Asymptotic stability of large-scale systems with application to power systems. Part 1: domain estimation

Asymptotic stability of large-scale systems with application to power systems. Part 1: domain estimation Lj T Grujic University of Belgrade, Belgrade, Yugoslavia

M Ribbens-Pavella University of [_i6rje, Li~@;, Belctium

Slabili O, o f nonlinear, stathmary, large-scale and contim~ous-time syslcills is treated withitt the general context ~ff the deeompositk)n aggregation method, attd estimates (if the asymptotie stability domain are estabfished. The results o f the study are intended for transient stability analysis o f electrical power systems; tho' must therefore be the least conservative possible. To meet this general requirement o f relaxed stability eonditions, the paper discusses stabiliO' conditkms at the subsystem level as well as estimatk)n (if" the overall httercomwcted s3,slern. I. I n t r o d u c t i o n During the last two decades, much research has been devoted to stability analysis of large-scale dynamic systems in genera/and o f power systems in particular. Within this framework, the decomposition aggregation method based on Bellman's approach seems quite attractive I - 4. The following outlines this approach, which will be used in the present paper.

bet a large-scale system, described b y the nth order differential equation = f(x)

(1)

be decomposed into s interconnected subsystems of the nith order Xi = f i ( x )

= gi(xi)

+ hi(x )

V i = 1, 2 . . . . .

s

(2)

The vector x is partiti(med :is x = (x(', x ~ ' : . . .

, XsT )7"

fi(O) = 0

gi(O) = 0

We introduce s scalar functions V i : IF{ni > []:={, associated with tire ith disconnected subsystem "~i = g i ( X i )

V i = 1,2 . . . . . s

~, )/"

Vol 1 No 3 October 1979

By so doing, we transform the s/ability analysis of equation (1) to the problem of testing appropriate sign properties of v and of z). More precisely, this analysis comprises the following steps: •

system decomposition into interconnected subsystems

•

choice of the aggregation functions Vis

•

sign tests of z, and z;

•

suitable estimation of the asymptotic stability domain.

The first two steps, considered separately from the last two, provide several possibilities. Essentially, the problem consists in choosing such an aggregation decomposition form and V function that sign tests of v and z) can be effectively performed together with useful estimation of the asymptotic stability domain.

(3)

easier construction of Vis for tim subsystems than construction of a scalar Liapunov function directly for the overall system

(4)

a reduction of the order of the aggregation matrix to the number of the subsystems

and a vector function V = IRn --, IF{s V = ( V l , V 2. . . . .

x ( O , x . ) - x,,.

The above short review reveals the following itnportant conceptual advantages of the 'vector' approach over the standard scalar Liapunov method:

and the vector functions f and gis ()bey f(O) = 0

which we shall use to aggregate the system in equation (1). This function will essentially represent a vector aggregation function of equation (1). In addition, we shall introduce a scalar function ~ in V: t~ [V(x)] will be denoted by v(x). Then, we shall determine the total time derivative f, along the system motions in order to examine the time evolution ~' Ixtt: xo)l of v, where X(/; x,, ) is a motion of equation ( I ) ,

0142-0615/79/030151-07 $02.00.© 1979 IPC Business Press

151

•

the derivation of effective criteria for the (asymptotic) stability of x = 0

•

the consideration of structural properties O1" the system and the influence of variations of its structure on tile stability

•

the possibility of inferring an estimation of the (asymptotic) stability donrain of x = 0 from those of Xi = 0 Of disconnected subsystems and therefore the possibility of easier parameter sensitivity analysis

•

easier analytical estimation of tim stability domain.

Despite these potential advantages, the 'vector" approach has two main drawbacks, namely: tile number and roughhess of nmjorizations, and the requirement for the exp()llential stability of x i = 0 of the disconnected subsystems in order to o b t a m a good estimate of the asymptotic stability domain o l x = 0. Tim present paper attempts to overcome these drawbacks by reducing the first and eliminating the SeColld.

