Stability of Large Scale Systems Under Decentralized Control

1:2 , 1 - 1 I..\R(;(', S(\I.I, S\SIT\IS : ,\'\,\LYSI.S, S I,\BIUT\'

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(:0»\ right \(lIlIi(

ST ABILITY OF LARGE SCALE SYSTEMS UNDER DECENTRALIZED CONTROL I. P. Popchev and S. G. Savov 11/ \/;111/1'

11/ I }/d// \/J'inl f.\/Ii"! Jlf"/,( \ (/1/1/ nll/lllfir" . 1J1I1.!},"fIlHIII \( IId/'/ln 11/ .\, Inil n. , 11111/, (" /i"lIlill " \11 .. l if, 2, Ill" '\"/111, /i1i/Pll'liI

Ah st r ac t. 'Jl1C proh lcm Jea l t I,i l h ill thi,; pape r i o-; the decen t r:1i i:c' d ,; ta h il i ~~: lt ion oJ' I ill, ': ; r coiiTlimolls-t ime Ltrge ,;c li p sys t l'III'; ( I.SS) In- q:lt ic' ,; t a te Ccedh:lcks. Th t' ';llggc s t l'cl :lppro: I,'h con,; ist,; in a n ;Idditional dec ompo,; it ion oJ' t he 10,':11 s lIh c;"S tl' III';, lI S: lgl' oJ' hOlllhl,; o n tilt' ~oll1ti oll or 1.:';.Il'lIno,' matr i\: el{u;lti on , ,'ec'tor 1." :lpuno\- {'unction ;lI1d th e cOll1pari';OIl principle . Clur 111:1 in purpose is to OH' r C'OrlK' IhO irllport:11lt ,;ho rtco!llillgs the (':\ i Sl in~ :'1'pro:1c'l1L's : till' rcqll i rcllIcn t 1'01' S
or

KI')l\'OI'ds . Pecentl';! l i ~ cd contro l; 1.I';l jlI 1l10\" S 111('Thoci ; dl'c omposit ion; C01!lp' ll'i ":on pl' i rKi pl ,' ; sYi:lng ci I' 1110\' i ng \T h i c l es . r I)

I. I ST 01' SY\u\OI.S

'l iT cons idl'l'L'd in tilt' pil pC' 1'. ,\S"ll11l(, I h" t r" , i" dc,'\lII'po s l"ll int o \ intl"l'Connl'\,: t cd ~ llh~y~tl'Jll ~ 1 Si) LlL'~I..:ri ­ hl'd I,,'

the ~1J;lce or 1'l'i 11 rrL,n matl'icl' ,'; 11(7~J<1!I1 mn id entity m:rtri "

1I

Cl A , 1\) I

A\ II \ 1

m<,l:C: \ , ( \J

.\ ( \ )

min )" ( \ J

1

'i ( \J

Sin~lI l:rll' ~ III:I\:,

i 1\ )

~ mln"

1\)

"ml\1

11 \ 11 , c, (\ 1

1

C\ II \ I c (\ ) III

\': ll lI e elf a m;!tr i \: \, '

: / ~ ( \T\ I

~

'\11\1 :l

1II1nc , 1\ ) 1

t:

open l ert h:l l f or the comp lex pLI in 0

I' 1'C':1 1 nC'g:l t i \'C 11l lmhcr,; intl'~cr

\\h('l'e'

In,

,, ~ R r, It. ~ I~ 1;

, 1 '

'

1

\

.\

' ~ ln,=n;'~ l m,~llI . I'hc 1'1'01>I ··

1

)-

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icm 01' dlY('n tr:lli:cd ;;t : l h i li :~ 'lt i on 01' ,:-;, \ i:1 s l :l t i,' ,'ontro l kl's i - t o f i nd :1 "e t o f .\ 1",: :11 s tat e

\ .

