Free Response Characterization Via Flow Invariance

( : ~I P \ l lgll1 :[. 11-\(

Hud " I't"'r .

11 1111 1..; _11\ .

~ " It

I ll!" lIl1l.~ 1

,," ,ql d

( "ll·.! I ~'·"

1',':-.1

FREE RESPONSE CHARACTERIZATION VIA FLOW INVARIANCE M. Voicu

Abstract. It is shown that the flow invarianoe can be an efficient tool tor a more detailed oharacterization of the dynamic processes. Por the linear constant dynamical systems described by ~ - Ax, t~o, with A • (aij ) a (nxn) real matrix, and for the time-dependent state interval I( t) - {z€ Rn, Iz I~Y( t}}, t~O. as flow InTariant set (1·1 and ~ signify1ns componentwise absolute value and inequality respectively). where .(t) is differentiable, a componentwi.e oharacterization is developed. A nece.sary and sufficient condition such that Ix( t )I~ Y (t) for each. to;;:=:O' for each Ix(to)/~Y (to) and for each t~to 111 that IY(t)~ ,et) for each t~o. I has the elements a 11 and laijl. i"j. Por Y (t)-O as t--oo the componentwise asymptotio stability is defined. Such being the oase_a neoessary and suffioient condition for the existence of yet) is that A be Hurwitzian. The results of the paper may be also used for solTing the oomponentwise stabilization problem. Kerwords r SY8tem theory , linear systems , time-domain analysis ; free response componentwise charaoterization , stability , componentwise asymptotio stability. IJl'fRODUCTION

charaoterisation for the temporal eTolutlon of syste. (1) ls desirable. An example of suoh oharaoterisation maf be that oertain oomponent8. or all. of (J). satisfy Ineqmalities of the form (4).

Let us consider the linear constant dynamioal system (1)

The purpose of this paper i8 to develop a componentwise oharacterization for the free response (J) of the system (1) and subsequently to .tate the componentwi.e a..,aptotio stability problem. In this respect we prove some simple and easily applicable results. by using flow inTariance methods (Crandall. 1972; Martin. 197J; Nagumo. 1942; PaTel and Vrabie. 1979; Pave 1 , 1982).

where A-(aij ). a ij ER. with the initial oondition (2)

It is known that the asymptotio stability of the triTial 801ution in the sense of Liapunov is oonoeiTed on the basis of the norm in Rn. Thi8 .eans that the temporal eTolution of the solution of the Cauoby problem (1).(2). which has the form x(t) _ eA(t - to)x ' o

t~to'

FLOW INVARIANCE OF A TDIB-

DEPENDEN'r IJl'fERVAL

(J)

We begin with some basic notations and definitlons. Let v-(vi ). wa(w i ) be two Teotors in Rn and let C-(cij).D-(dij ) be two real (nxn) matrioes. In all what tollows we denote by ITI the Tector with the coaponents ITil. by /CI the matrix with the elements /c i ;11 and by ~ the matrix with the element. 0li and /olj/' i"j. We al80 denote by T>W (T.",W) and by C>D (C~D) to signify \>~ (vi~ w1 ) and c ij > 'tj (oij~dlj) for all 1.j. Let yi(t»O, t~O. i .. l •••• n. be n differentiable 80alar !unctions, whioh define on each unit vector basis of Rn the veotor !unotion Y(t»O. t~O. with the oomponents Y i(t), and the time-dependent state InteJ'Tal

is eTaluated Tia the soalar funotion Ux(t)U. t~to. A well-known result in this re8peot is the follOWing r The system ~l) is a8lmptotically stable if and only 1 Ulere exist 11>0. >0 suoh that for eaoh to ~O. for eaoh Xo E Rn and for eaoh t ;;:;;.t o

r

IIx(t>l~ IIl1xoue-Nt - to)

(4)

~.

In oertain theoretioal problems as the tlOl 1nTarianoe ot the solution (3) with respeot to a oompaot sub8et I C Rn or In oertain applioatlons. espeoially in system projeot and engineerlng.a .ore detailed 1421

~t.