Notice Ihal eqllal toll (5) defines a s ; s l e m ag~lcgu! (,ill 1! .. links those ol'relerences o and 7. In comparis, m wi~l~ reference 7. it allows lhe Iunctions ui to he :rob p,~sill\c definite functions on the sets [1_.i lathe1 than to he /[i,>,< dependenl on Itxill + and related to scalar Liapun{w Ikiilc lions (`il disconnected subsystems in equation (3) and simultaneously to be strictly increasing, l'hc pl~qx~>'d system aggregation in equation (5) also allows a l-sl,,* aggregation and stability analysis of the over'all sbstcm. In conlparlson with rel0rence f~. it provides a lalgel ,'sl imale s01 of lhe 'asynlphHic slability dornahl. In section IV we shall also use Assumption • and Lcmm:~ related to a Metzler matrix A = (oeii), i.e. such lh:tl c,i/ >z> 0

i :~ j

|'i{gl2Xi)

Througlioul the paper, the following two assumptions will be used: I1.1 A s s u m p t i o n 1 There exist a connected neighbourhood ILi of x i = 0, functions V i = IB ni ~ [R and ui: IR "i ~" IB, and real numbers Oti/, satisfying the following: is tile unique equilibrium state of the system equation ( I ) i n IL

(a)

x = 0

ot

(b) V i is positive-definite and different(able ('ill ~-i Vi = t , 2 . . . . . s ui is positive-definile on I1/

(c)

V i = I, 2 . . . . .

s

(d) the total time deriwitive ~/i of Vi taken along molions of equation (2) (,boys

(,)

Vi=l.2 ...... Vk 1.2 11.2.1 Lemma 1. If A i s a M e t z l e r m a t r i x , the lifllt>win~ are equiwltent : (a)

A is stable

(b) AsatisfiesSevastianov

(

Let IL i be a subset of []:l"i and I1 be the Cartesian product e l a l l II/s, I1 = IL I X I L 2 x . . . x [ L S.

iff 0 <-~/at <2 g*2

V~kx,~ INi(IL,

~'11 , O712 . . . .

II. S y s t e m aggregation Consider the large-scale system of equation ( I ) and its decomposition into s i n t e r c o n n e c t e d subsystems shown by equation (2). A motion of the interconnected subsyslelll denoted by Xi([; xo), Xi(0: xo) -= xi~,, is supposed 1o exist for every x<, G Ilq n and to be tinique and coi]tinuous ill t ~ []:::1~ for every x , C ~ n V i = 1 , 2 . . . . . s.

. . . . . ~.

11.2 A s s u m p t i o n 2 The functio]] I i of Assumption I is radially increasilLu Oil :m di-neighbourllood :~ IN/( [t-i, ~i) o1 l1 i, ['i(,U I Xi) <~

This paper has been divided into two parts. Although the theory developed in the second p a r l is quite general, it ix intentionally oriented towards power system stability analysis: in facl, some of the results presented here have already been used in previous studies uiidertaken by tile authors (see references 1 to 10 of refererlce 5). In addition to the complete proofs which are presented lor the first time, tiffs part contributes several theoretical refinemenls, such as Theorenl 5 and estimates ~2 and '~/illti-oduced in section IV.

V i. i = I , 2

t i k [a'211 c~22 ..... !,

tlick'sconditions

, Q'lk

a!_,k

~" 0

V

k = 1.

2 ......

~

iU)

i

(c) if b = (/)1, f)2 . . . . . h~.)/' is determhled from

ATb =

(7)

c

allen c > 0 implies b > 0 (dr lhere is a diagonal matrix B with positive diag~mul such that ATB + BA is negative-definite. Proof of the equiwllence between (a) and (b) is ,.!iw:u i! reference 8,, between (u) and (c) in reference 9. aml hclwecn (a) and (dr in reference 10. For tile possible link betweeu (c) and (dr see Appendix 2.

III. Stability criterion A c r i t e r i o n for tile asymptotic s t a b i l i t y o l x O.icqu:~I,,i ( I ) is established ii! this section by referring to) Jele,cm:c~> u and 7. ~ II1.1 Theorem 1 If Assumption I holds and there is a diagonal mairix B wilh positive diagonal such that A/'B + BA is negaiive-deliniie, then x : 0 of the system in equation ( 1 ) is asvmploiic:~ll} slable.