Sl' tlof C' i gC'Il\';ll lIcs ot a Sqll: ll'l' IIK Ill'i, \

set ul

);11 n,

, 1

,' ( \ )

:'et

1 ~)

\/5;

,

sltch Ih:,lt 11ll' (}\'(' r;111 closed

I

I> /il\i + i,;, 1 \ / ,i; i =I-,\

:di

reil p,rrt of :l l'olllp l c, C'igell\':lillc' o r reil c i gcl1I':l lu (' 0 1' :1 sqtl:l l'l' 1'1;1 ) ri, \ 1II:1'C, 1\ ) , I ,

i"' J

\

i 1 :'pectr:i1 nOllrr 0 1 :1 m:ltri, \,

=

+ '\'1 x'I + g.tl. 1 r

fl' cdh:lcks It,~-l\" I pop ,;\'stC11I 1 I

1

',\(1\ I

I!I

.\

positi\'c dcfinit e nes,; ai' ,1 ';Ilnrnctri c m:ltri:\ \ :ero nllmh c l' 01':1 : cro rrK1tri" \\'ilh r es pect i,'C' dilllcns ion ,Irbitr:ln' IlKltri:c: hl o,'~ h' ith 1'(''':I'"','1 in' d illlcn si oi1 e igel1\'a Itlc of ;1 Sl'rrullet r ic IIkl t I' i \: \

nur;!Ill' r": m, l1\+ I , ...

, nl r :1 ~ n

r

T\TW)l1lJCl lO\

'111 (' rl' is presl'ntl y ;I gre:lt de' ll of intercst in till' ueccntrali:cd control of LS S and thc dc\'c lopmcnt or crfic ient desi gn mcrhods 1'01' them . ,\s i':u :l~ high dimcnsionalit\' 01 the model is :111 ou tst:mdin g [l'a ttlre l l'ading 'to s i gn ificant comput:l t iona l dil'f i c ultie,.:, :I "'idl' h' tI ":l'd appro:lch 10 1' so hin g th(, st::l h i1 j t l' prob l l'1II i s ha s ed on t h E' c ent r a I conc l'pt o r dl' C011lPO;; it i on and ] ,\'apunO\" 0' ~ec on ,1 11Iethod .

i s :In :1;;I'mptoticaill' st:lhl(' onc . It is "ell kllOh11, that :lh"el1l'c 01' un " t:1h! l' fi,ed ul1dcr de,'['lltr:11 i ~ l't1 control 1~I()dl's is ;1 ncc[' ss'IJ'\ ' COlldil iOI1 fnr 1'1'01\ kill " o lulioll l'"io-;ll'I1C l'. '111 i s ,' '''l1ditiol1 is ;11,;" ,I surt' ic iellt OI1C', i I' lhll:llT1ic ,'onITol krs ;nl' ;I!'p] ied. ,\ " rl':1t mrmher or ,lcsigl1 ll!C' th od,.: h:lSl'ci nn t ill' lIS;I)!l' (l i' :1 hci"h Tl'ci SIUlI !. \': Ipu no ,' l'lI l1Ct i OI1 ( ') il1"h , 1 ~ 'S IJ, and the ,',',: t or !.\': Ipuno\ i'tlnction L :-; ilj :IJ.., l " - SJ , [Ccrolllcl ;md Ik n1u SSOII, I ~ ) - ~) l app r o:k' lll's :Irc :1\" 1i la h l (' . \0 m:l tt cr ,d1ich o r t hcw is aplll ied, thc\' c xh ih it t,,'O COllllllon s ho r tc' omin "s , Fi r s tl l', th e' l'csIll'CI i ,'c stabi I i t l' c' ond it ion in cl lldc s (mo re ':l'nc 1':1 1 1\' s lx':lk in g' \ \I I'il :lI1d \n ll't l , ,,'here I' i ' i~T,\, arc ,;ol ut ions of eitI1C'r Ril' l'al i 0 1' L":lllUllm 111:ltri.\ cqu:l tiol1s, :Issociat cd hith c: ld1 ~lc~' ou)llcd 1\1. hpel' imcllt:11 c\:pcl'il'I1CC "h OhS , that "hcn \)1 Pi) » m(I' il for SOlllC, or :I l l i , ,,'hieh h!' (;11' i" nut :1I1 c,ception, the stahil i:in" :Il go rithm ol'ten di\'ergt's C\'l'1I ror ,,'cakl:' cOllple'd sull '; I'stems . Sl'e'ond !;', sol ll ti ons or \ Riccati or L!':!pllno,' cqllations arc I'c<\tlir cd :I t e:tch it c l';]tion, "hich 111;11' I'll' tilllt' - COIl " Ul~lill g rCc r olllc I :1Ilt! Ik llltl S:'OU , 1'1 - :1] :Illd : il ":I\'~ IC: lds t o (omplicatcd comput ;] ti oJ1 ~ , rhi.s p:lpcr IS ;]11 ;]tt ClIlpt t o O\'l' rCOIHC' t hcsl' shortcoming,;. Thc basic [llc;] i s to Ilse hotlnds 011 thE' soll1 -