14:::!2

\' o i cu

l(t) -(v ERn; Y(t)-lvl~O}, t~o. (5) For the free response (J) of (1), whioh belong to let) on a certain time interval, the following definitions oonoerning the flow invarianoe are available.

~

Definition 1. let) is flow invariant for (1) if for ea'c h to ~o and for each Xo E l(t o ) there exists tf>O suoh that the free response (J) satisfies x( t) E I( t) for each Xo E I( to)

(6)

and for each t E [to,to+t f }. Definition 2. I(t) is globally flow invariant for (1) i f for each t 0 ~ 0 the free response (J) satisfies x( t) E I( t) for each Xo E I( to) anel for eaoh t

~to.

As for each t~o, I(t) is oompact and for eaoh to~ 0 the free response (J) of (1) is defined and oontinOl.ls on [to'+OO), one may prove the following.

,

1 .• A neoessary and suffioient con tIon such that rct) be flow InvarIant for C fs that f{t) be sloballi flow nvar an for 1. pro~sition

r)

Proof. Sufficiency is obvious. Necessit{. tit"Us oonsIder T.. (t f (to+tf.+OO);X(t). (t)} and t-inf T. Clearly, (7) holds if and only if T=~ (t-+oo). Suppose by oontradiotion that t<+oo. If t ET. then x(tH '* I( t). But x( t) E I( t) for each t€ [to' t) and therefore for t .... t. x(t)--x(t) E l(t), what is impossible. If t T. then x( t) E E I(t). But t cannot be an isolated point of now invarianoe, what implies that t~inf T. This contradiots the definition of t. Hence t=+oo and T-~.

*

Remark 1. Acoording to Proposition 1 the In! Hal time t 0 ~ 0 and the length t f > 0 of flow invariance of let) for (1) can be arbitrary. Therefore. from the basio result of Pavel and Vrabie (1979) we may derive the following. Theorem 1. A neoessary and suffioient condition such that rct) be globally flow invarIant for (1) is that lim h-ld(a + hAz; l(t+h»

h ..... o

for eeoh t

= 0 (8)

~

0 and for each z E l( t) •

We denoted in (8) d(v;I) ..infllv-wU for wH. Using Theorem 1 explicitely for (1) and l(t). we oan establish the following more detailed result. which is useful for analytic and oomputational purposee. Theorem 2. A necessary and sufficient conditIon such that rct) be globalli flow

invariant is that 'ht) ~ Ir(t) for each t~o.

Proof. It is known that (8) is equivalent

ro-

Iz+h(Az+a(h»I~"'(t+h)

for each t?:O, (10) for eaoh z EI(t). for h> O. small enOl.lgh. and for oertain a I [O,+oo)_R n • with a(h)· --0 as h'-O (Pavel. 1982). As Y( t) is differentiable, there exists r : [0,+(0 ) __ Rn , with r(h)--O as h ..... O such that Y(t+h)- Y(t)=h Y(t)+hr(h), t~O. In view of (10) the .tatewent ~8) is equivalent to Iz+h(Az+a(h»l~ Y( t)+h ..y.( t)+hr(h) for each t~o, for each zEI(t) and h > 0, small enOl.lgh.

(11)

Substi tuting z successi vely by ±(Y l' g12' •• , gln)····±(stl'···sti-l·Yi·gii+l···'stn)·· .'±(~'."Snn-l'Yn)' where Yi=Yi(t) and gij=YJ(t)sgn a ij , i.j=l ••• ,n. j~i. we asoertain that each row of (z+hAz) reaohes its maximum value (for +). respeotively its minimum value (for -) and simultaneously each row of (11) can be simplified by h >0 for fixed t ~O. for z E I ( t) and for fixed h > O. small enough. Consequently (11) is equivalent to I Y(t)~ ~ Y( t)+r(h):;:a(h) for each t ~O. In view of the fact that a(h)- 0 and r(h) -0 ae h ..... O. it follows that (8) is equivalent to (9). Por a further characterization of the behaviOl.lr of (3) regarding this special oase of flow invariance, let us consider the system

t .. Ay,

t ~ 0,

(12)

Y E Rn.

with the initial condition y(t o )

= yo'

( 13)

to~O.

