-STABLE HYBRID ADAPTIVE CONTROL(U) YALE UNIV NEW HAVEN i/iCT CENTER FOR SYSTEMS SCIENCE K S NARENDRA ET AL.JUL 82 8286 Ne@04-76-C-8e7
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KOPU1 S. Ratmoot s To ZRy . Ekast
'1illlmN 2 aw a Wik~ hbft *dt~ catv. of .4g*1pu Ss.
S4 b .U 9% " ~I t Is sbou UM* .*ftttw. 4a~~wd s 4fte
oou A io t etvtys'Wa oh mst~.% sv
0"sA l -to e I 1-to doh "~*to~ -tt do1Wft f oto
7w 1aS 7
tft* O~fwf't 001 o~
a. ia~~uta~~u tuft ~ aftell ft n t" :U )
goaf r_ UU~~~~s o Stm*W o, tale ktm a
A* W eua f satot af
$"w atittu "nw
~~0"o *fl. UO*f CItt ad ttqOVli U1f) ue atteqeed to sha tat ler
weltn &eW diac eatat esVtaw by dtecws*a the differmeta equatioc
n Strt an *16ci the Gcatrel parmetrs roat earnstat. Ibis procedure
to mfla*4a af .tIably Vf*1* dpeed on the rat at Witleb permters
40 Znmd.T tb" "Pet a Al hybrId ~Iva alonitls for adjusting
dftt 1*a *zm** Vttr itee W0Vfishdc ms te "#oa stoblily
W I" S-#flflh *"to twaftedat Of '01 fteqinqt with *Leh the cstret
eeru10to a otnte.
PeT a*= eetd oas*i. attmt of a0 dWaflsblttyp of hybrid control the
madft I* sontro to tio pqsw by Ellott (fl8).tebiddaivatl
Vgbba 16 dtrnd t S atnf a atd ttftnmfv teat m "AW s dIn Uazadra
t ~ ~ s &a wawx a . Infwnsu4 *dow4 at AoStt
ON~ Xy. obwl1 vtls of1 a pqnf a
ft asrsb:Aun) 34 Ef* Ik agw 3. 40ot atb Is al""ee to
Th~~~t,*'E Af namwki. as 1a04 to I& z
ft~at~v"" 09W *"Sa mm"rd f ad of the pouer
t*6 ww1 "WD"mn l
£ Oaflsm tf seJm P tofU fltS Is OWlety 1srme by the
* opn t t4yl( t)4) am *a b6 Fmldby a tfliawartast Oeta
Ob~at a g (W eerie, ad d b, ane a-"wtn. Ibs tramefer fustia or
~aa N 44~c 6"00
" 7, "1
I IP
40Pa tt? Ito fortbet
(L do 404*#pla
'm w .4* ; '01 .* ,o u
#A 4me to"" "aatr
,~ TOO,
Av + -biA y
pC)
*Ono c 1 1) (2) +ui~Wnc* %'r* ~,"p v :1. la adAI
~ (W~v C',?(t)v~~(tI; QO 1%(tA(t)9 %t)d (t)1 (7)
"mlb ceutmi "-tt th PlfMtt& Of'Ps s
AW (Srt WVtf ffx oIU*m.~
Afes4o"04* ->040000 tvt.*,msnata .n OU tat aM
ciwafe~ pumbi M~) to~w t evfl
isa at Met.
t"4t a OWN Ooft *a d te.16 am do I.nt s0 qe a
" - SS~m*ptkt lass ef itS OW4 $a sfffge amble to abuao"
~~.esa ~ U 91qi lamin t nt in y iitit* oSeefd emc
(toat*itt 7 t±1 bSU SfflbSlks teasrallepsaaw1tflbe~ll b ydd
Il"t
5
As is the comtanuous Case cistral to the stability analysis of the hybrid
~IVt* COnRol PO*IMare the sur Models. if A7h0. the error models relate
the emattivitty Vegto" 9, the aemtrmr vector ;and the- output error i
IQ 86900416 and l&W419R2 )eral erro modl, hae been analyzed and stable
bybr alsorithes hae bees derived. One o.f those algorithm Is awaysed Insoam
detail In the followIdg section. At In used Is Seetisn 4 to adjust the control.
pemter vector I and, the glaol vtability of the resulting Sstem Is established.