~/i(x) - [grad V i ( x i ) ] T l g i ( x i ) + h i ( x ) ] -i. iixll (xTx) ] " :!: ~i([Li,~ i) = {x:inf(!l* Yli:Y ILI) ":~:il,¢'j '0 '~ All proofs are qiven in Appendix t =

152

L Gijlli(Xi)llj(Xj)

V

i

Vi-l,2

I

X ~ ~. . . . . s (5)

Electrical Power & Energy Systems

Notice that this the,,H-enl is also valid when i.e. when A is nol Metzlerian.

O~ii <

is the largesl possible estimate o f D that we can derive for chosen V. In order to establish c o n d i t i o n s under which V is an estimate o f D , we introduce the matrix E : (cO) by

0, i~j,

In partictl]ar cases determined by negative-definiteness of AT + A, it is sulTicien! l o s e ) B = I the identity s x s matrix.

I I I. 2 Corollary If Assumption 1 holds, A isa M e l z l e r m a t r i x a n d o h e y s relationship (6), then x = 0 o f the system e l e q u a t i o n ( I ) is a s y m p t o t i c a l l y stable.

('ij : [ r i l £ i ] + r j 2 ( 1

infllti(x

ri2 =

IV.

I, 2 . . . . . s (12)

i ) : x i ~ a~,/i ] l/f, t

(13)

sup l u i ( x i ) : x i ~- ~/il

17'

IV.2 Theorem 2 I f Assumptions I and 2 hold and if Eg°<:.0 then the set ~ / i s an estimate o1" D .

Estimation of the asymptotic stability domain

In this paper, 'domain" denotes a non-elnpty c o n n e c t e d set l h a l Call be. hill need not be, open. A definition and notions needed in tire second part s will he presented here.

I V. 1 Definition A set IF C IF{ is an estimate of D the asymptotic stability domain o f x = 0 o f e q u a t i o n ( l ) i f f

(i)

Vi,/=

ttere [Sij is tire Kronecker symbol, all Vis are assumed b o u n d e d and ril =

A positive result on a s y m p t o t i c stability o f x = 0 is often not sulTicienl l'or engineering applications such as p o w e r system analysis. Knowledge, or at least an estimate, of the domain of a s y m p t o t i c stability is needed. ( F o r its definition ':
~Sii)]O~ij

IFclD

This theorem presents quite sin)pie c o n d i t i o n s for estimating [D. No stability test o f E is required. However, since E is a Metzler matrix, E g ° <, 0 can be satisfied ilT E is stable Lg. Itence, stability of E is implicitly required. [hider the condition E V " < 0 we gel the besl possible estimate o f [D for chosen V. When the c o n d i t i o n s of T h e o r e m 2 are nol met, we look for other estimates o f [D under relaxed conditk>ns. Certainly, these estimates cannot be better than the '~'

estimate. If there exist positive numbers r/il and T/i2 such thai

~illl2(Xi) ~'< [',.(Xi)~gTi21l~(Xi)

(ii) IF is positivelyinvariant with respect to motions X(/: x,,) of equation ( I ) (in short, IF is positively invariant set of equation (1)): x,, ~ IF i m p l i e s X(t: x o ) ~ IF

V x i ~ [Li V i = 1,2 . . . . . s(14)

then the matrix K

I,ii:lr~i2

V t ~ IR~

I/2

=

(/{ii) is constructed

8,i+rbl"2(I

~ii)lc~ii

Vi,/=I,2

..... s ils)

Lel tire b o u n d a r y

0 IL i o f []-i exist

V i = 1 ,=,...,~

s

and T h e o r e m 3 is proposed.

In order 1o define tentative estimates IF] and E 2 of D , tire l'olh)wing t2` L:~are introduced:

Ti = rain It~,V~': i = I, 2 . . . . . .

u l ( x ) = b T v t x)

[ l"i (xi) :u2(x)=max/~w--i=

L h?

T2 =rain

['"' t)2

i

IV.3

vl

] 1,2 . . . . . s

1. ~

s

]

theniF z is an estimate of D.