2

I. P. Popchev and S. G . Savov

tions of Lyapunov matrix equations . Vnfortunately, the available bounds are restricted In application to some special cases . This is the reason why an additional decomposition of (5.), which makes possible to overcome these restrittions is performed . Our main result is a sufficient condition for stabilityof (3) . Some inportant features of the approach are commented. The error regulation problem for a string of moving vehicles is solved and clea~ ly illustrates the flexibility and simplicity of the present approach . Before that , we give some typical previously obtained stability conditions for decentrally controlled LSS.

~ i -'X';1' ll i =VI.UI. , - i i +B'u'x i i' i=(J.';1'.,u U~U. 1 1 where 'Xi = UIAiU i , Bi

RESULTS

I

AT

-1 T

A

(5)

are associated with CS' 1.) , where A.=A. 1 1 + aI n.1 , R.>O. 1 Provided that, Qi=LILi with (Li ,A i ) is an observable pair, the solution Pi is a positively definite matrix. Let the local control for CS i ) be -1 T

u i = -Ri BiPix i , i=T;N • The following are two typical sufficient stability conditions for the overall closed loop system , ob tained via the weighted sum Lyapunov approach [ Hassan and Singh , 1979) N

minAm(Zi) > 2~xAM(Pi) i~l 1

1

, Ei = diag{YlCB i )}, l=l,m i

Without any loss of generality it is assumed that

Consider the interaction free CS i ) CS'i) Xi = Aix i + Biu i ; i=T;N. (4) Let all CS'.) be controllable . The following Riccati equation 1 AiP i + PiAi - PiBiRi BiPi = -Qi' i=T;N

[_~L]

m.

Ei ERm

1

SOME PREVIOOS

UIBiV i =

N

j ~1 UAij 11 2

Hi

[Soliman and Fleming, 1981J N

L>O, L ~ RN ' L = {lij}' Am CZi) IAMCP i) , i=j liJ = { -II A U -fl A ll ' i;"j ji 2 ij 2 -1 T

where Zi=Q i + PiBiRi B/ j • The stabilization procedures are based on an iterative pushing the eigenvalues of the closed loop decoupled subsystems to the left of the imaginery axis by a distance a . Another group of similar stability conditions can be obtained by mR.king use of Lyapunov equations associated with C$.) , which are sta bilized by some control design approach , e . g . pole placement . No matter what type of equation is used, all approaches based on Lyapunov's second method exhibit two main disadvantages. Firstly, since nothing is a priori known about the solutions P., i= T;N the respective stability condition may ne~er be satisfied even for weakly coupled systems . Obviously,the worst case is Ivhen AM(Pi)>>AmCP i ) , i=T;N. Secondly , a solution of N Riccati CLyapunov) matrix equations is required at each iteration, which may be time consuming . The main purpose of the suggested approach is to overcome these two disadvantages . and to provide a stability condition fo r the class of LSS under consideration, which is more flexible, than the existing ones. ADDITIONAL SUBSYSTEM ' S DECOMPOSITION Orthogonal transformations can always be applied to

where r i = rankCB i ). Let us denote 1i = diag{Y l CBi)}' l=l,r i • Then in order to put B.in a form which eliminates the difference betweenlit's singular values and which is of importance for our main result, we make the substitution

13.u. = B. 11 1

[-~~~ 1 0] ;'1. =[~~L i 0 ] ;'1. = 1i.;'1.. 0 . 1 0 , 1 11 I

I

(6)

The transformed decoupled subsystem CS' 1.) becomes

t 1 = ft: 1.';1'1. + 1i.;'1. 1 1

.

(7)

R~rr~rk . The transformation of (S'.) in (7) can be 1 achieved via nonsingular transformations as well, but since the determination of U. and V. is based 1 1 on numerically unstable algorithms it is preferable to use orthogonal transformations for the purpose .

