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The Adaptive Control of Unidentifiable Objects
Copniglll © 1FAC h 'aluatioll of Adapti\'e CO ll trol Strategies. Tbilisi. l'SSR. 19H9
THE ADAPTIVE CONTROL OF UNIDENTIFIABLE OBJECTS M. M. Kogan and Ju. I. Neimark The Chair fill' Thl'on 01 COl/lro! al/ri DYI/(11l1irs 0( Mochil/f's, Corh Slalf' Cl/il 'I'rsit)'. (;"rkr. L'SSR
Abstract . ':ii th the help of averac ing method there i s stu d ied the behaV l our of adjustab le ~a rameters in adapt i ve contro l sys te m with identifie r. There has been found a comr l ete class of contro l strate g ies c alled locally- optima l; for this cla s s the desene rati on of information ma tri x and the bia s of estimc tes fo r unknown ra rameters of the objec t do not ~reven t from i de ntifiability of ~2r a meters of these contro l strateeies . Keywo r ds . Ad2~tive control ; i den tifi ab i li t y ; l ocal ly- ortima l control ; avera~ ing me thod .
le as ~squares
metho~
I1:THODUCTION use of c ontro l and i dent ification is known to c ause Rome definite difficu lties. Fi rst , avai labili t y of feedback entails a dec enera tion of in f ormat ion matrix and consequently a non i den tifie bili t y for initial I ararne t e r s of the object . Second , in the ueasurewents of outj.ut there occurs some correlated disturbances th a t r e s ults in the bias of est i ma tes obt a ined . Hence they r e2 ,rt to v8r ious l'l ays of sUj:-pl i rnentary influence ur on the object ensuri ng its i dentif iability( Sa ridis, Lobbia, 1972; Anderson , Johnso n, 1982 ) , nnd to rreven t from b i as they turn to various extended method s of i den tification (j,strom , Eykhoff , 1971 ; Chen iian I'u , 1982 ).
l i mit man i fold i s determ i ned by the kind of functional derend ency of regUlator rar8r.lete rs uy.on the objec t parsrne ters . At this it turns out that wi thin the entire lin'i t manifold the r arm"eters of local lyoptimal control stra teGies obtain the values similar to the real . In such adaptive systems some kind of ad,iusting regu l a tor p::rame t ers occurs , ;:..nd in this sense the,e systems do no t a t sll di ff er from di rect r y s t ems of adartive cont r ol .
J~tual
The resu lt s we d i scuss are arranged in the pqrer as follows : first we d i scuss the study of averaGing equstions d es cribing the dynamics of estireates in the identification proc ess , we have found the equa tion of manifold they tend to ; i n the next section there a re considered the ada r tive locally- opt i ma l con trol s trate puramete rs of which wi thin the g ies entire limi t monifold assume the values c6incident with the r ea l; in the l ast sect ion the r esu lt s obtained a re ap plied to the caE;e of nonl":easurable st a te .
In this reto rt we d i scuss the resu lts obt a ined by u s in the study of adapt ive contro l of linear descrete stochast ic objec t s (Fe i r:lark , 1978 ; EOGe n , J: eirr.'lrk , 1986 , 1987, 1989 , 1990 ) . Our m ~ in re su lt includes the fo l lowinG • ..'1 thin the cla~;s of f e edback relations i t becomes pos s ible to distingu i sh the contro l st rat eGies ca l led l oc al ly- optimal for which , de s pite the non i dentifiabil ity of the obje ct 8nd the bias of estir.:ates , there appears the identifiability of [8rarneters of these contro l stra t eGies , and hence the contro l [ roblem is solved .
TIlE S'l'UDY Q}' I lJEI:TIF L\B I-
T..ITY BY THE iETi iOD
AVE i~ .\CIr:G
~
S
1et the vectors E Re and AtE R to be subjec ted to measuremen t s be fo r al l t =C,1, .,. interconnected by relation
Th i s ~ tate~ent h ~ s been made ul on the appl ic ation of everae ine method to the e quations descr ibing the ope l~ ti on of adar tive system . In these equot i ons the recurrently obtained estimates change s l owly as cor;\p ared to the r ha se v ari able s of the object . The analy sis of deterrdn istic di fferent ial equations of slow vari ab le s has revealed t hat within an estimate siace there is ava ilable some asymr t o tic ally stab l e man ifold of equilibrium s tates . The av a ilability of such man i fo l d reflects the fact of non i dent ifiability of the objec t, with the feedback f.resent in the syst em. The equation of
(1 )
vlhere Q* and {tt)
i s an unknown matri x eX$ is a s equence of inde ;:.enden t
random vectors, and
Mtt - ~
Mttt/'a le
Acco r ding t o the leas t- square method , we define the sequenc e of estima te s for matri x .Q1f through the following re current rela tion
Ili:)
M. M. Kogan and Ju. 1. Neimark
166
jiff' .. SJt
+ It
~ ~ ( At+{-.Q;tfp ~ (
2)
is s ome a rbitrary matrix where J2q RX$ and the mat rices It are given by the formula
I t +d
=
It
rl-.
'T'
~ > O.
qJt '"'t,
1-
O:ote that vector It>t pend u~on the estimate
(3)
in (1) can de-
.ot
)
.