J

(o)

~/i = { X i : 0 ~ - Vi(Xi) ~ I"~'~j

{10)

be a c o n n e c t e d subset o f I1 i which contains x i = 0. Besides 0 ~ ) will denote the b o u n d a r y of ~/i-

requiring A to obey relationship (6). ttence, it is interesting to discover under what c o n d i t i o n s the A malrix car) be used for the a s y m p t o t i c stability domain eslimale. One case occurs when u i = I~,L'2

N o w we can define IF I and E 2

Ti ~ (0, +oo)

If in addition K V " ~ 0 then ~ / i s an estimate o f D.

Theorems 1 and 2 impose conditions on systems through E and K matrices, respectively. In essence, these matrices should <)bey relationship (6), which is m o r e stringent than

tlere ['ff is the ith c o m p o n e n t of V °. Let a set ~'i,

I F i : { x : v i ( x ) -'~ %.}

Theorem 3

If A s s u m p t i o n s 1 and 2 hold and the matrix K fulfils (6),

V i = 1,2 (11

exponential stability domain estimate of x = 0 of equation (I). (See also reference 14.) The set ~/, \V C IL,

V=V~xV2x...x V~,: {x:0
1,2 .....

s

Then: 'r/il = r/i2 = I

Weissenberger Lr presented an excellent analysis of the

Vi-

Vi=l,2

..... s

and therefore K = A in T h e o r e m 3. In such a case, the c o n d i t i o n s o f T h e o r e m 3 are m o r e relaxed than those of reference 13. A n o t h e r case when A can be emph~yed for estimating ID is shown by T h e o r e m 4.

153

IV.4 T h e o r e m 4 If Assumplions I and 2 are fulfilled and the rnatrix A7'B + BA is negative-definite for a diagonal matrix B with positive diagonal, then the set E I iS all estimate of D . This criterion requires knowledge o r B for lhe sot F i 1o be completely determined. For the asymptolic stability dolllaitl esiimale, it is m)t etloilgh It) k n o w lhal there exists a diagonal matrix B with positive diagonal for ,,vii)el/ ATB + BA is negative-definite. Tile first trial lo determine B can be wilh B = !. The second one is with

Vl. Appendix 1 Stateineni 1. The function v l e l equ'atic>n (,xl i~, pc,sllt~,' dot)nile arid ditferentiable on I I i due to ( b ) o f Asstililpi I, !li I '
1)I(X ) ~; I tl7'(X)(ArB + BA)u(x) u=(ul,u2

..... u~ j

Vx~:IL

Nole that

B = {diagbl,h2 ..... b d

u(x) = 0 on [k i f f x = ()

b i being the illl element of b determined by equation (7) for c > 0, provided thai A is Meizlerian. Then, c o n d i t i o n s in rehllionship (6) are to be verified for mairix A (see A p p e n d i x 2 and reference 10).

I f i w o , or all thrcc, of Theorems 2 4 are satisfied, the final eslilnz.lte o f D is the unioll of esiinlates deternlined by the theorems. Following the above approach, the proofs of Lennna 2 (see VI.21 and Theorem 3, and referring to reference 16, we can easily determine estimates of [Ds the domain of stability of x = 0 of e q u a t i o n ( l ) . ( F o r lhe definition of [Ds see reference 1 1.) Notice that IDs is to be a connected n e i ~ a b o u r h o o d of x = 0. ~ ' will denote the interior ol" ~x/, and E(.)the interior of El. I.

due to (c) of Assumptioil 1. Vl.1 P r o o f o f Theorem 1

I.Jnde~ the conditions of the theorem and f]om the stale ments I and 11 above, it results that ul is positive-definite and t)l can be majorized by a negative-definite tunctioi~ on I1. Since []_ is a connected neiglibourhood of x = 0. Theorem I is)rue. Let []:ll -~ = I V : 0 % b r V - % 71 li IP2y= ~V:0 ~ + %

7

IV

V i = 1.2 . . . . . sl

(1+7)

Vk = t,2

(17)

alld I V.5 Theorem 5

IFkT-{x:O%vk(x)%?}

If Assumptions 1 and 2 hold and (a) if there is a vector b > 0 obeying Kb ~< 0, then I1~2 iS all estimate of D s,

be conneclcd sets containing the origins. Hence,

IEk~ = ; x : V ( x ) ~

(b) if KV" % 0 , t h e n ~/ is an estimate of D.+,

IF~ =~v'ifft'~'b,:2=7_,

if there is a diagonal matrix B with positive diagonal such that ATB + BA is negative-sere)definite, then II~I is atl estimate of Ds.

