The Adaptive Robust PID Controller

Copyright © IFAC Identificat ion and S\stem Parameter Estimation, Beijing, PRC 1988

THE ADAPTIVE ROBUST PID CONTROLLER B. Bartlomiej and L. Andrzej ~\'arsml'

Technical l: nil'ersit)', Electrical Engineering, Dept, [SEP , \1'arwwo, Poland

OO~62

Abst.ract. : The paper describes t.he . adapt.i.\Ie PID cont.roller. it.s designing procedure and con~rgenceanalysis . The concept. includes an explicit. identif'ication .... ith recursive paramet.er est.imat.ion and a regulator design f'ormula . A plant is assumed t.o be a SISO second order dead-t.ime lag . The regulator is based on plant poles direct. cancelat.ion. so t.he plant must. be st.able . In order to improve robust.ness of' est.imat.ion the model of' a plant. is chosen t.o have equal degr ees of' numer a tor .M)d denomi na 1.01" pol vnomi al s. Then the est.imat.es are used t.o design t.he regulat.or set.t.ings· t.hrough solving t.he nonlinear equat.ions . The met.hod of t.he regulat.or paramet.ers selection ensures reliabl e st.abili ty i. e required phase and ampli t.ude margins and also trap condit.ions t.o be sat.isfied . This t.echnic combined ....it.h const.rained paramet.ers est.imat.ion ensuring t.hat the est.imates lie inside desired region leads to assympt.ot.ically reliably st.able adapt.ive cont.roller . Keywords :

Adapt.ive cont.rol.

PID cont.rol

I NTRODUCTI ON

and

sa ti sf'i es

ccnd1tic~

a

t.r ap

pr'cpc~~j

by

condi t.i on . Kur' n'~n

(1984)

unf'ort.unat.el y ....e cannot. gi ve any ref'erence in Engli sh ) has some ....hat. heurist.ic formulat.ion : a Nyquist. curve of' t.he op&ned-loop t.ransf'er f'unct.ion af't.er approaching t.he Circle CO.1-10- ACVC:v) should remain in it. The mot.ivat.ion f'or int.roducing such requirement. is t.o prot.ect. t.he syst.em st.abi li t.y in t.he presence of' high f'requency unmodelled dynamics. Not.e t.hat. even small deviat.ion in t.ime delay can produce signif'icant. phase shif't. f'o~ high f'requencies . C

", ........"

One

possible approch to achieve robust. cont.roller is t.o st.art. ....it.h robust. cont.roller and t.hen malee it. adapt.ive as e . g in Song e~ al. (1980) . Thus incorporat.ing described cont.rol law ....i~h explicit.e est.imat.ion of' t.he plant. para~~ers one obt.ains indirect. adap~ive cont.rol syst.em. The conv.,-gence and t.he st.abilit.y problem of' t.he adapt.ive syst.em can be sol ved in analogous ....ay as it. ....as done by Good ....i n and co . (1984.) if' one ensures t.hat. all est.imat.es in every st.ep lie in t.he region of' st.abilit.y. It. can be done by const.rained paramet.ers est.imat.ion e.g using const.rained paramet.&rs recursive least. squares met.hod. Ho ..... ver .also ot.hers t.echnics ensuring convergence of' adapt.ive syst.ems can be applied .

The model of' a stable plant. is described by a discrete transf'er f'unct.ion of' the second order .... ith equal degrees of' polynomials in numerat.or and denominat.or and a t.ime delay. As one takes t.he second order model i t means in practice t.hat the reduced order model is used . Ioannou and Kokot.ovic. 1983 sho ....ed that equality of' orders of' polynomials in numerat.or and denominat.or can hav e st.abilizing ef'f'ect on an adapti ve syst.em in the pres&oce of' unmodelled dynamics . BeSides t.hanlcs t.o t.his f'eat.ure partial t.ime delays can be included in t.he model.