The solution of (12).(13) is yet)

= eACt

- to)yo'

t~to'

(14)

Notice that the elements of I. whioh do not belong to the first diagonal are nonne6ative. Via a classical result (Bellman, 1960) it follows that (14) satisfies eA(t - to)yo ~ O. for each to~O. for eaoh

Yo~O

and for each

t~to'

(15)

Theorem J. A neoesstfY and sufficient conditIon such that rc be globally flow InvarIant for (1) is that let) ~ eA(t - ~) Y(&) (16)

for each pair &. t E [0. +00 ) , , Proved by I. Vrabie.

(9)

t

~~.

Fr ee Res pons e Ch a r acterization

Proof. Sufficienoy. Substituting eI(t-&) oy-rts Taylor expansion around the point teS, (16) may be brought to the form [Y( t)-Y( &)] I( t-&) ~ AY( &)+I2 ( t-&)Y( &)/2 r+ + ••• , t >e-. ObTiously for e-.... t one obtaiXlJl (9). Neoessity. For each oontinously function v : [O,+oo)_Rn , with v(t)~O and suoh that i'(t)=AY(t)+v(t) for each t~o (i.e. (9) with " .. " in plaoe of "~"), one deduces Y(t) • eI(t-&)y(&) + jt eI(t-t')v(t')dt. afor eaoh pair &, t E (0, +00 ) t ~&. According to (15). it follows that (9) implies (16). Remark 2. Taking t o.0 and wbst! tuting y 0 in (15) sllccessively 1:u (1.0 •••• 0) •••• (0 •• •• 0.1) one concludes that elt ~ 0 for eaoh t ~O.

1423

Remark 4. One may equivalently restate (9) as Yi(t)

~

aUli(t) + [ j laijl"Yj(t) (21)

for each

t~O.

i .. l .... n.

where ~j signifies the BUm for j=1.2 •••• 1-1.i+l •••• n. Aocording to (19), it follows that for each 1=1 •••• n. there e.:z:tsts ti ~O. i=l •••• n, such that '( i(ti)0. i.l •••• n. from (21) it follows that ( 22) Thus, we proved the following.

(17)

Corollary 3.1. Each real (nxn) matrix A satlsl'Ies ( 18) Proof. Let Y(t)=eIty(o). t~O. whioh iin'ifies (16) for each y(O) >0. In Tiew of (3) and (7). for t o EO. one may write Ix(t)laleAtxok;eIty(o) for eaohY(O»O. for eaoh I Xo I ~Y( 0) and for each t ~ O. Clearly. for xo=±Y(O) one obtains (18), beoause Y (0) > 0 is arbitrary. COMPONENTWISE ASYMPTOTIC STABILITY To define the componentwise aaymptotic stability via the flow invariance it is natural to suppose that Yi(t). i.l •••• n. have also the property

From (16). under (19). one can easily derive an existence condition for I(t). Theorem 7. A necess~ and sufficient condition for the eXis ence of I(t) such ttft (1) componentwise ayrrptotrcarlY sable wi h respeot 0 y (t s that A be Hurwitzian.

re

~.

Sufficiency. If I is Hurwitzian. then one can take '( t) .eIt y (0). for which (16) and (19) are satisfied. Necessity. If (1) is oomponentwise asymptotioally stable. then (16) and (19) hold. Suppose by contradiotion that A is not Hurwitzian, i.e. lim sup lIelt 11>0. Taking t-co

a-=O and using (17) and the inequality '(t»O. from (16) it follows that limIIY(t)II;;;'lim sup IleItY(o)/1 > O. t-oo t-oo whioh oontradicts (19). COMPONENTWISE EXPONENTIAL ASYMPTOTIC STABILITY By speoializing the funotionsYi(t). i.e. Yi(t) • 'l"ie-rt.

t.

Remark The oomponentwise aaymptotio stabi1i y with respect to Y(t) is equivalent to the globally flow invariance of I(t) (given by (5) with (19» for (1). Aocording to Theorems 2.3 one oan formulate the folloWing.

t';:!:O. i=l .... n.