To analyse the system (1) - (3), let us introduce s ome small j::aramete r c> 0 -"f by rep lac ing the . I .f, by;~ f we obtain the followlng: -1
~#i .Qi - c~#{ ~ +eI- 1 CP. ~", t+.f
Ii1
et; (-'?t -.0,..) ~
( )
4
fo
(5)
i >t
7'
~ I- e CfJt 4Jr .
( 5)
The Egs. (1), (4), (5) comp letely describe the identification [ rocess with resp ect to the assiened t • AT-p lying the averaging method (Bogo ljubov, I.:i tropo l'skiy, 1963; Ljung,1977; Kogan , Ne imark , Ron in,1 983) to these vectors wi ll y ield dif f ere nt i a l equ a tions of s l ow variables
91,2
.Q - -l-{P(.S2)(.Q-.Q*)
(6)
j - P(!;). The matrix function P(.Q) in (6) is dete r mined through the rel at ion .Ir'-d
P(.Q) .. !,im.N-.fI
ct;(.Q)ctfC.Q)
(7)
./I..,.. 00 t=O in which the sequence {4it (Q)j is pro duced from the initial s equence [qDt) if in it for al l t=O, 1 •.• there is de. Fo r the case c l ared the S)t = Q when vector q5t in (1) does not depen t ul,on .Qt Vie obta in CfJ/.Q) ~
CfJi
I f the matrix P(.Q) is determined positively, then the system (6) obtains a unique asymptotic stable equilibrium state Q c.Q... corresI;ondinr; to the object's identifiabili ty. In the t; ene: a l case , the manifold of st a tes for the sys tem equilibrium (6) with respect to.Q is described by the e quation
P(S2) (.Q - J)*)
=
o.
(8)
When i dentification is acco m~ lished in the course of control [rocess there occurs a degeneration of matrix f)(J.?) due to some de~endencies existing between components of vector ~t:
~ -: (CP ,'1' Cf.J."7') t t, t , where matrix
(Y(.5J)
cp = B(.Q)cP.' t
K
t
t,
(9)
is determined by the
selected control strategy. This fact yields the fo l lowing representation of ma trix P(.Q) :
P '(f)) 6)(.Q)
f)'{Q)
P(~) = ( [;7'(.Q)f)(SJ)
)
1Y{.f2)P(Q)B(.Q) ,
(10)
PCQ) >0.
will f converee to the man ifold defined from(8 ) by the equation In this case the variables
si +6)(52)J)" = Sc>; +6)(.Q)Q:
( 11)
.Q 7' = (.Q ''': .Q'I'1') • The Eq. (11) in c; eneral c a se rer:resents some [-lane to which the ma trix S)t will tend when t increases. Thus , in case of adartive control, general ly sr:eaking , no identification of matrix oc curs but it becomes evident that the limit mu trix satisfies the Eq.(11). From the noni de ntifiabili ty of t:atrix 5.'. it does not however follow t ha t ma trix .Q,.. is uncapable to r:roduce any functions t hat on the contra ry a re identified. The function S(52,..) will be such iden tifiable function if and only if it is of the kind (Kogan, Fe i ma rk,1 989):
(;}(.Q,.) =(8 -.o;)!{(.Q: HII,
(12)
where H i s sor.le arbitrary matrix, such that the ntGtr ix.Q:H i s square and a degenera te d one, and S is an arbit rary ma trix of corres ~ onding dimension . The identific a tional fe a tur es srotted in the course of control proce s s and during the r elated degeneration of informati on m8.trix retain inherent not only to the least-squares method but a lso to its mod ific a tion, in which
It+;
1;+.( Ee , dt/i 1t +ff ~ ;2, t;, > 0,
and also to the i de ntificntional Irocess done by stochGstic approximat ion method , in which
~+/ '"' Jt+/E!, 1;1-/ -;; +1, liere
Et
i s a unit rna trix
THE IDENTIFIABILITIY OF LOCALLY-OPTn-:AL ADAPTIVE CONTHOL Let us consider the object of control desc ribed by the equation
tt ,
~t+{ '"' A*~t + B~Ut f in which ,xi E R117 is the
(13)
st a te to be measured; and {~t} are the nonobserved random disturb a nces; andM~t 0, M§tt;= /(; and A* and B* are the unknown If x ID and m x k matrices. Se lect the following form of the control strategy
Ut
= 6)
'1'
(.Qt)X
t
(14)
Adaptive Control of Unidentifiable Objec ts
where IY(.Q) is an assigned matrix func7' tion, and ~~t is the current matrix estimate 5.>. = (A .. , B.. ) obtained from equations (1)-(3) where ",
",
'1'
.xt+! = t +{, ~ = (Xt , Ut ) • If the ~t = S} fixed and i f the control process (14) with J)t ~ ~ ensures an asymptotic stability of the object, then there exists the limit _( -t'-{ "'\.
a:
tim AI
..N-~
Z'
t=O
~f
XI'" .. P'(Q) ~
where
and the matrix P(5J) obtains the form (10). Under the abovesaid, the ~~ in this case will tend to the plane described by equation (11) that in this case will be of the form: ".
",
(15)
=
7'
(1 6)
G) (AN, B~)Xt
provided it, in acco rdance with (12),will be of the form 7'
7'
7'
-{
7'
@(A*,~= (H Bit) H (S -
A.. ) .