adap~ive

The designing of' the non-adapt.ive PID cont.roller is based on direct. cancelat.ion of' t.he plant. poles and t.he procedure ensuring reliable st.abilit.y of' t.he closed- loop system. By reliably s-t.able syst.em we underst.and a system. ....hich has suff'icient. margins of' phase ~ and Cto amplit.ude ACtoC in practice it. is suffi-

The paper is organised as follows . In Sect.ion 2 t.he designing procedure of' t.he · non-adapt.ive PIDcont.roller is present.ed in det.ails. The next. sect.ion present.s t.he algorit.hm of t.he aciapt.1ve cont.roller and · t.he convergence analysis . Finally some · simulat.ion result.s are discussed in

cient. t.o have ~ -30·- 45· and A -0- 12 dB) CIo

al so

A tr-ap

The concept of a simple adapt.ive PID cont.roller based on direct. plant. poles cancelat.ion and margin of' phase I' t't.,t1+ r t't""""'~. w,.~ p,y ",",v" ",,,,k y .. t at . . C19~~) . Beliczvnski and Lukianiuk (1986). obt.ained t.h~ ext.endent. version by adding a margin of' amplit.ude crit.erion and improving t.he method of' solving of' a nonlinear equation. ....hich is a kev in eval uati ng a gai n of' the cont.roll er' . . Thi s paper describes f'urt.her development. of' t.hat. concept on t.he ....ay to ....ards robust. adapt.ive cont.roller.

CIo

997

228

B. Bartlomiej and L Andrzejz

Sect.ion 3 . At. t.he end of' t.he paper in Appendix t.he met.hod of' sol ving t.he Ieey nonlinear equat.ion of' phase is present.ed. ·

So we DlUSt. specify

. { s1gn b 0 sign P ,.

for Cl + y& + Yz)O (7)

0

THE

DESIGNING

OF THE NON-ADAPTIVE CONTROI..J...ER

PlO

The process under cont.rol is assumed t.o be well aproxiJll&t.ed by second order t.ransf-er funct.ion in t.he form

0

for Cl + Y, + y z)(O

.But. our Dl&in goal is t.o det.ermine Po such t.hat. t.he closed-loop syst.em is reliably st.able wit.h proper margins of phase and amplit.ude as it. has been ment.ioned in I NTROCUCTI ON.

8pCz - ') Gp( z - i ) ..

-sign b

Using st.andard subst.it.ut.ion

Z -le

ApCz-') (8)-

b

,

+ b z

0

,

1 + a z

.

-l

Z

0

Z

-z -& + a zz

1 + Y, z

b

+ b z -z

1 + a z &

-, -&

+ Y. Z + a z z

-le

where x e [O,n) and T is a sampling t.ime, t.he opened-loop t.ransf~ funct.ion can be expressed as

-

(9)

-z z

-z

-le

Cl) (10)

where z-& is a backward shif't. operat.or yt"'b,/bo Yz-bz/bo C bo" 0 ) le is a t.ime delay It. is also assumed t.hat. t.he process is assympt.ot.ically st.able i.e. Ap has no root.s inside and on t.he unit. circle .

Cl1)

The PlO cont.roller placed in t.he Jll&in pat.h of' t.he cont.rol loop, which t.ransf'er funct.ion is given by t.he formula

PCx)=-----------------------2sinCx/2)

(12)

QC x) :,-----------------2sinCx/2)

(13)

(14)

GrCz

-l

)

z

1 -

z

-&

1

-

z-& argcpcx) + jQCx)) Cx)-

~

1 -

(2)

z -&

[ argCPCx)+jQCx))-n

p

(HS)

can be det.ermined according t.o t.he f'ollowing met.hod . Polynomial PrCz --, is chosen t.o cancel t.he pl ant. pol es Thus.. one obt.ains PrCz-&). PoApcz-'). p Cl + a,z-' + azZ- z ). o -& -z ) (3) • p Cl + p'z + p'z & z o

,

p'

-, a

p' - a z

P, • a,po

•

Pz .. a.po

C4a)

-

z

-,

First. of all one should not.ice t.hat. t.he necessery cond1t.1on for st.ab1l1t.y of cl osed-l cop syst.em 1 ~ posi t.1 vness of t.he int.egral gain of (3oCz ':i i.e +

Y.)

