-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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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

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(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.