The other alsprto In ftreatra and th-lifa(962cam also be used An a similar
inrto destgn stable. adaptive Stem.
The armo Ao*l demoxibed, Is this seetis, Is. a ismlwa.e tim syste In whick
te, the set of pe.At~v reee. ~f4a sam ei :4 an. pedoaime comm-
tims fumnei'saf am d wl be, mgee 0.4 the Ismad awps twhosms
raesiwyOf the, '26W ft1g esemda ~~b e sm e
fined ineationf (7) theS ut1Pn esm e ttm SOdie m I pawhlont de
In Section 4
Aet {tjl be an IebwSw in Samly Imsweet Seqeesse got
*?,ISTIS TIMn.IO 1m -eretIt1 omt. 6the do llSSiMs seetions Itj Vill
be referre to as the "Nopll" #:IteA is a piecomise ot~ant
function, amd Sa.MO 4616 wiwe
there Is a constant vect~or. -rhe error model of interest ft this paper Is thew
deseibe bythe equatiou
-
#k w(t) slt1 t) teltk'tk~l) (10)
key
6
It Iseasmad that ;0 10 Unknown while al (t) and W(t) can be aured for
al e e. The objective in to detoruis. the adaptive law for choosing the
Sequms hereA7#so that Ila *I(t M 0.
commide~r the Lyspumov function candidate.
I Ts
aV(k) A (h4.1) - V(k) - [i;k+AylJ&;) (12)
A~ kl ; (13)
(9bseefs the adaptive Um as
A~k*-k J c()I) dT (14)
"k
AY(k) f t j 121 - Le'kI1 k(5
t k+1
% , . IQ ( __ _ (_ )k~h~i tk J lT?~T r(6
elm 2k~b91 Is a positive ami-defisite matrix with eigeavalues loe than unity,
7,4. t :-w.~ . . .* ~ *- * * - *
7
a&@ mot (1 > ' fat for e contant 0:11.
at J S() is LyeOMv fuatia Md sequi then boundeduess of #k~I If
111.1 itbfded. rap (17.) it faos at
(17b)
I ~ - 0
VIT4Ut. 4j@ (16
*.*SP
almf I If :Wa Iaaholy bo"u$ Is .Ov) It+foljSstat al
alt *WA It 542at m (14) chat *I. 0(t) a. Iwose for
6.AS"m wth £ bsMM dbietnt,, 141t teas to
foi"uk-S- a h.
& t if iN OWit tO bate umiosnly beaded, 8Is
"attt~t1~fis" (gmaad Iseem. 1977) over mny
Sm*9*4ft U5 1mp T a s that gh+is positive definite
ike431hiS Afk)4 sd bairn. the painter error vector
ft - h.*
", ' T-IT
z'7
9
adaptive law can be modified to
- r k+ (~ TA7#k u 1 1 ()r (T) dT (20)tdk
where r is a positive definite diagonal matrix with
elpvalues within the unit circle.
4. GWOBAL STABILITY OF T HYBITRID ADAPTIVE SYSTUI
The proof of stability of the hybrid adaptive control problem follows generally
along the same lines as those for the continuous case discussed in Narendra et al
(1980). Bowevr, since the adjustment of the parameters is done at discrete
instants, j(t) is discontinuous at these instants and hence the arguments have
to be suitably modified. Only those features which are pertinent to the hybrid
control problem are discussed here and the reader is referred to the earlier paper
(Harendra et al, 1980) for some of the details. The basic mathematical concepts
as wall as Lee 1 essential for the proof of stability, are developed briefly
in the appendix.
For ease of exposition the stabilicy problem is discussed in two stages.