~

of

~

p

CX)

funct.ion

+arct.g~---------------------

(1~)

Cl+y +y )-Cl-y +y )t.g z (x/2)

,.

'z

where arctgC ' ) is evaluat.ed in t.he range C-n.nl. because of phase cont.inuit.y requirement. . We are gOi ng t.o Jll&xi mi ze gai n KJ wi t.h siDlult.anously ensuring t.he reliable st.abilit.y condit.ions. so t.he crit.erion t.o select. Kz is as t.he following : Cind sup Kz Culf'iling t.hr . . condit.ions :

Such simple syst.em is a base f'or f'urt.her analysis t.ending t.o det.ermine remaining paramet.er Po '

p b o Cl + Y, o

form

2C l-y z) t.gC x/2)

C4b)

Then provided t.hat. ApCz --, is asympt.ot.ically st.able t.he opened-loop t.ransCer Cunct.ion i . reduced t.o

1

specific

The

def i ni t.i on is · a COl'lseqtrene. o£ cornU t.i on (7) . The formula for ~x) can be obt.ained in a . more convenient. form when using z - ' _ - )"=<:1- jt.gCx/2))/(l + jt.gCx/2)) subst.it.iut.ion inst.ead of (8) . The phase ~x) can be also expressed as ~x) - -nr2 - CIe+lr2)x +

0

KJ)O

1 . phase margin condit.ion ¥XL where

I

x,.

n+~xL )~~..

denot.es

a

and L' CX )(0 , (17) l

frequency

for

which

of

phase

LCX )·l L

~..cCO.n/2)

III&rgi 1'1

is

a

chosen

value

The Adaptive Robust PID Controller a. amplitude margin condition

229

-cosC x/2) [ Cl +y +y ) z -4y Cl -COSx)· 1 •

CUD x_n denotes a

where

frequency

ftCx_n) "-Cai+l)n. X >0 is a Cl

3.

for

I

I

L'CX)~------------------------------~

2sinl(x/2)

which

i-o.l •.. .

1

x

chosen value of amplitude margin

trap condition C1Q)

V xe Cx c .nl where

x

is

the

smallest

x

cond1~1o~ L(X).10-~~O

is a strict.ly decreasing f'unct.ion or f'irst.ly decreasing and then increasing funct.ion

satisf"ying

The problem of evaluating Kx for a simpler model i.e. when yz-O has been presented in

Cpe) LCX) is a decreasing funct.ion in t.he range CO. v). where 1 arccoscl_1 +,.,,+r·1 )

Beliczynslei and i:.uIcianiulc Cl98tD. In similiar way as there above problem can be solved although it is much more complicat.ed . F'or further analysis i t is very useful to present. the properties of' the phase and the amplitude functions . Properties Q[

~ ~ ~

for "'z>O and

a~

1

+"'.+"'.1 12~

functions. 1

CPl)

boundar y val ues of ftC x) ar e

for

C"'z >O and

I/JC 0) a-n/2

"'z:SO

f'or Cl+y.+yz»O and

-len

Cl-y. +y.)l!;O and

l"'zl:S 1 -(le+1)rr

for

C~

ly.l
C1+r.+,.,.)Cl-,.,.+1".»0

CPG) first. derivative of' I/JCx) has the form

~

2)or

LCX) is an increasing funct.ion in t.he range (v.n). where v is as above

let.·s introduce some auxiliary Now notat.i.;m . Let
•

of' equation

,

fCX) ,p'(x)---l' CX) z where

ftC x) --n+,p Cl

(20)

•

where,1 is a number of' sol ut.i ons . Let. x_nbe the smalest. solution of equation

f' CX)"1-2lc+2y Cl +2le)-,.,·C3+2le)-,.,zCl +ale) + I • • z

ftC x) a-n

-4y Cy Cl+le)+lelcosx-4,.. C1+2Ic)cos·x •

+""+"'ZI

1 a~

Proofs of' these propert.ies are i_diate af't.er some calculations .

for (l+1".""'z)(O

-Cle+2)n

< a

I

Z

i. e

x·

-n

is

t.he

C21 ) smallest.