(17)
The control strate GY (16),(17) m ~n~ m~zes in each st ep the conditional ma t h ena tic a l expectat i on for the efficiency function T/"
Y
_
_
3
7'C
(X tf !) - (Xt+1 X t +!)
_
3f1 8)
(Xt"/~'I'
with the hist ory of the proce ss ava ilable. In (1 8 ) the nonnega tively defined ma trix C C 7' sa tisfie s t h e equality H = [JBN 3 ' and oX t+1 make s up the value of referenc~ traje~tory assigned by the equation ,xf:+' = S t:Ct where S i s the arbitrary matrix of related dimention. Thus, when choosing the control strategy of the form (14),(17), in t h e adaptive sys tem under discussion there is set up a de s ired locally-· optimal strategy of control
=
THE IDENTIFIABILITY OF LOCALLY- OPl'HIAL ADAPl'IVE CONTROL WITH UNr,IEASURABLE STATE OF THE OBJECT Let u s con s ider the one-input one-output object described by the equation of the form
Jltfl + = -G;,*Ut
..
*
a{!lt t, .. +am
+oo,+
!It-m+! -
-B;-./-
+.{
or represented in the operator form ~ t+! = 1jt )
( -1'!!
{,
A*(i!)jr 2
-1
1
-1
V
BIf(i!-,)Ut + G(l! ') t ,
l -"J
/!_(-I in which AJf (Z -I'), Bit i! , er % Jare the f polynomia l s in 2- ; and {Vt} , a sequence of independent random values ;
2 /IfVt
/If Vt = 0,
2
= () .
Vf ,.,.
-7'-
7'
= Cft CCJ>t, C = C •
lPt :: (1/t-mf-I'"'' Vt, lLt-mf-{, "', "-I-I) minimizing the conditional mathematical expectation of increment pe r a step a long the object trajectory, with the history of the process Given , i. e .
u; = aUI min M(~+ll{ja, .. ,.rt).
The Eq. (15) ensures only partial information about the matrices A.. and 8* though for the case zan.-:B.= K this in format ion will b e sufficient to formulate a desired limit strategy
Ut
+itVt + .. , + ~m Vi -m
For such objects, in (Koga n, Neimark,19 8 7, '199 0 ) there is formulated a locally-optimal con trol s t rategy with respect to some efficiency function
,
A + Bf) (A,B)= A. ~ 8,1f t9 (A,B).
167
(19)
Ut £* It has been found that for St~ +
d
-1,.0 this strategy i s descri bed by the equation
[$., Bit (;c-') +.D(£)G-(2-)}Uf =
(20)
= [S1 A*(i! -I) - $(2-)G(r-')jrYt , and the related characteri s tic polynomial of the closed system wil l be equa l to
L1(~ -J",G(ii'[S(Z-?B*(i!-1) +])(i!-)Aft(z-~l (21) In (20), (2 1) the coefficients Si and t/.;' of the polynomial s S(Z-I) and .1J(i!-') are_e xpressed through elements of matrix RIf of the C an d t h r oug h par ame t er v_ object. 0
First we consider now the particular c ase when the efficiency funct i on is equa l to = ~'f fand the locally- optima l con-t"rol s trate gy i s de fine d by the equation B.. (2- 1) Ut [Ai Z-I) - G{i!.-I)}:i!'Yt (22)
lit 11;,
0
=
and t he characte ri s tic polinomial (21) i s equal to
As it f ollows f ro m (21) the stability of the object under locally-optimal ~ontrol (2 0 ) requires the polynomial G(Z1) to be stable, i.e. the object has to be a minim~phase one with respect to its excitation. If not so , we take a control strategy in the form (20) where the polynomial ;;: -I 0-( ~ ) is replace d with the polynomial (2-') • In the_tbelow the line above polynomial from z will denote the substitution of roots ~ injit ( for which /ZdJ> 1 ) by roots z~ , i.e. if
G-
168
M. M. Kog-an and
G{;i'} =
n
and
1~/<1
/~/~J
0{
n
=
r U) .5) = (.5)'7' ~ If)
= ('
If,
~
if
'0
_I .JI~1
7'
t
(JJ) ,
(28)
PljJr{J)) =
(Q)
where ~(S))=
1 G-(~-? ~ (fJ) .
(29 )
(30)
In (28), (29) the components of t he vector
'I{'T'(52) = (J't-mt/RJ, ... , dl 1/ (JJ) ,U (-0, ... t t-m+1
r
7'
are determined as follows
,N-I
Let us represent the equation of object (19) in the form r fltt! Sc>. ~ of- ~t/$Vt +"'+im~-mfo{ ,(2 3 )
cpt
PSO$l'(.Q)
lo ffJ/.Q)
(I-i~).
/~/
This control strategy also provides s tability for the objects being nonminimum phase with respect to their excitation.
where
P.9l1' (.Q)
q,,(.Q):,I!!.':!,-'"
then
G(2- 1) = n (I-;i~~'J
I. Neimark
In these equations the matrices
(I-i/~)n (1-i!~!7fi),
1~/n
.Ill.