(c)

Notice tllal U (or '~/) is the best possible estimate of for chosen V.

ID+

V. Conclusions The research reported herein corltribuied to the relaxation of the stability c o n d i t i o n s I\)r large-scale system analysis. The main achievements are as follows: •

[P,~. x C I1} Vk=l,2,

a one-shot aggregation arid stability analysis which essentially .reduces the n u m b e r of stability tests

•

c o n d i t i o n s l~r the best possible estimate of the asymptotic stability domain for chosen V

•

c o n d i t i o n s Ior good estimation of the asymptotic stability d o m a i n w i t h o u t requiring any type (e.g. exponential) of stability of the equilibrium poim of the disconnected subsystems

•

an aggregation form suitable for both stability analysis and estimation of the asymptotic stability d o m a i n without using comparison functions of the class ,~t7 new estimates of the stability domahls of large-scale sy st enl s.

154

(IS)

Vi =1,2 .....

s

(1'~)

Let ;)* [Ply - IV: b r V = 7, V ~ [Ply }

(2O)

0SIDe'>)='IV l'it~ i ; - % V ~ [P2"t; an d

OE2~)i= { x : V { x ) C O+~2yi , x#. E2V ].

t211

Under Assumption 2 the following is obvious for 3[EI~ the b o u n d a i y of El.~: Statement lll. l t ' x C - - E i , r t h e n V ( x ) ~ o + l P l ,

itTx

i)E~v.

l_et 0[E2-r denote the b o u n d a r y of [1:::2~.Now. Statement I V . x (

0IF2-~ i f f : l i ~ - [ l , s l

such thai x ,

01F2~i

i.e. OlF2.~ = ~L aE2.ri

i=

I

Statement g . Under Assmnption 2 and for e,,, = m i n { 4 / : i = 1,2 . . . . .

•

Ek~t~ = E~

s}

V(x~ -~ V ( y ) < V(z)

Vxc-IFk- ~

Vy(

OlFkv

V z C ~]( Eke, c,,, )\ Ek~ Z=~X

~c_ IR

Vi-

1,2

Vk=l,2

Electrical Power & Energy Systems

Statements II1 V ilnply Lemma 2. Vl.2 Lemma2 If there are: a compact n e i g h b o u r h o o d [Li Of Xi = 0, l'tln¢liOllS l/i aild positive numbers &i such that Assunlptions 1 and 2 hold and il" ahmg inolions of equation (1) 5k(X) < 0

V x ~ OEk, ) k m {I. 2}

7 C (0, 7,,,-] (22)

d~en IEk~ is a posilively invariani set of equation (1 i.

Ill the second part, s we accept B = diag{bl, b 2 . . . . . bs} satisfying s t a t e m e n t X and b = (b 1, b2 . . . . bs) T. Hence: u T ( x ) ( A T B + BA) u(x) < 0

V (x :* 0) ~ V.

Now, the following is obvious: Stateinent XI. Both vn a n d (

b I)arepositive-definiteon~'.

Since V is positively invarianl set and statement XI holds, it results thai

Sialement VI. Slaten]ent I holds aiso for tile ftmction v 2 of equalitm (9) duo to (b) of Assun)ption I and :ill h i > 0.

limlllx(l'x,,)llt~'+°~]=O

(26)

V x,,~V

1lence: Vl,3

Proof of Theorem 2

Assunlption 1 and Ihe definition of E in equations ( 12) and (13) imply: A u ( x ) ~ EV"

V x C: 3 ~ '

Statement XII. ~ / i s an estimate of the domain of attraction o f x = 0 ofecluation ( I ) . l l ("~_.~)

Fronl equation (5) and (c) of Assumption I it results that:

sign (i(x) =

o%u/(x¢

if x i :~ 0

V i = 1,2 . . . . . . ~

(24)

Vx~ll

where sign i7 = ~:1~1 I for ~ :~ 0 and signO= O. Equations (_.~) and (24) and EV ° "~ 0 (tile c o n d i t i o n of Theorem 2) iinply

~i(x) < 0

VxCO~/,xi~eO

V i = 1,2 . . . . .