1"requency.

f'or

which the phase intersect.s -n .

l' CX)-2CC1-y )z+y·+2y Cl+y )cosx+4,., cos·xl Z

and

Z.. fzCx)~

"Ix

Z

and

•

equation f,CX)-o

50_ at.t.ention has t.o be paid t.o problem of solving equat.ion of a type

has no more than two solutions

ftC x) .
•

l' (X)-C1-y ) ( y (l~y +yl_,.,I)+

•

z,.

I

,

-8y (l+y )cosx+4,.,,.. cos·xlsinx Z.

,

I

f .(x) as above

f.c

and f!tqllat.1on solutions

in

x.1 -0 has no more tha.n

the

C22)

This is a nonlinear equation and its solutions can be found only it.erat.ivel y. Fort.unately if ,ps- C-n.-rr/2l then t.he

CP3) second derivative of ftCx) has l' .CX) the form ,p' 'Cx)---flCx) where

the

solut.ion always exist.s and the number of solutions is not great.er than thr... This is because ftCx) has no more than two exlremal points Cs . . CPG)) so • 1:S 1:S 3 . The ~hod ot solving eq. (22) is proposed in Appendix • ..mere the ex1st:ancl! of solutions and the convergence of it.erative process are also proved.

two Let
i-l • . .. • 1

•

range CO.n)

denotes

the set

of

gains sat.1s1'ying CP4) if -n:S,p :S-n/2 then s

,.

,.

V O
I

"'"

where x.p. is the smallest solution I/JC x) a,p (Pe)

i.

i.

(23)

LCKx •• x,p.) - 1

ftCx)~,p••

By

or

Kx..

Kx.

satisying as

•

let' s

denote

the

gains

approp~iate

Ca4) f'irst

derivative

of'

LCX) C2!5)

B. Bartlomiej and L Andrzejz

230

Defined above gains have simple i nt.er pr et.at.i ons. Assuming t.hat. L<1,x) is a decreasing funct.ion t.hen every KI~ is t.he gain sat.isying t.he margin of phase requirement. . Sim1lary. if L(X) is a decreasing funct.ion t.hen KI. is t.he gain ensuring t.hat. t.he closed-loop amplit.ude margin equal t.o >" .. ' int.erpret.ed as t.rap condit.ion.

t.he

gain

syst.em has KIs can be

sat.isfying

t.he

Theorem 1 says how t.o choose maximum sat.isfying all t.he condit.ions (17-19). Theorem 1 If t.he r equi r ed

mar gi n

of

ampli t. ude

KI

>.. ..>0

tf

tI>C ''L ) +1l ~tI> co (2tla.>

2

SUP

{ KI •

=

sup

{KI'

1

\. =:i •. . . • I

,z.l <0'

.KI » IKIL
*

~ where L(KI ,x )=l A l is t.he great.est. gain ensuring reliable st.abilit.y i. e t.he closed-loop syst.em fulfiles condit.ions (17-19) c

bot.h (18) and (19). Finally if one t.aJc:es KI =inf{KI .KI > t.hen equat.ion LeKI .X)=1 A :r!l • • lA has unique solut.ion Xl and Xl
addit.ionally

sat.isfied t.hen fulfiled and KI

SUP

If

for

KI

A

t.he

all =KI

•

tl>Cxl)+IT~tI>

condit.ions

is are

A

margin

of

phase

is

not.

reached. t.hen because Lex) is ,t.rict.ly decreasing funct.ion in range [0, Xl ]. it. is always possible t.o obt.ain t.he required phase margin t.hrough lowering .t.he ga.in. In such case one select. s'rP KI: but. lower t.han KI

c

A

Thus having calculat.e t.he

~h.

value of KI one paraJnet.ers PO.P1' P Z of

can t.he

cont.roll er (2) from eq. (11) • (7) • (4.b). And t.his compl.~.s ~he regula~or design problem.

Al cor i t.hm 1

tI>

. . >..