Under the assumption that all CS'.) are controllable subsystems , without loss of genefality we assume that oCft:.) G R-, et ('X.) ;" e Cft:.); t,p=l,n i , t;"p, 1 i 1 Pi 1 since a pole placement technique can be always applied to get distinct real stable eigenvalues . Note also that oCA .) = ocfi:.) , since the respective eigen1 1 values are left invariant under the applied transformations . Obviously, there exists a nonsingular transformation } . = T.y. by means of which (7) is transformed into 1 1 1

y. = A·Y· 1 1 1

+[:!~ 1 0 1;'11'; S2' • 1,

A.=diag{e Cft: i )} 1 qi qi=l,n i ,

(8)

where Sl' and S2' are matrixes, obtained by suita1 1 -1 . . ble partition of Ti accordIng to whICh Ti

1

=

[ :!~21 l' ~], SliE R:~, S2i E R~~-ri 1 1

(9)

Let the local control for (7) be chosen as u. = 1

[ :L~ELo Jj y.=1 .:'i<.y., 0 1 1

ri

G . 1; R 11 r

i

(10)

3

Stability of Large Scale Systems

We recall that the nonsingular transformation applied to CS i ) as a whole i s xi = UiTiYi = RiYi. Consider now the interconnected subsystem CS i ) and xi = RiYi applied to it . The interconnection mat r ix A.. becomes 1J

With ( 10) applied, the i-th transformed interconnected subsystem can be written as Yi =

[~l~=~l~:+~~l~_~-~-l Yi + -S2i S1iG1i ; 1I2i ( 11)

BOUNDS ON THE SOLUTION OF LYAPUNOV EQ.JATION The Lyapunov matrix equation plays a fundamental role in a variety of fields and especially in control theory for stab ility analysis, design of fil t ers, estimates of transient behaviour, etc . Due to i t s importance, there have been developed many numerical methods for efficient numerical solution. Al t hough this equation is a mere linear one , considerable difficulties may arise when the dimension of the matrix increases . Sometimes it may be more necessary to have at disposal some bounds on the solution, such as minimum and maximum eigenval ues , determinant , trace, etc ., than to solve the equa t i on itself. A summary of the bounds on the solution of algebraic matrix equations in control t heory may be found in ( ~lori and Derese , 1984 J from where the following result is taken. For "m CP) and "M CP) , where P is the positive definite solut i on of T

r. n.-r. HI' l ' (; R 1 , HI'1, 2'J E Rn.1 -r . , Hr l ' ER 1 1 1, J rj 1, J rj J J

n . - r.

H .. f R

21,2J

11

nj-r j

Let N2~N denotes the number of CS ) with n ::i:2 and i i ri
are ordered as follows: S1 , ..• ,SN , ...

,~.

The

sta:e[v~c~orTYJiTcan

be ~~COrdinglYR2n~:~~it10ned as Yli': Y2i' Y1i t R 'Y2i€ ,1=1, N .

Yi -

For a sake of simplicity it is convinient to introduc e new subscript notations , accroding to which 1i and 2i, i=T:N become: 1 1 ~1 , 21~2 , ... , 1 N2~2xN2 -1, 2N2~2xN2 ' N2 +1 ~2xN2+1, ..• ,N~2xN2 +N1.

y.1

L H Y p~Ie Ip p

pril+1,if It;I o 1 € I* (12) L H Y PEI* 1+1,p p Pfl L

pile

H

y,

1+1,pp

-

T I T for - CA + A »0 , where u2C . ) = 2 "M{( .) + C. ) } de notes matrix measure induced by ').!1 2. Unfortunately, the above restriction imposes serious limita tions on the validity of the bounds. However, ,.men A is transformed in the quasidiagonal form A = diag

1 e 10

+

(13)

p;tl + I

whe re Io = {I, 3, . .. , 2xN 2- l}, l e = {Z , 4 , •.. , 2xN 2} , 11 = 2xN 2+1 , 2Xl\l2+N1' I* = IO Ul 1 · Thus by means of a nonsingular transformation applied to each CS i ) , its state space is decomposed into two subspaces with dimension r = rankCB i ) ~ i mi ~ n i and n i - ri ' respectively . As a consequence the overall LSS is additionally decomposed into j * = NI + 2Xl\l2 subsystems , described by (12) and ( 13) .