(23)

where q:i>O. i.1 .... n. are the oomponents of the vector Q' .(GYl'.' ."'n) >0 and f'> >0 is a soalar. one may develop a more exp1ioite charaoterization for (3). Definition 4. The system (1) is oalled com~onentWi8e exponential asymptotioallY sta le it there eXist Q' ::>0 and ~ >0 such that for each t ~O and ?Or each Bt 0 Ixo I~ GY e -,- 0 the free response (J) satisfies (24) Ix(t)l~ GYe-r t for each t"d o ' Proposition 2. A necessary and sufficieut

1424

00

a

a

M. Voicu

tion woh

11

Proof. Suffioienoz. If (27) wHh

liOICli. then there exists ~1II&X(C\"1" •• a'n»O

o

woh that (25) is verified. Booe.sitz is obTious.

Proof. SUffiOieDO{ ill obTiou •• Beoe.sitz. 1000rdiq to the he ore. of Perron - Proben1u. (Bellaan. 1960). there Is tor eaoh t ~ 0 an unique eipnvalue e -rt > 0 of olt ~O (eoe (17», with r> 0 (X 1. Hurri:hian). whioh ba8 croatoet ab ••luto value. Thi. olsonvaluo 1. .aplo and it. uaooiated olsollToot or ~ be takon v > O. Thu •• one oan write oI~v.e-rtv.t~O. Integrating thi. rolatlon on ~.+oo) and then ~ltipl;rins 117 o-pt, with 0< f!'~r. one obtune oa.117 that d(o-~tV)/dt ~ I(e-~tv) for eaoh t~O. 'fhiellleane that (9) is satiefied for yet) siTOn by (23) with Cl'-v. 1.0. (24) holds. In this oontext. aooording to Tbeore. 4. one ru::T prove. o onse a that

exponen

o<:r~~n(-al1

-Cl'ilL;jlaijlQ';j).

(25)

The proof follow. illlllediatel;y fra (21) by replaoing Y i ( t) si TOn by (23). Corollary 8.1. Por eaoh real (nxn) matrix I HurwHzian thore exist Cl';> O. p> 0

J. wHh

wOli'"l'ha t leU

ex:> 0

o

I~

eIt ::::;

Cl'~'e-~t for eaoh t~O.

ex' i. the row veotor

i

(c( 1 •••• «;1).

The left-hand inequa11t7 ls already proved (see (18». If I is Hurwltzian. then. aooording to Theorem 7 and Propo.ition 2. 1178te. (12) Is componentwlse exponontial a87lllptotioall;y stable. In Tiew of Definition 4 and taking to.O one may write

~.

leItYol~ Cl'e-~t for eaoh t ~O. Replaoing now Yo successively tu (CI'l'Q •••• O) •••• (O •• •• 0.CI'n) one deduoes eallily the right-hand inequality. Remark 5. Por fixed ~j' 1.;j.l •••• n. the maximum value of ~ dependll on C1'i::> O. i.l •••• n. A. a matter of faot one can define tho funotion raax( C\"l' ••• "n)

4 n( -al1- cri l Lj laij ICl';j ) • ( 26)

equivalent statement for Theorem 8 ill the following.

An

oon-

(27)

Remark 6. If we try to use Theorem 9 we hive to prove an existence result tor the inequation (27). or equivalently for

1

0(

(28)

< O.

where 1.[1 :-I n )'. In ill the unit (nxn) matrix and [.J' signif10a the transposition. 1 0 An existenoe condition for the problem (27) may be eall1ly derived from Theorem 7 and Proposition 2. 2 0 Por the problem (27) we oan also use the notion of M-matrix (Ostrowski. 1955). An M-matrix is defined to be a real (nxn) matrix C suoh that Cij~O' i~j. posseasing one of the following equ1valent properties I a) There exists v>O suoh that CV> >0, b) C ia none1ngular and all elementa of C- l are nonnegative, 0) All prinoipal minora of C are positive. The util1ty of thill defin1tion lies 1n the following , and wffioient oondit1on for A neoess e e s enoe 0 a 110 u on 0( :> .:::.::o;:r~.;::..,\,,,,, 1a that -I be an M-matr1x. Hote that the propert1es ~d 0) 81iow to construot a solution «>0 for (27) (d1reotl;y with b) and Tia the Gauss1an e11minat10n process w1th c». In terms of property c) one oan est1mate the maximum positive eisenvalue l/~IIl&X(Cl'l •••• an) of (_1)-1 ~ 0 by u.ing method. derived from the theorem of Perron - Probenius. 3 0 An existence result concerning problo. (28). whioh allows also to oonetruot a solutionC\'. was proved by Dines (1918-1919) I A necess~ and sufficient condition for the eXistence of a SOlutIon « for (28) i. that the I-rank of 1 be greater thin zero. For the defin1ifon of flie I-rank of a (mxn) real matrix see Dines (1918-1919) or Tschernikow (1971). In terms of Remark 6 (2 0 and 30 ) and for oomputational purpose. one may prove.