, e(O)

OF'

THE

The Cert.aint.y Equivalence Principle al·lows t.o const.ruct. t.he indirect. adapt.ive cont.roller ensuring reliable st.abilit.y of t.he closed loop. The est.imat.es of pla.nt. parAJllet.ers can be used as if t.hey were

0'

1

•

0

1

•

using RLS met.hod i. e P( t.-2) vC t.-1)

A

eet.)= e<:t.-l)+

x

1 +vC t.-l) "r P ( t.-2) vC t.-l) ry
Sc t..-1)]

(28)

PC t. -2) vC t. -1) "r vC t. -1) P( t. -2) PC t.-1) = PC t.-2)1 +vC t.-l) T P( t.-2) vC t.-1) (29)

where

v(

t.-l) = [-yC t.-l) • -yc t.-2) • u( t.-Jc:). u(t.-Jc:-l). u(t.-Jc: -2)] T

If t.he est.imat.es lie in t.he specified convex region of st.abilit.y i.e c~ndit.ion 2 0'

A

A

-Ia/t.)

10'

~ a:ret.) ~ 0'

2 (30)

is sat.isfied t.hen QO t.o st.ep 2 . otherwise Aevaluat.e t.he new vect.or of est.imat.es e'(t.). for which condit.ion (30) holds. Simult.anously. t.he new vect.or must. sat.~sfy t.he condi~ion : A A llV-e' (t.) Tpc t.-l) -le' (t.) -ee t.) Tp( t.-l) -lee t.)~0 This can be solved by using Const.rained Least. Squar es Al gor i t.hm ( see GoodWi n and Sin .1984. p . 92), which ret.ains all propert.ies of t.he RLS met.hod . In our case due t.o simplicit.y of condit.ion \:,30). comput.at.ional effort. in evaluat.ion e' (t.) is not. increased t.oo much. St.ep 2 : Using t.he est.imat.es evaluat.e - Xl solving eq. tl>Cx)"-n

-n

Choose

KI

-

{xt/>a> sol ving eq.

-

{KI >

.up

1

,KI

KI

z

according

par~t.ers

of

po(t.).p.Ct.)'P:r(t.) ALGORI TIlM AND CONVERGENCE ADAPT! VE CONTROLLER

, P(-l) •

St.ep 1 : ~t.imat.e t.h~ model para~t.ersA e(D= [a (D.a (D.b (D.b (D.;; (D]T

~omput..

TIlE

1

The last. assumpt.at.ion means t.hat. t.he plant. does not. include derivat.ives . The algorit.hm of t.he adapt.ive regulat.or is present.ed below.

x

The,t.rap c~~~on (19) requires t.ha~ if L(x )~ 10 t.hen for any xe[ x • IT] -IT ->.....reO -IT also L(X)~ 10 L(X) is a st.rict.ly deer _ s i og 1'unct.i on or it. posseses onl y one minimum point. (P5)-(P7). If it. is t.he decreasing funct.ion. t.hen t.he t.rap condit.ion is fulfiled aut.omat.icaly. If L( x) posseses a mi ni mum poi nt. i . e. it. increases before reaching n. t.hen t.he point. X 2 IT must. be t.est.ed. The smallest. value of KI. and KI!I ensures fulfiling

•

<

0< 0'

•

b +b +b .,. 0 where b . b . b are o 1 • 0 1 • coefficient.s of t.he numerat.or of t.he plant. . t.ransfer funct.ion.

Given :

ot.herwise (20b)

L • (Xl)
i =1 • 2

•

(A2)

gain K1sup such t.hat. i f

The plant. is asympt.ot.icaly st.able. linear. st.at.ionary syst.em of second order wi t.h Jc:nown t.i me del ay descr i bed by t.ransfer funct.ion expressed by (1) . Addit.ionaly let.·s assume t.~ t.he poles of t.he plant. %1 ' %. sat.isfy

(A!)