"', l1t - 1 (.5))) are determined by the followinc equation s :
,
052'7' * ... Cf.t*", -&m-I, . .. • .f..,) If . * = (- am' it
1-
In these denotations, with due account of the abovesaid, the control strategy (2 2 ) will be converted into the form
J
Ut
,
7'
= - 6:'" (.Q~ + f) 'It
where compongnts of vector
j
7'
J
(24) E Rflm-.f
;: (jm •... , 1/ , 0, ... , 0)
will be the related coefficients of polynomial G(i!~") . Having the transfer functions of the object a priori unknown with respect to their input and excitation, we nO'ii,.define the estimates of unknown vector .sltt through the equations
.Q i+1
= Rt
-f
+
rn
rrrF-,
I t+{ '-1f (!it+J- ~ '¥t J,
(25)
I tt/ = It + ~ £Pt ,
Q;r YJ
_.1 I nlr
-v;,
.JC
(26)
it
~-~
(27)
etI dt
-{
./
f01f)= _i~~m-'i !lpif(.Q)(~m_,,~~-~~/.l'.Q~J (t d~
I~
-
The limit s in (28),(29) exist for fallin g into the area of system s tability (31) with respect to parameter o Upon fulfilling the frequency condition
I
f~':!, io(i}J2(t} -
t In contrast to t~e above discus c: ed case, the excitation in Eq. (23) is a correla ted one, and it entails a change of the averaging scheme. As shown in (I(ogan, l' eimark,1990), in the case under discussion, the equation of slow variables obtain the
d
.m-,,... 8;)
IJ q Jl4lEf:l,.l]
-.2)
(32)
along the trajectory of the system (27) there is fulfilled the equality
and the control strategy we choose from rr the condition .Qt ~ = 0, Le. in the form
= - I~t)
",.
Q =(-anz,''''-a.,•-G
Re [ G (eitJ )
7'
Ut
,r
Sc> = (.Q ,I),
j
I
-6;," (S2* +1).
(33)
Note that in (Ljung,1977; 3in,Goodwin, 1982) dealing with identific a tion by extend e d least-squares method the condition (32) is used to prove the convergence of estimates for correlated excitations. Hence, despite the nonidentifiability of parameters.o", and in spite of the correlated excitations resulting , in the process of identification, in some biased estimates, there appears some control strategy (24) in the adaptive system we discuss. Above there has been discussed the adaptive control strategy based upon the locally-optimal one with respect to the simplest criterion equal to the object output dispersion, this strategy providing the stability for objects remaining stable under locally-optimal control (24). The results obtained can be applied to the case of more general criterion, with due modification of the form of the control strategy and that of identification algorithm (Kogan, Neimark,
Adaptive Control of Unidentifiable Objects
1990). At this, the condition (32) is reduced to the form
Re C
.1
.
CS (e Jl.J)G-(e'4I)
-
.!....]>o
,2
tr't,}€I:1r.1"] ,",
,
and the class of adaptivity embraces those objects (19) for which the polynomial
S{~-")B*(£/) + J)(E-J)A*(Z-f) remains stable. CONCLUSION The main conclusion in this report is that an effective adaptive control does not necessarily need a complete identification of the object. For the revealed class of locally-optimal adaptive control strategies, the problem of formulating the control strategy is solved without identifying the object. REFEHENCES Anderson,B.D.,and C.R.Johnson(1982). Exponential convergence of adaptive identification and control algo rithms. Automatica,18,1-13. Astrom,K.J.,and P.EykhofIT1971). System identification- a survey. Automatica 7,123- 1 62. BogoIUbov, N. N. , and Yu .A.I.li tropolskiy (1963) Asymptotic methods in non-iinear oscillations theory. Nauks, toscow (in Russian). Chen,lI. F . (1982). Self-tuning controller and its convergence under correlated noise. Int.J.Control,12,1051-1059. Kogan, m.I',i. ,and Yu. I.HeimarKT1983). About substantiating of the mean method for stochastic systems.In Dinamika sistem.Upravlenie,adaptatziya,optiffiiZatziya,Gorky,Gorky State Univercity,pp.42-67 (in Russian). Kogan, ~;; . LI . , and Yu. I .Neimark (1986). Adaptive locally optimum control. In Preprints of second IFAC symposium on stochastlc control,Loscow,Vol.2, pp .8-12. Kogan,I,I .T!:.,and Yu.I.Neimark(1987). Adaptive locally-optimal control. Avtomatika i telemehanika ,8, 126-13~ (in Russian). Kogan,l.i.L;. ,and Yu. LNeimark( 1989).A study of identifiability in adaptive control systems by the averaging method. Avtomatika i telemehanika,2,1 08-116 (in Russian). Kogan,J.I.~,;. ,and Yu. LHeimark (1990). The identifiability of locally-optimal adaptive control in indirect observations. Avtomatika i telemehanika, 1 (in Russian). Ljung7L.(1977). Analysis of recursive stochastic algorithms. IEEE Trans. Autom. Control,22,551-575. Ljung,L.(1977). On positive real transfer functions and the convergence of some recursive schemes. IEEE Trans. Autom. Control,22,539-551. Neimark,Yu.l.(1978).-rrynamic systems and controlled ~rocesses.Nauka, Moscow (in Russian • Saridis,G.N.,and R.N.Lobbia(1972). Parameter identification and control of linear descrete-time systems.