{r e1

l-w1

W1] e

, ... ,

[e 1 ss w ,

-w

I

s

e

s

es
~ "mCQ)e~1CA) ~

"mCP)

~

"MC P)

~

-

~ "MCQ)e~1CA) C14 '

Then an equivalent presentation of the overall LSS is

+

w

A P + PA = - Q, A€. Rw ' o(A)e C ,Q>O the following bounds are valid

n.-r. 11 . € R 1 1 21 n.-r. 1 1 r.

VECTOR

LYA~OV

FUNCTIONS

111e concept of vector Lyapunov functions was introduced in 1962 , when Bellman and Matrosov indepen dently proposed an approach in the framework of differential inequalities and the comparison prin ciple, thus providing f l ex ible tools needed to ob tain new important results in stability theory. Since the present method is based on this approach 'lie shall pay some more at t ention to it. The set of N* differential equalities (12) and C 1 ~) can be written in compact matrix form as

y

=

Ay , YIi: R\'*

( 15)

Tt is well knOlYn, that ( IS) is a stabl e system if and onl y if

(16) is a s tabl e one . From now on we study s tability of (13) through stabilit y of (16) . 1Now, suppose that in (12) and (13) we set Gl = Pi ' where PI is a solution of the Lyapunov equation - (11 + BII)P

l - PlC lI l + BlI ) = - 2s l sI, l ~ I*

( 17)

Let us suppose al so that all Slare nonsingular. This supposition will be commented l ater on, in order to show that it is not a restriction for the

4

I. p, POJlch e\' and S, (;, Sa\'()\'

validity of the main result. It is obvious, that if we choose tha scalar pa 'c ameter ~ l > - em(Al) ' then PI

"'1+1 ~ el'1 + v l+ l + Yl +1{ 1 1

i s a pos itive definite matrix. Consider the set of N* decoup l ed subsystems in (16) -1 T 9'1 = (AI - PI 5 l 5 l )Yl ' 1 6 1* ; Yl+l

l:

+

pEle

A1 +1Yl+1'

1 € 10

l:

p~ I*

Yh v P 1+1 , p p

pn

Y- 1h v } , l eo, 10 p l+l,p p

( 25)

p~1+1

(1 8)

where

and the scalar Lyapunov functions vI =

?

y~l {>,~ 1 (Pl)YiplYl}1/,2 l ~ T 1/2 _ Yl +1{Yl+1 Yl+l } ,11:: 10

( 19)

(26)

e~ 1

,

: 1 +1

1*;

Then (2.)) and (25) can be I,Titt en in compact mat rL,{ fonn as

v~

\\Iv.

Y E: R , y >0 , l=l,N* , associated with them res l 1 pectively . The proposed Lyapunov functions can be evaluated and presented respectively as

111e fonohing diffcrentirll (comparison) s),stem corresponds to it

-1 vI ~ Y1"YIIL , 1 E 1* ; vl+ 1 = Yl +111 Yl +,u, l E To

f

(20)

T -1 -1 T (AI - 51 51 Pl )P l + PI (AI - PI 5151 ) = - 2~lPl '

Cl

(m

1

(21)

By making use of ( 12) and (13) I"e compu t e

VI = - ~lvl ' 1 (; 1* ; vl+l =

llf, f E R\* .

According to the cOllllKnison principle' , ~tabi 1it )' of (27) implies also stabil it)' of (L~) :lJld ( 13) . .\kh exper i ence is not required to conc lude' that ie 8

Equations (17) can be relvTitten as

H I*

=

are all I'ery high, stilb i 1 i ty oC (:7 ) may never be concluded and this is a gene'ral problem for all design appr oacl1C's of the t\1X' . Of course , one Illa)' usp RicCJti equrlt ion s , but this h' il1 not ch,lIl gl' things .