In the fLr case the high frequency gain K of the plant is assumed to bep
know while the more general problem with K unknown is discussed in Case (Ii).p
a) Case I 0
With no lose of generality we assume that K - 1 so that
c - 1 and only (2n-l) parameters have to be adjusted. Defining
IT(t)[c ,OT(t)], wT(t)-(r(t),wT(t)I and *4(t)-[o,0T (t)I
the input and the output of the plant can be expressed as
10II
u(t) - r(t) + *T (t)W(t)
yp(t) - w,(s)[r(t) + ,T(t)w(t)] (21)
As in the continuous case an augmented error 1(t) is generated by
adding an auxiliary signal 7a to the plant output, where
Ya(t) a [8T(t)W (e)I _ W (s)OT(t)]w(t). (22)
The augmented error is then given by
'1(t) e e(t) + Ya(t)
T (t) C(t) (23)
where W (s)w(t) - t). The adaptive law for adjusting #(t) is
generated using il(t) and C(t) in equation (23) but it remains to
be shown that equation (21) will be globally stable with such an
adaptive law.
In the hybrid control problem e(t)-ek and 4(t)-#i over the
nterval [tk,tk+l), keN where 8 k and *k are constant vectors.
Hence equation (23) can be expressed in the form of the error model
described in Section 3 as:
0 T C(t) - ;1 (t) t[tk tk+ 1 )
(24)keN
The corresponding adaptive law is given by (14) as
k 1 tk+ 1 (T)(T) d T (25)
kk 1 1 +tk
Again, to avoid obscuring the principal results the adaptive gainmtrix r is not included here.
*4 .. . . . . . . . , . .". ' ,,,".. ... , .:.. '. ,,. :. ,.:,. ,i, ,,- ,,- . ".,.
.:6 .7 41* -.- - ss.j.-
11
It follows from the discussions in Section 3 that #k will
be bounded and hence the plant output yp as well as the state
variables of the entire system can grow at soot exponentially.
Again from Section 3 me have
J;l(t)J - r u pT 11M11 (6
or the augmented error Ili(t) will grow more slowly tban the
norm of the vector ;(t).
b) Case ii (K Unknown)
The error equations n this case appear, at first sight, to
be considerably nore involved than n case (I). However they
can be reduced to the form of the first error model by a change
of variables. The input to the plant is
u(t) = T(t);(t)
The main difficulty arises since Kp II !.1 in general, resulting
n a plant output which can be described by
yp(t) -W()r(t) + -(tt)(27)
and an error equation
0l(t ) = W M;(s) ;Tf(00(t ) .(28)K T
In generating the augmented error, an additional parameter
#1(t) has to be used making a total of (2n+l) adjustable
parameters. Defining the augmented error once again as
il(t) - el(t) + ya(t)
1.2
ya(t) *I W 91 (t)S)ITt] (29)
weobtain
.1 (t)W m)t + #(tct (30)
xI#1 (W) + #: ad
".W mmwt(.z - V.)S(t k(o (330
Def ining
(32)
,The n ted erMr GquaBtio my Ie serem s
i1 (t 7(t)t(t) (33)
which again corresponds to the mro Mdel descrIbed In Section 3.
In the hybrid control problem the (20t41) shinto of the control
vector 0 and hence of the parinemor vector are adjusted at
discrete instants of time and the adaptive law can be epressed as
Al -ATk k+ dT (34)k~k TkJ 1+ (r(r)
From the discussions in Section 3 It follows once aasin that ;k
Is bounded and hence the plant output as well "s all variables of
the system grow at most exponentially. From the results of
Section 3 this Implies that
lii(t)u'o [stujI Z(@)j or equivalently o 4sp TI()llsince kI bounded. (35)
. . ...
13
C) Prof f lobl !tablity
bsm ms reslts of the analysis carried out so far, which are
cetral to the proof of Slobel stability, my be summried as follows:
(1) the hybrid adaptive equations assure the boundedness of
all the parepeters s that the signals In the system can
armw at Mt exponentially.,
(11) the aegmented error 114 (t) Ioan Pow only at a rate slower
than that of the siativIty functions (equations (26) and
(35). UslaS these results It tosebos, In this section
that the output of' the plat as well as all the relevant
signals of the ~AVe systm winl remain bounde" for all
tal.
Let the plant output y l, and gow In an unbounded
*fashio. Big" te Paam ter er oetr 1.bouaded,
YVcan Po at meet ftsocetially (NNNatd Ot al. 1O) .