KI

and required margin of phase O..
KI A =i nt'{KI ~ • KI.>

t.rue t.o design t.he PlO regulat.or described in previous sect.ion . t.he begi ni ng 1 et. • s consi der t.he At. simplest. case i .e pure det.erminist.ic system under t.he following assumpt.at.ions .

(11). (7). (4.). cont.rol signal

z

is

1

and

cQnt.roller equat.ions

comput.e

+p (t.)e{ t.-2)

e{t.)-r(t.)-yCt.)

Theor_

PlO from

UC t.) -u( t.-1) + Po (t.) eC D

where

•

to

Finally,

tl>Cx)=-n+t/>.. from (23-2e)

act.ual

+ PlC t.) eC t.-l) + ,

(31) t.he

act.uat.ing

231

The Ad aptive Robust PlO Controller signal. rC~) is ~he reference signal. y<~) is ~he ou~pu~ signal . S\ep 3; go ~o s~ep 1

Remark Some

GpCz-~"

1.

modifica~ions

Algori~hm

~Q..

adap~ive

(33)

1

GpCZ-~2 Cl+3s)Cl+!'5s)Cl+O. !5s) e-i. and

l!&llrCt.)-yCt.) 1"0 c Proof i Part. Ci) of t.he t.heorem can be est.ablished immediat.ely aft.er using t.he result.s ot' Goodwin and Sin. lQS4 pp . 212-213 . •He!::t . i nat.ead o!' t.he desi gned pol ynomi al A Cz ) h~;--e ~ -, -, ~ ~ -, - le ] A Cz ) -AC t.. z ) [l-z +p 0 (t.) BC ~. z ) z

wa

(32)

where ....

ACt..z )-l+a,Ct.)z

_,

....

+azCt.)z

-2

cases were si mulat.ed. The of ~he plan~ were es~ima~ed by usi ng ei t.her model Cl) Ccase a / ) or simpler model Ccase b/) having t.he !'orm b +b z-' o ,

t.wo

in~egrat.ion.

0

, One can concl ude ~hat. proposed adapt.1 Ye con~roller leads t.o Ci) s~able syst.em wit.h all signals bounded (ii) ~he st.eady st.a~e error equals t.o zero Ciii) due ~o Const.rained Least. Squares Algorit.hm every st.ep of t.he es~ima~ion procedure does not. make worse t.he es~ima~es and if t.he input. signal is sufficent.ly rich. ~hanks t.he designing procedure t.he syst.em become asympt.ot.ically reliably st.able . SI truLATI ON RESULTS

A few simulat.ion resul~s.as ..,. believe t.he most. in~erest.ing ..,. are present.ing in ~his sec~ion . In each experimen~ discussed here. t.he margins >.. ...8dB and (/>... 4.0· were

wa,

up. and t.he covariance mat.rix chosen as a diagonal mat.rix wit.h 10 val UH on t.he di agonal. The sampli ng t.1 me was al ways 1 sec .

se~

For t.he first. experiment. cont.inous having t.rans!'er funct.ion

plant.

s--------, -z l+a , z

Z

-le

(35)

+a z z

Not.e t.hat. out.put. signals shown in Fig . 2 are different. very lit.t.le . But. t.he est.imat.es in case b / drif~ed Cwi~hout. dist.urbances) in observed numbers of samples and t.hey s~eaded 1n case a / . Fig. 3&. b present.s an example o!' behaviour of a,Ct.) . This experiment. shows ~he need for t.aking model Cl) rat.her t.han (35). Finally we invest.iga~ed t.he performance of t.he cont.roller in rat.her d1!'ficult. condit.ions . Cont.rolled plan~ wit.h s~ep dist.urbances was modelled as Cl +s)C 1 +3s)C 1 +!5s)yC s)· =( 1 +1 . ls)e - i . 75"U(S) +d(s) (30) The reference signal (also s~ep) was equal ~o ~he d1s~urbance signal r(t.)-dCt.)-o . l for ~~ O. Resul~s of ~hat. simula~ion are shown in Fig. 4 . The mos~ in~el'_~ing was ~ha~

act.. z-')-(b Ct.)+b Ct.)z-'+b Ct.)z-z 0' z The cons~rai~ param&~eps es~imat.ion ensures ~hat. AC t.. z -~ is al ways st.able. The pol ynomi al in squar e br ack I~s _ ;. s designed t.o be st.able. so A Cz ) is st.able.Part. (ii) follows ,from ~he fact. t.hat. ~he cont.roller includes