Fhis result, equation (24), Assumptions 1 and 2 and Lenuna 2 prove that ~x/is a positively invariailt set of equation ( 1 ). Fllfther,

ltence: Statement Xlll. ~ is an estimate of the d o m a i n of stability of x = 0 of equalion ( I ).ll St:ltements XII and Xlll prove the theorem. II

V" > 0

and

VI.4

eii>0

Vi, i=

i;ei

"Pi=PilPi~ VII.

Vi = 1,2 ..... ~i ~ (0,

I )

s

i °ziJlt/(xt) % L kijl/o1'2(xj) /

OQ/ = qtio~ii~ii + o~ii ( 1 .....

6ii)

s

Now, statement VII implies: Statement V I I I . A :7 Aq' :is soon as all

ogii<"-'~()aild(*ii----:O

i~i

I

V xc[L Vi=

~ k i / { s l " 2 ( x / ) < ~ ' ) ' l / 2 ~ . kijb / i / i

1, ~.:, . . . . . s, (27)

Slalemenl IX. Stability of E implies stability ot A . Because A'l' is Metzlelian and stable, there is d ~ ~ , d ~- 0, such that Aq'd < ()is which hnplies, together with Statemerit V I I , thai A d < 0 :ind (reference lO and (d) of Lenuna l): S t a t e m e n t X. There is diagonal matrix B with positive di:Le,onal such that ( A r B + BA)is negative-definite.

V X~ (3[E2,) i, 7 ~ (0, "Y2[ V i = 1,2 . . . . . . s' (2g)

Let b = (K) l c f o r c > 0 . Now, b > O b e c a u s e K o b e y s relationship (6) and K is a Meizler matrix. This choice of b, equations (27) and (28) and Assumption 1 yield

I>/(X) ~"/I'2Hi(Xi) ~

ki//~ /

J J

V i , i = 1, ,~. . . . . s

When E obeys rehilionship (6). A 'I, also obeys rehitionship (6) due to the definition of A'I'. ltence:

Vol 1 N o 3 O c t o b e r 1 9 7 9

i

S

i Vi,/=l.2

I

Furlher,

V i = 1 , 2 .....

Lel

A q' = ( o~t'--/!)

Proof o f Theorem 3

Equations (14) and (15) yield

1, .~. . . . ,s

i n @ y stability of E, that is, that E = (eli) obeys relationship (6) (see reference 15 and Lemma 1). Let

S[;itciiiellt

clue to the positive invariability o1 '~i/and s t a t e m e n t X1. Notice that 7 = 7(e) is c o n t i n u o u s and strictly increasing in e 6 JR+, and that there are positive n u m b e r s "7* and e* satisfying V • Igl,y * C ~,- *. Therefore, x , ~ V guarantees X(t:x<,)~lB~ v/~IR~ V e c [ c * - , +~o).

s (25)

EV" < 0

Vt~IR~

X(t: x<,) C IBc

( 0ifxi=0 sign I ~

Let ¢ > 0 be arbitrary and 7 > 0 be a n u m b e r such that IFty • I]3c, IB<: = {x : [lx[[ < e}. Then, x,, (- IFny r>V implies

V x(- OE2.,i V i = 1,2 . . . . . vvc(0, v2l

s (20~

Tile following s l a l e m e n t s restllt from e q u a t i o n s (18), (20) :rod (21): Statement XlV. For ever5,' ( v =s:0) C IR', y is partitioned = 7 ( y L and i@_ {1. . . . . sl such that y C 3 F27 (),)i" ill fact, 7(Y) = max { ['i ( y / ) ( b f ) l: i = 1 , 2, . . . . . vl, i.e.y(y) = o2(y ).

as x, there exist 7 > 0 : 7

Statement XV. Tile function 7

IR" ~ IR+ is c o n t i n u o u s :

7(x) c C(IR') Now. statements XIV and XV together with e q u a t i o n (20)

155

imply

~,lcquatmn(1) :2 Y ~ ~)lF>,(yl, ,\ l:;-(y)

V?<

Iktualiorl~ (341, (35) and xi:ltemcnl W i l l

-'--::7 "(YlUi(y i) ,--. kii @

(30)

where ) met_ins "such that', and A the:ms 'and'.