The output eivr st (t) 1.gives, by equete (28) as
k"N
Since I()(t) 1(t) and by (17b) Ak *5 a k-0-0, w
have by LeMs I in the appendi:
- ws)~;(t) IIosulcI)] (36)
or
a (t 41t + 0. 1 P()I]1(7
wow."uoas (a) a" (3)igo km
afmi 1II~)II~~ 5(c) j I. tm tb otput of ais model
Wow fs nmflbmm (loo (M) &t(36) -bas
tun)and (30) St SOUsi VWn Pit) am R(t) i.
0kso* -bom aSI Na -f doOSaL
*am5 4"*sAU vsft *
at 40U*, ,* ,)f. t uft, a."
ta e"uuls *t d. afnt *""A as Uw^t by ."nism
(* ft (5 topt bs an enst fejtrfa.ItI u
suahmao thewm ty()to.
(i)ai tms arn ot i "Oabst inrsla~ of a diessuu gaa
mria "* ft9P tit Sqefle is go afdent the
=a"'~t of a"a sactin.
~~~AeS4~~~-.m QWRe~ 1!.i bi Ufa (M~2).
2ass" -his eSl "4"s the tiOMSOf of tm t"'m s3.o th.at moe
mo %? eT d~t the period, I"e us lutswnt tis *1 Staations
tIs th eb .ff.t of T e the ad"Cigg pra. to bthI oxwe. jm reawitg for
-0 UNWAORhe to the anwtn cow (uaifa emfficia4~ SO=i w2w OfT)
-o ftelsei fr Puse of oor"
110i error Oda 4"~ by 4*84 (10) awaseI 4 004"Ws
sadb~ "w "d1t ofW tk uptvot (t) an.
10 *-4C tbs.fe Eu f.ats wato aitu
*Aa to the atoome SIe*.. I"e oZ" moe. ir ~st chi
~; ON dmotte 014 sm the low a 1.0 i,et. be espected**S~aftttr. *aft to amm T Ua usmeavaed to ibta a
On Algil amo o tftt at tuogjmcae
,440
14~44~444 of
lot
MINIM
ToJ do ths i~ atio hbrdaeply. conmi of a eastable
*~ ~ ~ ~ u aaoreflto me aosied. the pli~t traterw t~esk toe
*hee Gmay fhs bS* frepay aoft. is esd to be bern. the
Noewh*go*"u the 91stltpt hem to tOaw bee the stable ermefe
iSthe fizar U -do she mle etStne ac
* the omman of Atwea"& gf a obn*f a c(tt4 (e)
* *f asae"opoe'40 NOU it." low" ~ stny
too. *~to n enfn in
as P'*,s~ "ft.vc~ %a, aop urs *Inh pia au eee 4 to A
aspuuml ft" the pawen bans SnmM
~~u&~r'K RItY. pS
-S a.4, 4
.5
.64 * .431 II I
H 1-
- 04-,
0
S
V A
S Ia, 'I
K # C .
1 4WU
C-
)
3. 10
& S
U
aA
0
K
17
an the figure. Wile1 te. error el Is s..11 for all throe responses at t-7.5
sees, the splitude of oscillations when T-.25 see is relatively large.
This my be attributed to the fact that the plant is unstable. lNte, though
the overall system Is theoretically globally stable for all finite values of the
parmter T, the transient response say deteriorate with increasing T, particu-
larly if the plant is unstable. Bence, In such cases the desired transient response
will dictate the choice of T.
6. coN"Mu AnD COMMSON
An adpive algoritbm to presented In this paper whIrh assure the global
stability of hybriLd systems In which the signals are continuous but the paeters
are adjusted at discrete Instants.* The global stability of the overall sysem Is
inepndntof T., the Perto d betwen paramter adjustments Provided T Is
bounied. Simulation results Indicate that the choice of T will be dictated by
the dsred transient response, particularly wbon the plant to unstable.
Mwe adatve law depends upon the specific error model. analysed In Setion 3.
Several other hybrid error sodels have also been analysed by Vsreadwsa and Ksi
(1962) ad sinlaor result& can be Obtained uigthe adatve "M~cresodn
to these error EModel. The technique 41vel"Od, though apple only to'single
input-SAWgl output s""tm ($no0 here, "Trry over to mltivarieble systam as well.
frIer the sans approech cat also be used for the ad a control of discrete
limer systaws In which the plat output Is easured at a certain rate but the
GMtrMl 'PeeMeers ame adjusted at s1r rate.. She letter is ObVI~uel 8sgiicant
in se digital control of complex processes.
lb mark reported e m supported by the Office of leval Research under
Owmat N0014-76-0017.