1 +1vl+ l , l E 10 ,

hhere Cl l +1 yi+ 1Al+1 Yl +/ Il y l ll 2 . It is quite obvious, that stability of each origina l decoupled subsystem is not affected by the application of (10) since for the corresponding subscr ipts i and 1

'!lIe next thpo]'e1l1 provides :1 sufCic ient condition [or stilb il it)' of tllC' d('centn11y controlled LSS

T -1 0(.'\) = orA l - 5 5 P )

'1lIEORE.lI. Let all SI ' l E l a he nonsingular. The LSS (3) I"ith

l l 1

U

0(fl + ) •

1 1

The time derivatives of (19) a long trajectories of the interconnected subsystems in ( 16) are computed as follows

(3) •

Ki • V i

UF ( 2R)

- ~lvl + [ grad(v l ) )

+

T

-1

T

{- Pr 5 1S1Yl+1

l: HTl y + l: HTlyr } , 1 p~l * P P p ~ le p

pn

Cl

~ 1* ,

is JS),lnptot i cally stah1e,

(m

~ = {I''lp 'J , l\he]'D: ~

for 1 E; 1*

pn+l , ifl €o lo

-1 ~

Y l + 1vl+l + [ grad(v l +1)J T{ pf:l: I* HT p, I + 1 P

BI 61p - Yl

p~r

j

19 lp '

~!

Y 1hI, 1+ 1; 1 E J 0

+

HT Y} , 1 E T0 pEl e p,l+ l p pn+ l l:

'\., ,\ , \J* if 0 ( - h ) C C' , \\ E: R~ *

•

(23)

L P By taking into account (20) and

P

1'= 1+ 1

; P E le, p~I+1

and for 1 E le f

=

one gets a system of N* differential inequalities

-1

Y h v + l: Y hI v } , l E: 1* , pU * P Ip P p p pEle p pn +l, if l E lo pn l:

-I Y h 1 ; I' E 1 E' , p P

+ ~1)/(e + ~1) } 1/2 , lE 1* m l

8'1

{nl (e\1

1'1 1

1'1 \1/1'1 1ll1 ' 1'1 \1 '1

1

phI; p p f l* , prl

0; p=l - l {

+

( 29)

I)~ 1

'( Ph Jp'' I) E. 1* ,

-1 ' Y hl

,

')

?

(24)

hr , l+l

y;{Sl)' n ' 1111

1'1 - 1 (e ml ,Ill + r.' l )

T

II sl + l S I 11

(30)

pfl + 1

(31)

YJ~/Sl ) , 1 E. 1* 1 f 10 ,

Stabilit y of Large Scale Systems

Y ' ~l and hIp are defined in (19), (21) and (26), l respectively and alp is the Kronecker symbol . PROOF, Consider the set of N* differential inequalities (24) and (25). Let us apply (14) to get bounds for Am(Pl) ' A~l (Pl) and 81 , Having in mind(21) the following bounds are obtained

all required computa tions are carried out for lower dimension matrices, or at least for matrices of dimension equal to the order of the local subsystems. In this sense the proposed approach retains, or even improves the advantages of existing control design methods based on decomposition. It is believed, that this fact is of a significant importance, Imen a really LSS problem is faced.

By taking into accoun t the :lbove estimates anu (31), (24) becomes

4. The additionally perfonned subsystems ' decompoSItIon naturally brought to two more advantages, Firstly, there is no need to solve the N* Lyapunov equations at each iteration, but only once, after stability of (33) is established, in order to get Ki' i=~ . In fact this is not a problem at all . At the same time, this facilitates considerably the computational algorithm itself. Secondl y, it makes possible to use always val i d bounds on the solution of (21), which are applied here contrary to the general practice for design and not for analysis. This fact makes possible to go into greater details in the decentralized stabilization problem, when Lyapunov's second method is applied. It is evident that to solve the problem it is necessary to increase the parameters ~l' 1 E I*. This leads to two important consequences'; First of all hl,l+l' 1 E. 10 are also increased. That is the reason Imy the Lyapunov functions (19) have been chosen to incorpora-

(32)

te respective multipliers Yl' l= l, N*, by means of which it is possible to decrease the values of hl,l+l' l~Io. By means of an appropriate choice of this parameters, it is also possible to decrease the magnitude of the greatest off-diagonal elements

nm/Ce~1 + ~l) :;; Am(P l ) ~ \1(P l ) :;; n0'I/(eml + ~l); 81

:S

~l .

Consider no\\ (24) and the follQl,ing estimation , ob tained directly from the above given 10her bound on Am(P l )

IIp~ \

si +111 2 :;;

IfP~ 111 2 1\ SI si +I" 2 A~l (PI)

:;;

+

1:

p~l*

pn

yh

P

v

Jp

P

nl~l: (~\\

Il slsi+1 11 2 + ~ 1) " Sl S'i'+ 111 2 ' 1 (; 10 .

