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The Lyapunov matrix-function and stability of hybrid systems
.kmhn~ar Annlwn. Theory. Pnnred in Great Bntun.
Methods & Appli~mom,
Vol
-
THE LYAPUNOV
10. No.
12. pp. 151S1457,
1986
O362-M.Y.86 Pergtmon
_
MATRIX-FUNCTION AND STABILITY SYSTEMS
$3 00 - 00 Journals Ltd.
OF HYBRID
A. A. MARTYNYUK Institute
of Mechanics,
The Ukrainian
(Received
1 February
Academy
of Sciences,
Nesterov
Str., 3, Kiev-57.
1985; received for publication 20 January
Key words and phrases: Lyapunov matrix-function, whole), asymptotic stability, two-component system,
Ukr.
SSR. 252057
1986)
hybrid system. stability, uniform stability (in the property of having fixed sign of matrix-functions.
1. INTRODUCTION
THE LYAPUNOVmatrix-function [l, 21 allows us to apply construction of the direct Lyapunov method [6] to analysis of hybrid systems, the free subsystems of which can be unstable. In this case the outside of diagonal elements of the matrix-function must supply the more complete account of the effect of interconnection functions, which in case of the vector-functions application can be considered sometimes as a stabilizing factor. Let us also remark that attempts to take into account the effect of interconnection functions as the stabilizing ones, were made even earlier [4] while consideration of the weakly connected subsystems, but effectivity of the criteria obtained in this very case, can be defined by existence of the general solution of independent subsystems. In a number of cases, particularly for the weakly connected oscillating systems of the second order, these criteria are effective [4,5]. The method of analysis proposed here is based on a knowledge of the Lyapunov matrix-function. The present paper is the further development of work [2]. 2. STATING
THE
PROBLELM
We consider a system 1, which consists of the m-interconnected subsystems tiJ!..i E [l, m]. System 2 can be called a hybrid one if subsystems 4 (or free oj) are described by different types of equations (finite-dimensional, integro-differential equations of parabolic type with a strongly elliptical operator, functional-differential ones, etc.). The couple (ai, go. i E [l, m] determines an i-interconnected subsystem in system Z. We shall not concretize yet the type of subsystems 4 (subsystems 0;) and interconnection functions g, between them, but suppose only that solutions of the correspondent Cauchy problems for each subsystem are defined correctly, and the state of equilibrium of system Z in the direct product of Banach spaces B, x B2 x . . . x B, = H, is the point 0. The aim of the present paper is a solution of the problem on stability of the state of equilibrium 0 of the system Z in case when there are both stable and unstable subsystems among oi, i E [l, m] and stability of the state of equilibrium of the whole system Z can be obtained by the stabilizing effect of the interconnection functions gi, i E [l, M].
It is widely (Oi,
giP
E
[I,
known ml)
3. THE LYAPUNOV MATRIX-FUNCTION [7] that the functioning of the hybrid system 2 (of the t%ality
can be described by a differential
equation in Banach space E = n Ei, i=l
1119
1350
A.
A.
L~~ARTYNYLK
where Ei are Banach spaces, corresponding to subsystems (oi, gJ i E [ 1, m] with the norms ]/ . Iii. For a hyper-vector xr = (XT, . . . , xi)’ this equation has the form
z: g = X(t, where X : T, X H+
x), (X(&j) =
x0
E Ho c
E), X(t, 0) = 0,
(3.1)
E, H = fi Hk (a compact in E set), Ho = fi Hok, X is in general a k=l
k=l
discontinuous, nonbounded operator), defined on a set S2 = T, x H C I’; I- = T, X E. Under a solution ~(t; to, x0) of the equation (3.1) on T, = [t, + x[, t E R for t0 E T, C R and x0 E Ho we understand an absolutely continuous on T, abstract vector-function, which satisfies this equation almost everywhere on T,. Let us recollect that system (3.1) consists of the isolated subsystems ui : dt
=
XJt, xi);
Xi
E
where Xi: T, x Hi+ Ei, which can be united by the interconnection gi: gi =
such that g,: T, x HI x . . . x H,+
gi(r>
Xl
9 *
.
.,x,),
(3.2)
Ei, functions
i E [l, M]
Ei. By a couple (ai, gJ an i-interconnected
(oi,gi):%=X;(t,Xi)
+gi(r,xr,.