169
IEEE Trans. Autom. Control,11,52-60. Sin,K.S.,and G.C.Goodwin(1982). stochastic adaptive control using a modified least squares algorithm. Automatica,2,3 1 5-3 21 • --
THE ADAPTIVE CONTROL OF UNIDENTIFIABLE OBJECTS M. M. Kogan and Ju. I. Neimark The Chair fill' Thl'on 01 COl/lro! al/ri DYI/(11l1irs 0( Mochil/f's, Corh Slalf' Cl/il 'I'rsit)'. (;"rkr. L'SSR
Abstract . ':ii th the help of averac ing method there i s stu d ied the behaV l our of adjustab le ~a rameters in adapt i ve contro l sys te m with identifie r. There has been found a comr l ete class of contro l strate g ies c alled locally- optima l; for this cla s s the desene rati on of information ma tri x and the bia s of estimc tes fo r unknown ra rameters of the objec t do not ~reven t from i de ntifiability of ~2r a meters of these contro l strateeies . Keywo r ds . Ad2~tive control ; i den tifi ab i li t y ; l ocal ly- ortima l control ; avera~ ing me thod .
le as ~squares
metho~
I1:THODUCTION use of c ontro l and i dent ification is known to c ause Rome definite difficu lties. Fi rst , avai labili t y of feedback entails a dec enera tion of in f ormat ion matrix and consequently a non i den tifie bili t y for initial I ararne t e r s of the object . Second , in the ueasurewents of outj.ut there occurs some correlated disturbances th a t r e s ults in the bias of est i ma tes obt a ined . Hence they r e2 ,rt to v8r ious l'l ays of sUj:-pl i rnentary influence ur on the object ensuri ng its i dentif iability( Sa ridis, Lobbia, 1972; Anderson , Johnso n, 1982 ) , nnd to rreven t from b i as they turn to various extended method s of i den tification (j,strom , Eykhoff , 1971 ; Chen iian I'u , 1982 ).
l i mit man i fold i s determ i ned by the kind of functional derend ency of regUlator rar8r.lete rs uy.on the objec t parsrne ters . At this it turns out that wi thin the entire lin'i t manifold the r arm"eters of local lyoptimal control stra teGies obtain the values similar to the real . In such adaptive systems some kind of ad,iusting regu l a tor p::rame t ers occurs , ;:..nd in this sense the,e systems do no t a t sll di ff er from di rect r y s t ems of adartive cont r ol .
J~tual
The resu lt s we d i scuss are arranged in the pqrer as follows : first we d i scuss the study of averaGing equstions d es cribing the dynamics of estireates in the identification proc ess , we have found the equa tion of manifold they tend to ; i n the next section there a re considered the ada r tive locally- opt i ma l con trol s trate puramete rs of which wi thin the g ies entire limi t monifold assume the values c6incident with the r ea l; in the l ast sect ion the r esu lt s obtained a re ap plied to the caE;e of nonl":easurable st a te .
In this reto rt we d i scuss the resu lts obt a ined by u s in the study of adapt ive contro l of linear descrete stochast ic objec t s (Fe i r:lark , 1978 ; EOGe n , J: eirr.'lrk , 1986 , 1987, 1989 , 1990 ) . Our m ~ in re su lt includes the fo l lowinG • ..'1 thin the cla~;s of f e edback relations i t becomes pos s ible to distingu i sh the contro l st rat eGies ca l led l oc al ly- optimal for which , de s pite the non i dentifiabil ity of the obje ct 8nd the bias of estir.:ates , there appears the identifiability of [8rarneters of these contro l stra t eGies , and hence the contro l [ roblem is solved .
TIlE S'l'UDY Q}' I lJEI:TIF L\B I-
T..ITY BY THE iETi iOD
AVE i~ .\CIr:G
~
S
1et the vectors E Re and AtE R to be subjec ted to measuremen t s be fo r al l t =C,1, .,. interconnected by relation
Th i s ~ tate~ent h ~ s been made ul on the appl ic ation of everae ine method to the e quations descr ibing the ope l~ ti on of adar tive system . In these equot i ons the recurrently obtained estimates change s l owly as cor;\p ared to the r ha se v ari able s of the object . The analy sis of deterrdn istic di fferent ial equations of slow vari ab le s has revealed t hat within an estimate siace there is ava ilable some asymr t o tic ally stab l e man ifold of equilibrium s tates . The av a ilability of such man i fo l d reflects the fact of non i dent ifiability of the objec t, with the feedback f.resent in the syst em. The equation of
(1 )
vlhere Q* and {tt)
i s an unknown matri x eX$ is a s equence of inde ;:.enden t
random vectors, and
Mtt - ~
Mttt/'a le
Acco r ding t o the leas t- square method , we define the sequenc e of estima te s for matri x .Q1f through the following re current rela tion
Ili:)
M. M. Kogan and Ju. 1. Neimark
166
jiff' .. SJt
+ It
~ ~ ( At+{-.Q;tfp ~ (
2)
is s ome a rbitrary matrix where J2q RX$ and the mat rices It are given by the formula
I t +d
=
It
rl-.
'T'
~ > O.
qJt '"'t,
1-
O:ote that vector It>t pend u~on the estimate
(3)
in (1) can de-
.ot
)
.