1

+

~

I*

Then (32) and (25) can be written in matrix form as

v :s -i\fv .

of Vi. Secondly, 8'1 ' 1 E 1* are always decreased, IVhich is favourable. But having in mind (31) it becomes clear, that i f some local decoupled subsys tems eigenvalues are unevenly distributed, i.e. B «~~ , then it is necessary to increase the ml

The respective comparison s\'stem , corresponding to (32) and (25) is Z = -Ivz , (33) where \:;; is defined lw (29) anu (:)0) . It is knO\\TI (Si ljak, 1979] that 'stahil it)' of C)~) all,ays implies stability of (27) , sincE' l\' ~ \\ (th e inequali ty is taken element by el ement) ilnu consequently of the original LSS (:;1 h'ith Ki' i=l,'1 given by (28).

Remarks. 1. The requirement for nonsingulari ty of all SI ' l E 10 is not a restriction at all. Since all Ti are nonsingular matrices, there always exists a ri-th order non zero minor in Ti' h'h ich cm be placed at the required upper I eft pal't of T~ 1 by means of 1 suitable transfonnations. If there exist more than one such minors, it is recollUTIended to use that onc for which ill and/orIlS I +1 2 are min imlUTI or nl m is maxiffilUTI. This \\ill resul t in a des ired d~crease in the respective off- diagonal elements of 11.

si'1\

2. It IS "ell knO\\11 thclt (33) is s table, if and only if \\' IS an \1 Ol ,let: ler ma tri" since it is matrix with positive diagonal and nonpositive offdiagonal element s . Such matrices arc cal led ,I if and only if the follOlving equivalent conditions hold: i) iV- l exists and all i t s entries are R,0nnegative; ii) all leading pr.incipl e minors of 1\ are positive; iii) o( -\\') C C . 3. Due to the additional decoposition

perfol~ed,

I

1

respective ~l considerably in order to decrease ~1 in the same way . This obviously will result in high gain magnitude . lhus, it is possible to draw t he important conclusion, that in order to get a priori small ~l ' 1 E 1*, an appropriate pole placement for some ora11 decoupled local subsystems should be initially performed. 6. If we compare the suggested here approach with the simi lar ones, greater flexibility of this one is an outstanding features. Tt consists first of all in the idea to perfonn an initial appropriate pole placement. The choice of the transformation matrices U and Vi is by far not an unique one and i this fact provides another possibilities. The usage of bounds on the solution of (21) clearly indicates "'hat should be done in order to facilitate the solution itself. While, since almost all existing similar approaches are based on the idea of sufficiently stable local decoupled subsystems, to increase the local degrees of stability remains the only thing to be done. At the same time, the solutions of the respective Riccati (Lyapunov) equations are, not surprising a priori unkno,,~ and this fact may naturally lead to divergence of the algorithm, even for ",eakly coupled LSS. APPLICATION: A STRING OF MOVING VE-lICLES To illustrate the proposed approach, the error regulation problem of a string of moving vehicles is considered . We use a model proposed inCLevine and Athans, 1966), It is assumed that the system consists of a string of four vehicles . The motion of each of them is represented by two states: position

6

I. P. Popchev and S. C. Sa\"o\"

and velocity. After normalization the system ' s model can be written in the form ( 1) , where t he state vector is x = ( vl, dl Z,vZ , dZ3'v3 , d34,v4 J T, Vi being the normalized (dimensionless) velocity deviation of t he i - th vehicle and d 1, . .1+ 1 is the norma l ized (dimensionless) distance between the i - t h and the (i+1) - th vehicles. The control vector has four components, each of them being the normal ized (dimensionl ess) incremental force applied to the i - t h vehicle . It is required to stabilize the overal l sys tem by means of l ocal control laws that use onl y information about the respec t ive local states. The system ' s model is naturally decomposed to take the form (Z), Imere Ai

=

f- l

D

1,

rl' 1

B.l =

LDJ

L lD..!