C34.)

parame~ers

cont.roller

Theorem 2 The syst.em wit.h indirect. adap~ive con~roller CAlgorit.hm 1) applied ~o t.he plant. Cl) sat.isfying assumpt.at.ions CAl) .CA2.) is asympt.ot.ically convergent. in t.he sense ~hat. Ci) all signals remain bounded Cii) if lim rC~)=const. t.hen l-Q)

_~

75.

was select.ed . Fig . 1 shows t.he ou~put. and t.he I' et'erence Signal . The es~i_t.es st.ar~ing from 9(0)-(1.0.0.-1.0] converged t.o ~he ~rue values. Then ~he model of t.he plant. was changed t.o a have parasit.ic ~ime const.an~ as in t.he following

GmCz -i)

Now convergence of ~he can be est.ablished .

A

e-"

A

should be in~roduced ~o axoi;;1 a si~WI.~ion. when ~he ' es~ima~es bo.b,.b z do no~ sa~isfy assump~a~ion CA20. I~ is qui~e easy ~o do. so _ will no~ describe ~he de~ails. The only problem is ~ha~ ~he s~ar~ing vec~or of es~ima~es can no~ be one of t.he t'orm [a .a .O.O.O]T . But. t.his seems ~o be not , z ' very rest.rict.ive in pract.ice . Moreover. similar modificat.ions as presen~ed in 8elicz~ski and Lukianiuk. C1Q8e) can also be made t.o ext.end t.he class of cont.rolled plant.s of such wit.h int.egral act.ion .

~he

1

Cl +3s) C1 +!5s)

~he

es~imat.es

projec~ion

in~o

~he

assumed region Cp-O . Q) had been performed in every s~ep. wha~ had allowed ~he es~imat.es

~o

be

s~eaded .

CONCLUSI ONS In t.he paper we have present.ed ~he indirect. adap~ive PlO con~roller. I~ is based on direc~ plan~ poles cancela~ion and a design formula ensuring ~ha~ ~he closed-loop sat.isfies margin of phase and ampli~ude and t.rap condi~lons. I~ is shown how ~o choose ~he gain of ~he con~roller (Theorem 1) and how ~o solve ~he key nonlinear phase equat.lon . The convergence of ~he adap~ive con~roller ln ~he de~erm1nis~lc case 1s proved . As ..,. observed ~he proposed con~roll~ has some fea~ures of robus~nes. t.hanks lo (i) applied cont.rol law ;PIO s~ruc~ure and fulfiled condl~lons for asympt.o~ic reliable st.abill~y (11) chosen model s~ruc~ure ; equal degrees of pol ynomi al s 1n n UJDer a t.or and denominat.or (lii) or~hogonal proJect.lon of ~he es~imat.es ln~o ~he desired region ln paramet.ers space ~his allows ~o es~ablish convergence of t.he adapt.ive con~r 011 et' Acknowledgemen~si This work was suppor~ed by ~h. Inst.i~u~ of Sys~ell\S a.asearch. Polish Aca~y of Sciences (PAN IBS:> under CP8P 02 . 115/2. 1.7/87 .

B. Bartlomiej and L Andrzejz

232 References

~

G. , Bjorck A.(1974). Nymerical Methods. Pren~ince Hall . loannou P.A. , Koko~ovic P.V. (1983): A Reduced Order Nodel AdapUve Sys~ems. Lec~yre ~ 2D. Intormat.ion &DQ. Con~rol sciences . Springer Verlag. Gooc\win G.C, Sin K. S (1984): Adaptive Fll~ering Predic!,1on ADSl Con~rol. Pren~ince Hall, New Jersey. Banyasz C. , He~~hesy J . , Keviczlcy L. (1 Q8l5) . An AdapU Ye PlO Regul a ~or Dedica~ed For Microprocessor Based Compac~ Con~rollers . Preprint,. 2!:. nh IFAC/1FORS~· QD. Iden!,1fica!,.1on ~ ~ Parame~er Es~imat,.ion. Pergamon Press, York . Beliczyt\ski B, I:.ukianiuk A. (198e) : The Adap~ive PlO Regula~or wi~h Phase and Ampli~ude Margins . Preprin~ 2t ~ ~ QD. Microcompyt,.er Applicat,.ions ~ Aut,.omat,.ic Cont,.rol . Pergamon Press, Dalquis~