To analyse the system (1) - (3), let us introduce s ome small j::aramete r c> 0 -"f by rep lac ing the . I .f, by;~ f we obtain the followlng: -1
~#i .Qi - c~#{ ~ +eI- 1 CP. ~", t+.f
Ii1
et; (-'?t -.0,..) ~
( )
4
fo
(5)
i >t
7'
~ I- e CfJt 4Jr .
( 5)
The Egs. (1), (4), (5) comp letely describe the identification [ rocess with resp ect to the assiened t • AT-p lying the averaging method (Bogo ljubov, I.:i tropo l'skiy, 1963; Ljung,1977; Kogan , Ne imark , Ron in,1 983) to these vectors wi ll y ield dif f ere nt i a l equ a tions of s l ow variables
91,2
.Q - -l-{P(.S2)(.Q-.Q*)
(6)
j - P(!;). The matrix function P(.Q) in (6) is dete r mined through the rel at ion .Ir'-d
P(.Q) .. !,im.N-.fI
ct;(.Q)ctfC.Q)
(7)
./I..,.. 00 t=O in which the sequence {4it (Q)j is pro duced from the initial s equence [qDt) if in it for al l t=O, 1 •.• there is de. Fo r the case c l ared the S)t = Q when vector q5t in (1) does not depen t ul,on .Qt Vie obta in CfJ/.Q) ~
CfJi
I f the matrix P(.Q) is determined positively, then the system (6) obtains a unique asymptotic stable equilibrium state Q c.Q... corresI;ondinr; to the object's identifiabili ty. In the t; ene: a l case , the manifold of st a tes for the sys tem equilibrium (6) with respect to.Q is described by the e quation
P(S2) (.Q - J)*)
=
o.
(8)
When i dentification is acco m~ lished in the course of control [rocess there occurs a degeneration of matrix f)(J.?) due to some de~endencies existing between components of vector ~t:
~ -: (CP ,'1' Cf.J."7') t t, t , where matrix
(Y(.5J)
cp = B(.Q)cP.' t
K
t
t,
(9)
is determined by the
selected control strategy. This fact yields the fo l lowing representation of ma trix P(.Q) :
P '(f)) 6)(.Q)
f)'{Q)
P(~) = ( [;7'(.Q)f)(SJ)
)
1Y{.f2)P(Q)B(.Q) ,
(10)
PCQ) >0.
will f converee to the man ifold defined from(8 ) by the equation In this case the variables
si +6)(52)J)" = Sc>; +6)(.Q)Q:
( 11)
.Q 7' = (.Q ''': .Q'I'1') • The Eq. (11) in c; eneral c a se rer:resents some [-lane to which the ma trix S)t will tend when t increases. Thus , in case of adartive control, general ly sr:eaking , no identification of matrix oc curs but it becomes evident that the limit mu trix satisfies the Eq.(11). From the noni de ntifiabili ty of t:atrix 5.'. it does not however follow t ha t ma trix .Q,.. is uncapable to r:roduce any functions t hat on the contra ry a re identified. The function S(52,..) will be such iden tifiable function if and only if it is of the kind (Kogan, Fe i ma rk,1 989):
(;}(.Q,.) =(8 -.o;)!{(.Q: HII,
(12)
where H i s sor.le arbitrary matrix, such that the ntGtr ix.Q:H i s square and a degenera te d one, and S is an arbit rary ma trix of corres ~ onding dimension . The identific a tional fe a tur es srotted in the course of control proce s s and during the r elated degeneration of informati on m8.trix retain inherent not only to the least-squares method but a lso to its mod ific a tion, in which
It+;
1;+.( Ee , dt/i 1t +ff ~ ;2, t;, > 0,
and also to the i de ntificntional Irocess done by stochGstic approximat ion method , in which
~+/ '"' Jt+/E!, 1;1-/ -;; +1, liere
Et
i s a unit rna trix
THE IDENTIFIABILITIY OF LOCALLY-OPTn-:AL ADAPTIVE CONTHOL Let us consider the object of control desc ribed by the equation
tt ,
~t+{ '"' A*~t + B~Ut f in which ,xi E R117 is the
(13)
st a te to be measured; and {~t} are the nonobserved random disturb a nces; andM~t 0, M§tt;= /(; and A* and B* are the unknown If x ID and m x k matrices. Se lect the following form of the control strategy
Ut
= 6)
'1'
(.Qt)X
t
(14)
Adaptive Control of Unidentifiable Objec ts
where IY(.Q) is an assigned matrix func7' tion, and ~~t is the current matrix estimate 5.>. = (A .. , B.. ) obtained from equations (1)-(3) where ",
",
'1'
.xt+! = t +{, ~ = (Xt , Ut ) • If the ~t = S} fixed and i f the control process (14) with J)t ~ ~ ensures an asymptotic stability of the object, then there exists the limit _( -t'-{ "'\.
a:
tim AI
..N-~
Z'
t=O
~f
XI'" .. P'(Q) ~
where
and the matrix P(5J) obtains the form (10). Under the abovesaid, the ~~ in this case will tend to the plane described by equation (11) that in this case will be of the form: ".
",
(15)
=
7'
(1 6)
G) (AN, B~)Xt
provided it, in acco rdance with (12),will be of the form 7'
7'
7'
-{
7'
@(A*,~= (H Bit) H (S -
A.. ) .