(3.3) sybsystem
. *,X,)
is described for i = j respectively. Together with equations (3.4) for i E [l, m] we consider the real functionals j(z [l,m], vii: r + R and a “matrix-function” (matrix-functional)
B(f~x, = ivij(l,
x>l~j=l
(3.1)
Vij(r, x); i,
(3.5)
which maps r onto Rmx m. Elements Vij(t, Xi) for i = j can be constructed on the basis of equations of the free subsystems ai and Vii(t, x) for i # j can be constructed having taken into account the interconnection functions between i and j of subsystems ui and Uj. By means of a vector nT = (ql,. . ,, q,,J E R”, vi > 0, Vi E [l, m] we shall construct a scalar function (see [2, 141) V(t, x) = Il W, x)rl
(3.6)
for a hybrid system 2. Let us define the upper right derivative of the function V(t, x) in virtue of the system 2 by formula D+v(t,x)
= nrD+%(t,X)~,
D+93(r,x)
~{D+vii(‘,x)}~j_I
where
(3.7)
The Lyapunov
matrix-function
1151
and
D’ V,(t, X) = lim e-o+
SUP
0
f
{V,[t+
0,x+
&X(r,x)] - V,(t,x)},i,jE
[l,m].
(3.8)
If V,(t, x) are continuously differential in all their arguments, then D’B(t,
X) ’ {DVij(t, x)}Y,=~
and in detail V(i==j): DVii(fyX) = aVii/at+
VVii(t,Xi)r(X~(t,X1)
V(i#j):
m C, [VV,(t,~)r(X<(r,xi)
DV,(t,X)
= dV,/dt+
+ gi(ttXlj * . .,Xm))i (3.9) + gi(f,xl,.
. ..~m))].
i=l
Definition 3.1. Matrix-function B(t, x) can be called the Lyapunov one on T, X G. G c N if: (1) Function (3.6) is absolutely semi-continuous over any interval [to, rl] C T,. (2) D+V(r,x) 6 0 on T, x G. On the basis of function (3.6) and its derivative (3.7) for a hybrid system ): the theorems on stability are stated according to the scheme of the direct Lyapunov method. At first let us give some definitions. Definition 3.2. The function V(t, x): r + R can be called: (1) Positive semi-definite on T, if there exists a connected time-invariant neighbourhood N of the point x = 0, NC H such that: (a) V(t, x) is absolutely semi-continuous over any internal [to, ti] C T,; (b) V(t, x) is nonnegative on T, x N, i.e. V(t, x) > 0 V(r. x) E T, x N; (c) V(t, X) vanishes for x = 0 (Vij(t, 0) = 0 Vt E Tr). (2) Positive semi-definite on T, x cp if the conditions of the definition 3.2(l) are fulfilled for N = S. (3) Semi-definite in the whole on T, if the conditions of the definition 3.2(l) are fulfilled for N = H. (4) Negative semi-definite (in the whole) on T, (on T, x N) if (- V(r, x)) is positive semidefinite (in the whole on T, (on T, x N)) respectively. In definitions 3.2 an expression “on T,” can be omitted if and only if all their conditions are fulfilled for t E R [8]. Definition 3.3. The function V(t, x): r-, R is to be called: (1) Positive definite on T,, t E R if there exists a connected time-invariant neighbourhood N of the point x = 0, NC H such that V(t, x) is positive semi-definite on T, X A’ and if there exists a positive definite function W(x) on N, W: H-, R satisfying the condition W(x) s V(t, x) V(r, x) E T, x N. (2) Positive definite on T, x S if all the conditions of definition 3.3(l) are fulfilled for N= S. (3) Positive definite in the whole on T, if all the conditions of definition 3.3(l) are fulfilled for N = H.
(-I) Negative definite in the whole on 7, (on vvhole) on T, Con T, x .L’) respectively. In definitions 3.3 an expression “on T,” can fulfilled for T E R [8]. Thus. function (3.6) and its derivative (3.7) signs of which allows us to solve problems on
1. DEFIN[TIOSS
OF
Tr x ,V) if (- V(r. x)) is positive be omitted
OF
(in the
if and only if a11 their conditions
are the scalar ones. the property stabilitv of the hybrid systems.
