gene=LLi-ation of the circle criterlon for stabili-y
A.H.
,
.
.
a.
We view
[l]
thOut AR
ime (sec)
.
2.
I
Ith AWR
wtthte
AWR
,
0.
2.
'e(sec
Ficure .0:Saturated control inputs
or g;-.l closed Loop system and the
back. The top figure shows the fuel
and the botcCm one shows the non=le
for both the
one w:th feedflow response
area resoonse.