(17)
The control strate GY (16),(17) m ~n~ m~zes in each st ep the conditional ma t h ena tic a l expectat i on for the efficiency function T/"
Y
_
_
3
7'C
(X tf !) - (Xt+1 X t +!)
_
3f1 8)
(Xt"/~'I'
with the hist ory of the proce ss ava ilable. In (1 8 ) the nonnega tively defined ma trix C C 7' sa tisfie s t h e equality H = [JBN 3 ' and oX t+1 make s up the value of referenc~ traje~tory assigned by the equation ,xf:+' = S t:Ct where S i s the arbitrary matrix of related dimention. Thus, when choosing the control strategy of the form (14),(17), in t h e adaptive sys tem under discussion there is set up a de s ired locally-· optimal strategy of control
=
THE IDENTIFIABILITY OF LOCALLY- OPl'HIAL ADAPl'IVE CONTROL WITH UNr,IEASURABLE STATE OF THE OBJECT Let u s con s ider the one-input one-output object described by the equation of the form
Jltfl + = -G;,*Ut
..
*
a{!lt t, .. +am
+oo,+
!It-m+! -
-B;-./-
+.{
or represented in the operator form ~ t+! = 1jt )
( -1'!!
{,
A*(i!)jr 2
-1
1
-1
V
BIf(i!-,)Ut + G(l! ') t ,
l -"J
/!_(-I in which AJf (Z -I'), Bit i! , er % Jare the f polynomia l s in 2- ; and {Vt} , a sequence of independent random values ;
2 /IfVt
/If Vt = 0,
2
= () .
Vf ,.,.
-7'-
7'
= Cft CCJ>t, C = C •
lPt :: (1/t-mf-I'"'' Vt, lLt-mf-{, "', "-I-I) minimizing the conditional mathematical expectation of increment pe r a step a long the object trajectory, with the history of the process Given , i. e .
u; = aUI min M(~+ll{ja, .. ,.rt).
The Eq. (15) ensures only partial information about the matrices A.. and 8* though for the case zan.-:B.= K this in format ion will b e sufficient to formulate a desired limit strategy
Ut
+itVt + .. , + ~m Vi -m
For such objects, in (Koga n, Neimark,19 8 7, '199 0 ) there is formulated a locally-optimal con trol s t rategy with respect to some efficiency function
,
A + Bf) (A,B)= A. ~ 8,1f t9 (A,B).
167
(19)
Ut £* It has been found that for St~ +
d
-1,.0 this strategy i s descri bed by the equation
[$., Bit (;c-') +.D(£)G-(2-)}Uf =
(20)
= [S1 A*(i! -I) - $(2-)G(r-')jrYt , and the related characteri s tic polynomial of the closed system wil l be equa l to
L1(~ -J",G(ii'[S(Z-?B*(i!-1) +])(i!-)Aft(z-~l (21) In (20), (2 1) the coefficients Si and t/.;' of the polynomial s S(Z-I) and .1J(i!-') are_e xpressed through elements of matrix RIf of the C an d t h r oug h par ame t er v_ object. 0
First we consider now the particular c ase when the efficiency funct i on is equa l to = ~'f fand the locally- optima l con-t"rol s trate gy i s de fine d by the equation B.. (2- 1) Ut [Ai Z-I) - G{i!.-I)}:i!'Yt (22)
lit 11;,
0
=
and t he characte ri s tic polinomial (21) i s equal to
As it f ollows f ro m (21) the stability of the object under locally-optimal ~ontrol (2 0 ) requires the polynomial G(Z1) to be stable, i.e. the object has to be a minim~phase one with respect to its excitation. If not so , we take a control strategy in the form (20) where the polynomial ;;: -I 0-( ~ ) is replace d with the polynomial (2-') • In the_tbelow the line above polynomial from z will denote the substitution of roots ~ injit ( for which /ZdJ> 1 ) by roots z~ , i.e. if
G-
168
M. M. Kog-an and
G{;i'} =
n
and
1~/<1
/~/~J
0{
n
=
r U) .5) = (.5)'7' ~ If)
= ('
If,
~
if
'0
_I .JI~1
7'
t
(JJ) ,
(28)
PljJr{J)) =
(Q)
where ~(S))=
1 G-(~-? ~ (fJ) .
(29 )
(30)
In (28), (29) the components of t he vector
'I{'T'(52) = (J't-mt/RJ, ... , dl 1/ (JJ) ,U (-0, ... t t-m+1
r
7'
are determined as follows
,N-I
Let us represent the equation of object (19) in the form r fltt! Sc>. ~ of- ~t/$Vt +"'+im~-mfo{ ,(2 3 )
cpt
PSO$l'(.Q)
lo ffJ/.Q)
(I-i~).
/~/
This control strategy also provides s tability for the objects being nonminimum phase with respect to their excitation.
where
P.9l1' (.Q)
q,,(.Q):,I!!.':!,-'"
then
G(2- 1) = n (I-;i~~'J
I. Neimark
In these equations the matrices
(I-i/~)n (1-i!~!7fi),
1~/n
.Ill.