-~(/x,,),

[B<

V/~i ~, Vx<,, V ( ~ ((), #-~,)

ill|pl} I]3,,

I :,r,)

MolOt/Vel. MaiClllel/l XVIII implies:

Lcl

"((L x<,)+: IB,

Vs~ lB. Vx,,c V ~:+- it*. +~1

)

7 I x ( t : x,,)] : 1'(~: x,,

(0.72]

IF, f3-')

where c* is a positive lltlllli),2f ~;ticl/ liHil Statemonl XVI. Slatell]eni XV :.rod c o n i h l u i l y ol3~ ill l C IF{t guaranlee C()llliTltlily of F ill / ~ Fit V .vo ~ I}:{~L Let it be supposed that for arbitrary x<, G IF 2 , x,, ~ O, there exists Y,, ~ ( 0 . 7 2 ) such lhat IF2y,, • IF2 ~'( ~)

V t ¢ IFI,

( 31 )

IF : c IB, .

Now. equations (33) (37) pitlvc tilat IF 2 is :111c'stin]atc {~1 lhe domain of asynlplol it.' stabilily ~Hx : ( ) e l equation ( 1 ). If, in addili~m, K V " <: 0 we can s01 D~hi: = l'j'( I i ) ~. which implh.'s I1Fe = V bec'
Then, lhere exist ~<>= .(ix,,) ~ (0, +oo) and ~,, - <~(x<>)~: (0. +oo) such th:tt

~,)~,>

VI.5

V l G ~:~+ Vi-l,2 .....

s

because of slalen/ent X V l and equation (3 I ). ltence, lJeIX(I" X,,)) e.~ .~'<,~,> Ill:IX

{'~, f
f :!L

l. 2 . . . .

s

limllf'<(f:x, lN:[ -+col_-0 I

o~)

( 3;~, )

[]_

Vx<,< IF~

Let c < ( ( L + ° ° ) b e a r b i l r a r y a n d T = T ( e ) s u c h

(40)

U.'t <5 = a(e) obey

Vx,,~lF:,x,,=O

(41t

B~ • It:j~

where 1

i

(>))

that

E~ !2 [Bc rq IF~

This i m p l i e s / l i e l \ ) l l o w i n g absurd u2(X(t:x,,))
Vx~.

because of Assunq~iion I. This result. Assunlp/ion 2. positive-definiteness of :ill u i on ILi. negative-deliniteuess of ATB + BA and kemma 2 p r o v e / h a t nclw is a positively mvariani sol o f equation (2), V 7 ~ (0. ")'11. I lence, equalion (3g), positive-defiifiteness o l u j for b = (f) 1. D . . . . . . /'v )7`` /)i heing the ith diagonal elemenl of B, and tl~e last result prove

V <~ 0 ~3

The righl-h:.lnd side is negative due to lhe choice o f b, b = (K T) lc c > 0. Equation (32) C;.lli he intograled fiom l,, = 0 1o I. I)2()~(t: x°))-~<~u° + ~'
Along m o t i o n s ~1 equation ( [ ) the derivative ~)t e l ~t ~dw\ ~,

I)J (x } <-. u F(x)[ATB + BA ] u(x)

l

I. 2 . . . . . .

j~

Proof of Theorem 4

Positive invariability of IEIT, V 7 ~ (0, 71 ], and equations (40) and (41) prove

I

~( t: x . ) c [13c

which is a consequence of the assumption that 7 . > 0 instead o f the true value 7,, = 0. But. T,, = 0 implies

V t ~_ [B. V xo ~ [B~ V e C ( O , +oo)