A1Z =A Z3 =[ D

D] , A34 = [ 0]

-1 °

-1

•

i=T;3. The diagonalizing matrices Ti are computed as

[ 1_1 -e

-1 -1

eZ

l

1

, i=T;3

and when applied to each (\) , i=1 , 3 we get 1\ . 1

1 0 0 -e

[ -e

lh -h

1 Z

'

e21

el B. = f /(e l 1 Lezl (e l - e Z)

r- h ] ~ ] , A34 = L-h

A1Z = AZ3

,

where h=e l ezl( e l - e Z) ' When the local control laws (10) u.

1

are applied to the system , it is transformed 1I1 t he decomposed form (lZ) and (13) . Here Pi ' i=~ are scalar solutions of respective Lyapunov equations (Z l ) and are given by Z

Pi = ( e,l(e, - e Z) ] /(1\- e , ), Il i >e l , i \= 10;

For this particular case 10 = {1 , 3, S} , le = {Z , 4,6} I* = {1,3,S , 7}. The aggregate -~ i s constructed

["

iJ. 117

w = eZ(ll - e )/e , i E 10 i l l It can be easily shOlVll, that o(-~ C C- for all Il . 1 and e l , e Z (e . g. condition i i) for ma t rices M matrix. The gain matrices, stabil i zing the overall system are computed as

\ i=T;3; A4=-1, B4=1,

The rest of the interconnection matrixes are zero . Since all Bi are in the required form (6) , and a l l local decoupled subsystems are control l able , we go on with the next step in the decompos i tion scheme for i=T;3 . Before tbat , it is necessary to stabili ze asymptotically (~i)' Let us choose - e l and - e Z' e l >e Z>O as eigenvalues of all (for simplicity) closed loop (~.), i=T;3. Then it can be easily 1 shown , that

Ti =

~=

0 0 0 ° ° ° eZ -h -" -h ° 113 -w ° ° -h -h 0 e ° 0 ° 0 o 0 -h - h Z IlS -w - h -h 0 e oo ° 0 D 0 -h -h 2

K~

+

KiT~l

= [ Il l + e Z- 1 e zll l ] , i=T;3, If: 10; K4 = 1l 7- 1.

It is quite obvious, that one has an a l most infinite degree of freedom to choose the loca l ga i ns . At the same time any attempt to solve the problem by applying design algorithms , proposed in [ Hassan and Singh , 1979J , LSoliman and Fleming , 198 1] , et c . , have failed to give a stabi l izing sol ution and t he main reason for t hat was that AM(P ) were constantly i kept considerably greater than Am(P i ) , i=T;3 . CONCLUSIONS The present paper considers the decentralized stabilization problem for a c l ass of LSS . Our main result is a sufficient condition for asymptotical stability . The ma i n purpose is to perform an aditional decomposition of the LSS , thus making possible t o use bounds on the solution of Lyapunov matrix equations . One can distinguish three mai.n steps . 'The first one includes appropriate pole placement for some or al l decoupled subsystems. The second consists in performing the additional decomposit i on of the intercOlmected subsystems. The last st ep results in the construction of the aggregate matrix ~ and check the particular condition for stability. All computations are performed for low dimension matrices and since solution of Lyapunov matrix equations are not required at each iteration, but only once , the computational algorithm is rather simple . REFERENCES Geromel, J .C., and J . Bemussou (1979) . St abili t y of two- level contr01 schemes subjected to structural perturbations.lnt .J.Control Z9 , 313- 3Z4 . Hassan , ~1. F ., and ~1. Singh(1979).Controllers for linear interconnected dynamica 1 systems with prespecified degree of stability . lnt.J.Syst.Sc i., 10, 339- 3S0. -Soliman, H.M. , and R. J . Fleming( 198Z). Optimal scalar Lyaplmov function for decentralized stabili zation of interconnected systems . Electronics Let ters , 18, 993 - 994 . Morl,-r. , and I . A. Derese( 1984). Abrief summary of the bounds on the solution of algebraic matrix equations in cont rol t heory . lnt .J .Control , 39 , Z47- ZS6 . -Levine , II' . S., and M. Athans. On the optima 1 error regulation problem of a string of moving vehic l es . IEEE Tr ans . Autom . Contr .,1 1, 3SS - 36 1. Singh, M.G. ( 198 1).Decentral ized Control. North-Holl and , Amsterdam . Siljak , D. D. C' 978) . Large- Scale Dynamic Systems :Stability and Structure .North -Holland , New York . -