STABILITY
definite
,A HYBRID
of having
are fixed
S\.STEXI
For Banach spaces Ek and their direct product E we detine a norm \jxli = jj.\.,]i + ]/,y2]i+ + I/x,,,//and formulate definitions of stability of the state of equilibrium x = 0 of system IX. attraction and asymptotic stability takin 9 into account results of the work [5]. Definitiorz -1.1. The state .Y= 0 of the system Z is: (a) stable with respect to Ti if for every f. E T, and every E > 0 there exists 6(t,,. E) > 0 such that for ,Y~E I-2,. /]_~,,]l < 8( lo, E) an inequality ]l~(t; rU. s,,)/l < t‘ Vr E T,, holds; (b) uniformly stable with respect to T, if the conditions of the definition 4.1(a) are fulfilled and for every e > 0 the corresponding maximal value d satisfies the condition inf[b w(~, E): t E T,] > 0: (c) stable in the whole with respect to T, if the conditions of the definition 4.1(a) are fulfilled and a.Ll(f, E)- + x for E- + 2, Vt E T,; (d) uniformly stable in the whole with respect to T, if the conditions of the definition 1.1(b) and (c) are fulfilled; (e) unstable with respect to T, if there exist f. E T,, E E 10. i x[ and r’ E T!.. r > co such that for every b E IO, + x[ there exists x0 E H,. /IsUjj< b for which jl~(r*: to, x,,) 3 E. An expression %th respect to T,‘. in definitions (a)-(e) can be omitted if and only if T, = R. Dejkirion 4.7. The state s = 0 of system X is: (a) attracting with respect to T, if for any t, E T, there exist A(r,) > 0 and any c > 0 there is a T(t,, x0, <) E [O. + x[ such that for .yo E Ho, I/X,,]/< A(r,) an inequality (IX@;lo, x0)]] < ;
Vt E ]to + T(r,l. s,j, ;). + y-[
holds; (b) so-uniform attracting with respect to T, if the conditions of definition 4.2(a) are fulfilled and for any I,, E T, and any c E [0, + x[ there exist A(r,) > 0 and T,,(t,. A(t,). f) E [O. + x[ such that sup[T,,(to.xo,
c): x0 E B,(r,)
C H,] = T,(t,.
A(t,).
;);
c lo uni orm attracting with respect to T, if the conditions of definition 4.7(a) an’d !for-an,’,’ ; E 10. t x[ there exist A > 0 and T,(.r,. CT)E [O. i x[ such that SUP[T,(~~.XO.
(d) uniformly attracting fulfilled. i.e. (a) is correct
f):
are fulfilled
(to,Xg) E T, X Bl C T, X HO] = T,,(A. <);
with respect to T, if the conditions of definitions 4.2(b) and (c) are and for every < E IO, + =[ there exist A > 0 and T,(A. I) E [O. + x[
The Lyapunov matrix-function
l-153
-
such that sup[T,(t,.xO.
;): (fg.~~g) E T, x B, C T, x H,] = T,,(Ls. i”);
(e) properties (a)-(d) hold “in the whole” if the conditions of definition 4.2(a) are fulfilled for any A(r,) E 10, + =[ and f. E T,. An expression “with respect to T,” in definitions 3.5(a)-(e) can be omitted if
Defirzirion 4.3. The state x = 0 of system 2 is: (a) asymptotically stable with respect to T, if definitions 3.4(a) and 4.2(a) hold: (b) equi-asymptotically stable with respect to T, if definitions 4.1(a) and 4.2(b) hold; (c) quasi-uniform asymptotically stable with respect to T, if definitions 4.1(b) and 4.2(c) hold; (d) uniform asymptotically stable with respect to T, if definitions 4.1(b) and 4.2(d) hold; (e) the properties of (a)-(d) hold in the whole if the corresponding type of stability x = 0 and attraction of x = 0 hold in the whole. In definitions (a)-(e) an expression “with respect to T,” can be omitted if T, = R.
5. THEOREMS
OS STABILITY
According to the definitions accepted in Section 1 we consider stability of the state of equilibrium of a hybrid system with respect to one measure p = ]jx](.The presence of a matrixfunction B(f, X) and, consequently, of the function V(r, x) (formula (3.6)) with the derivative D’V(t, x) (formula (3.7)), allows us to formulate all basic theorems on stability and instability with respect to a measure. similarly as it has been done in the monograph [5] for the nonstationary systems. Moreover, numerous classical results, illustrated in monographs [6. 913] can also be extended onto the hybrid systems on the basis of a suitable matrix-function. 5.1. Let a hybrid system Z be such that there exist: (a) an open connected neighbourhood N of the point x = 0; (b) the matrix-function ‘Z(t. X) with the element
THEOREM
V,: T, x A’+
R,
V(i.j)
E [I, m];
(c) a vector q* > 0 with the element
‘7i> 0 Vi E [l. M] such that: (d) a defined by the expression (3.6) function V(r, x) which is continuously decreasing and positive definite with respect to a measure p on N(T, X N); (e) D+V(t, x) 6 0 V(r, x) E R x N [V(t, x) E T, x N]. Then the state of equilibrium x = 0 of system E is uniformly stable with respect to a measure p [on T,] respectively. Proof. Having fulfilled conditions (b)-(d) of the theorem we obtain the function V(t.x): T, x A’--, R which is a continuous. scalar, decreasing. positive definite on N (on T, x iv) one.
Together with condition (e). stability of the state x = 0 with respect to one measure follows from the theorem 2 of [l3]. The fact that the function V(t, x) is decreasing ensures uniformity of asymptotic stability on T, (see [j]).