"', l1t - 1 (.5))) are determined by the followinc equation s :
,
052'7' * ... Cf.t*", -&m-I, . .. • .f..,) If . * = (- am' it
1-
In these denotations, with due account of the abovesaid, the control strategy (2 2 ) will be converted into the form
J
Ut
,
7'
= - 6:'" (.Q~ + f) 'It
where compongnts of vector
j
7'
J
(24) E Rflm-.f
;: (jm •... , 1/ , 0, ... , 0)
will be the related coefficients of polynomial G(i!~") . Having the transfer functions of the object a priori unknown with respect to their input and excitation, we nO'ii,.define the estimates of unknown vector .sltt through the equations
.Q i+1
= Rt
-f
+
rn
rrrF-,
I t+{ '-1f (!it+J- ~ '¥t J,
(25)
I tt/ = It + ~ £Pt ,
Q;r YJ
_.1 I nlr
-v;,
.JC
(26)
it
~-~
(27)
etI dt
-{
./
f01f)= _i~~m-'i !lpif(.Q)(~m_,,~~-~~/.l'.Q~J (t d~
I~
-
The limit s in (28),(29) exist for fallin g into the area of system s tability (31) with respect to parameter o Upon fulfilling the frequency condition
I
f~':!, io(i}J2(t} -
t In contrast to t~e above discus c: ed case, the excitation in Eq. (23) is a correla ted one, and it entails a change of the averaging scheme. As shown in (I(ogan, l' eimark,1990), in the case under discussion, the equation of slow variables obtain the
d
.m-,,... 8;)
IJ q Jl4lEf:l,.l]
-.2)
(32)
along the trajectory of the system (27) there is fulfilled the equality
and the control strategy we choose from rr the condition .Qt ~ = 0, Le. in the form
= - I~t)
",.
Q =(-anz,''''-a.,•-G
Re [ G (eitJ )
7'
Ut
,r
Sc> = (.Q ,I),
j
I
-6;," (S2* +1).
(33)
Note that in (Ljung,1977; 3in,Goodwin, 1982) dealing with identific a tion by extend e d least-squares method the condition (32) is used to prove the convergence of estimates for correlated excitations. Hence, despite the nonidentifiability of parameters.o", and in spite of the correlated excitations resulting , in the process of identification, in some biased estimates, there appears some control strategy (24) in the adaptive system we discuss. Above there has been discussed the adaptive control strategy based upon the locally-optimal one with respect to the simplest criterion equal to the object output dispersion, this strategy providing the stability for objects remaining stable under locally-optimal control (24). The results obtained can be applied to the case of more general criterion, with due modification of the form of the control strategy and that of identification algorithm (Kogan, Neimark,
Adaptive Control of Unidentifiable Objects
1990). At this, the condition (32) is reduced to the form
Re C
.1
.
CS (e Jl.J)G-(e'4I)
-
.!....]>o
,2
tr't,}€I:1r.1"] ,",
,
and the class of adaptivity embraces those objects (19) for which the polynomial
S{~-")B*(£/) + J)(E-J)A*(Z-f) remains stable. CONCLUSION The main conclusion in this report is that an effective adaptive control does not necessarily need a complete identification of the object. For the revealed class of locally-optimal adaptive control strategies, the problem of formulating the control strategy is solved without identifying the object. REFEHENCES Anderson,B.D.,and C.R.Johnson(1982). Exponential convergence of adaptive identification and control algo rithms. Automatica,18,1-13. Astrom,K.J.,and P.EykhofIT1971). System identification- a survey. Automatica 7,123- 1 62. BogoIUbov, N. N. , and Yu .A.I.li tropolskiy (1963) Asymptotic methods in non-iinear oscillations theory. Nauks, toscow (in Russian). Chen,lI. F . (1982). Self-tuning controller and its convergence under correlated noise. Int.J.Control,12,1051-1059. Kogan, m.I',i. ,and Yu. I.HeimarKT1983). About substantiating of the mean method for stochastic systems.In Dinamika sistem.Upravlenie,adaptatziya,optiffiiZatziya,Gorky,Gorky State Univercity,pp.42-67 (in Russian). Kogan, ~;; . LI . , and Yu. I .Neimark (1986). Adaptive locally optimum control. In Preprints of second IFAC symposium on stochastlc control,Loscow,Vol.2, pp .8-12. Kogan,I,I .T!:.,and Yu.I.Neimark(1987). Adaptive locally-optimal control. Avtomatika i telemehanika ,8, 126-13~ (in Russian). Kogan,l.i.L;. ,and Yu. LNeimark( 1989).A study of identifiability in adaptive control systems by the averaging method. Avtomatika i telemehanika,2,1 08-116 (in Russian). Kogan,J.I.~,;. ,and Yu. LHeimark (1990). The identifiability of locally-optimal adaptive control in indirect observations. Avtomatika i telemehanika, 1 (in Russian). Ljung7L.(1977). Analysis of recursive stochastic algorithms. IEEE Trans. Autom. Control,22,551-575. Ljung,L.(1977). On positive real transfer functions and the convergence of some recursive schemes. IEEE Trans. Autom. Control,22,539-551. Neimark,Yu.l.(1978).-rrynamic systems and controlled ~rocesses.Nauka, Moscow (in Russian • Saridis,G.N.,and R.N.Lobbia(1972). Parameter identification and control of linear descrete-time systems.
169
IEEE Trans. Autom. Control,11,52-60. Sin,K.S.,and G.C.Goodwin(1982). stochastic adaptive control using a modified least squares algorithm. Automatica,2,3 1 5-3 21 • --