(_l)k

Ik

~ O.

k.l •••• n.

Il •••• ~ are the prinoipal minor. of

enoun

I.

1

!a! I-rank of 1 be greater than zero. Remark 7. Let u. oonsider Aa-diag(ail •••• a;1)Adiag(a1···.an ). where ai~o. i.l •••• n. are the component. of the vector a. Notice that Aa is siai-

Free Response Cha r acterization

lar to A. both having the SIllH speotrum ~(A). The a - Gershgorin's disos assooiated to A. i.e. the Gershgorin's discs associated to Aa (Bellman. 1960) are subsets of the oomplex plane C described by GAl(a) -

[sEC;llI-a11l~a11~laijlaj)'

~. Consider P.y- l • where Y is the modal aatrix of A over R. Sinoe lay-lA Y is the (blook) diagonal or the (blook) Jordan oanonioal form of A ove! R. aocording to (32). it follOWS that i is Huni tzian. EXAMPLE

(0)

i-l •••• n.

1425

Consider the system

which have the remarkable property ~(A) C GA(a). GA(a) represents the union set of GAl(a). i-l, ••• n.

! •

[-1 -32] x, -1

and find a function yet) suoh that ())) be ooaponentwise asymptotioally stable. exponen

Solution. Sinoe I is Hurwttzian one aay (1)

for at least one a> O. Proof. For a- er (er frOll Def1ni tion 4). rn-iieping with ()O). oondition (31) is equivalent to ~ max(erl' '" Ilh) > 0 (see (26». respeotively to (25). Remark 8. By analogy with the well-known stabi1ization problem one oan state the oomponentwi.e stabilization problem of the aystem t-Ax+Bu. where u € Rill and B i. a (nxm) real oOlUltant matrix. Aocording to Theorem 12 the solution of this problem oOlUlists in the assignment of the ex - Gershgorin t s disos in the half 0 _ _ plex plane Re s <0 via an adequate state feedbaok. DEPENDENCE ON VECTOR BASIS The o oap one ntwi se asymptotio stability implies the a~ptotic stability. Consequently each of the Theorems ~ - 12 is (mutatis mutandis) a ori er on or a..,mptotio stability. In this respeot we remark that the oriteria whioh oorrespond to Theorema 10 and 11 JD&Y be useful in some app1IoatIons.

ta fa

The asymptotio stability does not illlply the cOllponentwise asymptotio stability. beoanse the latter depends on the particular ohoioe of the veotor basis for (1) in Rn. In other words, there exist vector basis in Rn for whioh the flow invarianoe of (3) oannot be realised for any I(t). under (19). A natural question is that of the existenoe of sOllle veotor basis in Rn for whioh an asymptotioally stable ayste. is also oomponentwise a-.raptoti0&11y stable with respeot to Y (t) • A partial answer to this question is the following. Theore. 13. There exists at least one nonaisgelar transformation i.Px for (1) nch that the !ystem i.IX. t ~O. be ooaponentwIse &81fltotiOalll stabIi with respect to y( t) _ IT(A) c {sec; Re s
adopt Y( t) ..It Y(O) , whioh satisfies Theorea 4. One may easily see that 1 oe -bt + de -at