(42)

Further. for e* such that IF I C [Be*. c* ~ ( 0 . +oo). we have: limlE21,(ti:t-~+c°]

= {0} -,{(t: x,,1C IBc

This resutl proves lim[llx(t;x,,)ll:t

Vx<,~ E I

v e c: [e*, +oo) ~"+°°1 : 0

V x,, ~ IF2

(33)

Further. let c G (0. +°°i be arbitrary and 7c G (0, T2 ]' such that (34)

E2.), c Q [B e O E 2

Equations (30), (421 and (43) prove that E l is an estimate

VII. Appendix 2 VII. 1

[B~ • E2,~ Statement X V I I . The m o t i o n X(t; x o ) can leave E2.~,. x<, ~ E2.y, iff it passes through 311:27: - ] r G [ O . + ° ° )

(35)

7-: r ( x o , ' ) ' )

:i x(r, x,,) ~ alF2. r VT~(0,72] A s s u m p t i o n 2, statements I V and X V I I , and equation (29) prove : Statement X V I I I . The set 1::2.r is a positively invariant set

143)

of D.

Now. we define <5 = 5(el G (0, +oo) so thai

156

VtC~+

Conjecture

If a Metzler matrix A obeys relationship ( 6 ) a n d b = (b 1- b2- . 7" " b~ )7' is d e t e r m i n e d from equa/iou ( 7 for c > 0 then A "B + BA is negative-definite as soon as B = {diag b l , b 2 . . . . .

bs}

This conjecture fails for the following c o u n t e r e x a m p l e . let

' A =

20

'0 ) 201

(,) c =

1,

>0

Electrical Power & Energy Systems

6 Grujic, Lj T Int. J. Control Vol 20 No 3 (1974) pp 453-463

A obeys relationship (6) and b=-

(A T) ~c =

-II

>0 7 Michel, A N SIAMJ. Control Vol 12 No 3 (1974) pp 5 5 4 - 5 7 9

I tence,

B : diag {221, 11}

Gantmacher, F R The theory o f matrices Chelsea Publishing Company, USA (1959)

1lowever, ATB + BA =

( 2430

4422

t

is md negative-definite.

Persidsky, S K Avtom. & Telemkh. No. 12 (1969) pp 5-11 10 Araki, M J. Math. A n a l & App/. Vol 52 (1975) pp 309--321

For possible choice of b (and B) see reference 10. 11 V I I I . References 1 Michel, A N and Miller, R K Quafitative analysis of large-scale dynamical systems Academ ic Press, USA (1977)

Grujid, Lj T Int. J. Control Vol 22 No 4 (1975) pp 529-549

12 Grujic, Lj T and Siljak, D D IEEE Trans. Autom. Control Vol AC-18 No 6 (1973) pp 6 3 6 - 6 4 5 13 Weissenberger, S Automatica No 9 (1973) pp 6 5 3 - 6 6 3

2 Sillak, D D Large-scale dynamic systems North-Holland, The Netherlands (1978)

14

Bitsoris, G and Burgat, C Int. J. Syst. Sci Vol 7 No 8 (1976) pp 911 928

"3 Araki, M IEEE Trans. Autom. Control Vol AC-23 No 2 (1978) pp 129-142

15

Fiedler, M and Pt~ik, V Czech. Math. J. Vol 12 (87) (1962) pp 3 8 2 - 4 0 0

4

Grujic, Lj T, Borne, P, Gentina, J C, Bernussou, J and Burgat, C RAIRO (1978) Grujic, Lj, Ribbens-Pavella, M and Bouffioux, A 'Asymptotic stability of large-scale systems with application to power systems. Part 2: transient analysis' int. J. E/ectr. PowerEner.qySyst. Vol 1 No 3 (1979) pp 158-165

Vol 1 No 3 October 1979

16 Burgat, C 'Contribution ~ 1%4tudedes propridtds de stabilitd de syst6mes non-lindaires continus interconnectds' Doctorat d'Etat, L'Universit6 Paul Sabatier de Toulouse, France (1976) 17 Hahn, W Stability of motion Springer Verlag, W. Germany (1967)

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