From

~he

~ha~

~here

[Xj'X

j

+, ]

cons~ruc~ion

a

ex1s~

J

f '

solu~ion

i~

in

follows range

~ha~

and also [x .,x . ]

V x. sign

way of

I

J+i

sign

r'(X)"cons~

and

These condi~ions are ob~a1n unique solu~ion

'CX)zcons~.

surficien~

~o

o No~e

which

~hat

solu~ion

....

Is~anbul

Song H. K., Shah S . L . , Fisher O. G. (1980). A Self-~uning Robus~ Con~roller. Au~omatica , vol . 23,no . 3 Kurman K. (1984.). Technical Universit.y of Warsaw. priva~e communica~ion

Fig

ir one holds of (37) .

~akes ~hen

(4.0)

1.Simula~ion con~inuous

~he

smalles~

ob~ains

~he

j ror

smalles~

~he second wit.h delay t.ime

resul~s :

plan~

APPENDIX To solve nonlinear equat.ion t/IC X) =<1>. ' where <1>._ (-n/2, n) (37) it. illl more convinient. ~o int.roduce func~ion feX) defined as follows f( X) - - t/IC X) e 38) :z

and sol Ye

~he equa~i

<1>.+

{'(X) ..

n/2

on

+ (k+l/2)x +

2Cl-y 2)~gCx/2) +arc~g

2 -0 (39) (1 +Y, +y 2) -(l-y, +y z)t.g Cx/2)

•.*.....,.-r--.,.......,.......,.......,.-,......,......,......,..--.,-r-.,.......,...-+

L..emma 1 .

. ...

If ,. e (-n/2,-n]

~hen

one solu~ion in more t.han ~hr.. .

~he

eq . (39) has range

[O,n] ,

a~

Sill\ula~ion

con~inuous

no

bu~

• . to

2 .57

I.))

Fig 2 .

leas~

resul~s :

~he

~hird

10 :

or-der

plan~

c

~ Func~ion fCX) is a con~inuos func~ion . Moreover, f(O)(O and f(n)~ 0 (s. . proper~y CP1)) and fCX) has no more ~han ~wo e~remas Csee (P2)). c Now define : - a se~ of frequencies for which f'(X)-O <~ > - a se~ of frequencies for which f"(X)-O Alaor1t.hm 2; Eval uat.e and <~ > usi ng for mu! as given in CP2-P3). TakinQ ~he ele..nt,s of ~he se~s ,<~ >, crea~e s~rictly increasing serie Xi' .. . 'X ' mSO. While

'

.1.7.~:-"-.,.....-r--:'r::-...,....-,.-.,.........,...-r--...,....-,.-.,.........,..._r---+ ' .•0

d

c:ompu~ing

fCx,)

for

every x, find such

Xj

~ha~

t"Cx.)t"Cx . J

Then

J

~he

apply

+,

)$

. <0..1 quis~, 1 Gr74) (37) in

~he

~o t"ind ~he range [x ., x . l.

Thegr.m 3; The i~era.~ive

(~)

O.

seca.n~-~angen~

J

_~hod

sol uU on

ot"

J+i

process ~he of method used to solve ] eq. (3Q) in ~he range [Xj'Xj+, converges ~o ~he unique solu~ion of eq.(37) in ~ha.~ range . o ta.ngent-secan~

I."

t .C7

Fig ~. Si mul at.i on resul~lII : continuous plan~

t.S7

• . to

10 t

~h. ~hird order