Y( t)b

[ ge-bt_ge-at

f e -bt - f e -at] de-bt+oe- at

Y(O). t ~O,

where a.2+ VJ. ~.2- VJ, 0-3+ VJ. d-3- VJ. f • • 20. g-(3. For Yl(O}-(a-l)p andY 2 (0)- f. with arbitrary f >0. we obtain yet) -,[a-l

1] t e-bi,

t~O.

whioh proves that system (J3) is also oomponentwise exponential asymptotioally stable. CONCLUSIONS The notion of flow invarianoe proves to be an effioient tool for a 1II0re subtle oharaoterization of teaporal behaviour of the linear oonstant dynamioal systems. In fact. when the flow invariant set is a ti.e-dependent state interval I(t). a ooaponentwise oharacterization of the free response is possible. Suoh an evaluation ~ be useful espeoially when the state oomponents are of different importanoe for the normal evolution of the systems (for instanoe in Eleotrioal Eng1neeri~ (Voicu. 1984) or in Biology (Pavel. 1983». The silllply neoessary and suffioient oondition (9) allows to determine yet) for whioh I(t) is globally flow invariant for a given linear oonstant dynamioal system or to determine the matrix A such that a gi Ten I ( t) be globally flow invariant. The o omp one ntwi se asymptotio stability of (1). whioh is a globally flow invarianoe of I(t) for (1). with I(t)-O as t-CD. represents a row property of the evolution matrix J.. oOlUlisting in a oertain first-diagonal dominanoe (see (21». and holds if and only if X is Huni hian. This speoial type of a..r-ptotio stability depends on the veotor basis and it implies the asymptotio stability in the sense of Liapwlov. The results of the paper may be also used for the linear state feedbaok synthesiS (Voicu, 1981, Voicu. 1983) in the oomponentwise stabilization problem. In this

1426

H. Voicu

respect the solution oonsists in the assignment of the ex - Gershgorin' s disos in the half complex plane Re s < O. Aknowledgements. The author would like to thank Mr. N. Pavel and Mr. I. Vrabie for the very interesting discussions about flow invarianoe. REFERENCES Be llman , R. (1960). Introduction to Matrix Analysis. MoGraw Hill, New York, pp. 166, 172, 278, 297. Crandall, M. G. (1972). A generalization of Peano's existenoe theorem and flow invarianoe. Proo. AMS 39, 151 - 155. Dines, L. L. (1918 - 19l9~.~ystems of linear inequalities. Ann. of Math., 20, 191 - 199. MartIn jr., R. H. (1973). Differential equations on closed subsets of a Banaoh spaoe. Trans. AMB, ~, 399 - 414. .. Nagumo, M. (1942). Uber die Lage der Integralkurven gew~hnlioher Differentialgleiohungen. Proo. Phys. Math. Soc. Ja;an, ~, 551 - 559. Ostrowskl, l. (19 5). Diterminanten mit Uberwiegender Hauptdiagonale und die ab801ute Konvergenz von linearen Iterationsprozessen. Comment. Math. Helvet., lQ, 175 - 19&. Pave 1 , N., ana-I. Vrabie. (1979). Differential equations associated with

continous and dissipative time-dependent domain operator8. In Londen, S.O. (Ed.). Leoture Notes in Math., Vol. 737, Springer Verlag, Berlin, pp. 236 - 250. Pavel, N. (1982). Nonlinear AnalfSiS, Evolution EquatIons and ltPl oations, LeotUre Notea, uDiv. n o. • CUza, Ialjli, P.231. Pavel, N. (1983). Positiv1ty and stability of aome differential systems from Biomathematics. Proo. of Conf. on "Math. in Biology and MedIolne", July 1983, Barl Italy (to appear). Tsohernikow, S. N. (1971). Lineare Ung!eiohulfen. Deutsoh. Verlag der W18s., Ber n p.189. Voiou, K. (1981). State feedbaok matrices for linear oonstant dynamioal systems with state oonstraints. 4-th Internation. Cont. on Contr. Syst. and Oomp. Sol., June 1981, Vol. 1, Polytechri. Inst. of Buoharest, pp. 110 - 115. Voiou, M. (1983). On the determination of the linear state feedbaok matrix. -th Internation. Conf. on Contr.