Criatis aL sad L.V. "Mopoli (M96). Node Reference Adaptive Control :Tb.
NybtU Approach. Teceacal Report, Univ. of N aabsto.
Ul~t. . (1962). lybui Adaptive Control of Continuous Time System. EM
*me* Acts CMUntl 41"s3 419424.o
uithtp, ?.. (19). tbd Se fti Coitrolt ?moe. tin, vol. 1279
part D, ft. 15.
0e6.ws 0.CM * J. "'oS Mi P-.1. Oomu (19M). Discete TIm Mitivarlabie
Adsptiv Cotol U r t. Costrel, AC-25, 440-436.o
ft~w# A.?. "dit S. MisI 1 WM ft the t.Um Asatotle Stability of
000abt IAWW MA ~-Innt 0002 "norati" 1""m uiSUN 81J. control.
Nosm~04" LI IS. 6e t"ty of Pacemter-Adeptive Control System.
tUBA" amu. .Cetw.1. A-25t 433-4)..
Norbaes CB. ad I.S. SliUa (19M). Nmru Nudels for Stable lybrid Adaptive
Coat"l. Udo.al I -g i No. U04, Yale iversity.
8111d. W 1.1. am 1.31. L"n (9M). Stahl. Discrete Adaptive Control.* tU troms.
- Noada, L. *Le. &ft sad L.S. Valeaeit (196). Stable Adaptive Controller
NONSP, lust UIiiPin of Stability. tB ?eMi. hAt. Control,, W-, 440446.
.4 ......
20
Result 1: If the input to a Umser tim-invariant exponentially stable system
is x(-)cL and the correspoedimg otput is y(.), then
Iy(t)i - 0[Z t I1]T)1
iesult 2: If L(S) is a ratioma transfer function of a linear time-invariant
discrete system with all its poles and zeros witLn the unit circle and vith Inputand output X(-) and y(-), respectively, then
sup IX#) sup 1()
IAN"* 1: Let w(,), C(): + - Rn be the input and output, respectively of
a transfer mtrix 1(s)T where I(s) is a rational transfer function and I
is the (ai) unit metria. Let H(s) have all its poles and zeros In the openl ft half plaws. Futher suppose that there is a vector ,(t)eRn and 11.11
is uafermly hms and
t C t*%a
Where k Is a coetat vector and
ak as k- -
Them
[T (t) (s) - (s) T(t)] (t)- o su ] .0-) IZp (A-1)
According to Lease 1 If the input is w(t), the outputs of the two systemsT Tsup,-(ll . a--- 0 sTt) (s) and i(s)*T(t) differ by o OUrItwT)iI if 0 as
k "
k:
7KZT Z 7 -.; 7!-p
21
~1Proof: At time t -(n+1)TM4
*T(t) H(s) 10(t) 140. + ~ RW e1]
- ~(t)(A-2)
If the impulse response of H(s) Is h(t) Ae- Ih(t)I 4 be' for some positive
constants 0 nd r. I(i+)
H(s)# T (tW(t)jT V *I(B)IO(t) t 64 T J h[(*+l)T-rj.*r)dr5~~1 t-(a+l)T1-
iT
(A-3)
where C0 -(#n .) and Ci ~A#
Since the vector # is bounded C1(I0..n are bouaded. Further since
An -. 0,C n-->O as n --> -. From (A-2) ad (A-3) it follows that
I Cj h[~~)-TwTd
to(n+l)T £-iT
-V[(n+l)TJ
(A-4.)
22
n-i i+l) TI,,l("+lml-]I- C q r(n+l)T rT rll,.(TlidT
n-i
iT
n-i]S Y.0 sup I1,,€TllI 1C ' r(U+l)1 tZT 10 T
(A-4)
for some constant y1 .
Since IC nl-> 0 as n - , the term in the brackets as n--> , tends to
zero by result 2. Hence
o(t) vio ng eT
proving Lema 1.