THEOREM 5.2. Let a hybrid system5 be such that a vector-function on R x H (on I, x H). If there exist: (a) a matrix-function 9(t, x):
V,j: R X H*
R,V(i,j)
Xbe definite and continuous
E [I., m]:
(b) a vector
qr > 0 with the element 17,> 0 Vi E [l. m] such that: (c) a defined by the expression (3.6) function V(r, x) which is radially unbounded. decreasing and positive definite with respect to a measure p in the whole (on T,): (d) D+V(r,x) 2 q’D’%(r,x)q S 0, ‘v’(f,x) E R x H, [(t-x) E Y, x H]. Then the state of equilibrium x = 0 of system 1 is uniformly stable with respect to a measure p in the whole (on T,). THEOREM 5.3. Let a hybrid system X be such that: (a) conditions (a)-(d) of the theorem 5.1 are fulfilled; (b) a positive definite function v exists on N such that D’V(t, x) s -i+(x)
V(~,X) E R x N(V(r,x)
E T, x A’).
Then the state of equilibrium x = 0 of the system Z is uniform asymptotically respect to a measure p (on T,).
stable with
Proof. Having fulfilled condition (a) of the theorem 5.3 we obtain the function V(t. x) which is decreasing, positive definite with respect to a measure p on N (on T, X N). In virtue of the condition (b) its derivative D-V(t,x) is negative definite on iv’ (on T, X A’). Hence. V(r) s V(t,) - y(t - r,). Further on the standard arguments [13] lead to the statement of the theorem. THEOREM 5.4. Let a hybrid (a) conditions (b) a positive
(a)-(c) definite
system Z be such that: of the theorem 5.2 are fulfilled; with respect to a measure p in the whole function
D+V(t, x) s - I$(x)
w exists such that
V(r, x) E R x H(V(t, X) E T, x H).
Then the state of equilibrium x = 0 of the system whole with respect to a measure p.
C is uniform
asymptotically
stable
in the
Proof. Taking into account the said above in the proofs of the theorems of the theorem
is obtained
similarly
to the known
proofs
5.1.5.3 the statement (see, for example, [5, pp. 40-421).
6. STABILITY OF THE TWO-COMPONENT SYSTEMS ANALYSIS We consider a two-component system 2, (see, for example, [15, 171)
(6.1) (6.2)
The Lyapunov
matrix-function
where w(to, z) =
wO(z).,M(t,
d/az) wldR = ~‘(1. s),s
z.
E
aS?.R C R’.
X:ToxH--+R”;L:B,-+BI;~M:B,+B3.woEB,, L, M are some differential operators and B1,. . ., B, are Banach A hybrid
system
Z, consists
of the independent u,:
(72:
and interconnection
functions
dx z
X(r.x(r)):
=
Jg=L(t.
between
spaces.
subsystems (6.3)
7. a/a:)
bt
(6.4)
them
g,=g,(t.z.x,~):T~xRxHxQ+R”, g2 =g?(t.z.x. Let us introduce between them.
two assumptions
w): To x R x H x Q-R”. on subsystems
o,.
cr2 and
interconnection
functions
Assumption
1. There exist: (1) Functions VII(t, x) E %(R x N,), VJ2(t, w) E %(R x dV,,). ‘V, C C2,N,,. C Q are open connected neighbourhoods of the points x = 0, w = 0 respectively. (2) Functions qc:,([jx][), yi(][w]j) of the K-class [12]. i = 1. 2. (3) Positive values -cu,,. Cut;,i = 1, 2 and such estimations are fulfilled (uii&i]X]>
c V,,(t. x) s E,, gW).
(yT7~+u;(~~w~~) s v2& _-(4) Functions
w) s 6:: w3llwi)
V,,(t. s. w) = Vz,(t, X, w) V,Z(t, X,
(5) Such positive
values
W)
E
%(R X iv., X Lv,,).
-cy,?, Cu,? that
~,?~,(ll~ll)v/,(IIWII> G V,z(f,,Y, )V)s
LEMMA 6.1. If all the conditions
are definite
positive,
of assumption
~,~47I~ll-~!l~w~~ll~ll~.
1 are fulfilled
and matrices
then the function V(t, x, w) = qT3(t, x, w)q,
where
qT = (vi, qz) is a vector
with positive
components,
is definite
(6.5) positive
and decreasing.
1456
MARTYNYUK
A. A.
Proof.
Similarly to [14] we introduce the notations rT = (f7~l(ll-~ll>~ V~l(l/wlI))-qT = (vJ2(llxll)~ v2(IIwll))~ B = (Z’
“,,I.
Having fulfilled the conditions of assumption 1 for the function (6.5) such a bilateral estimation rTBrAIBrS
qTa(t,x,
~qTBTA~Bq
w)~
(6.6)
holds. In virtue of conditions of the lemma 6.1 it follows from the estimation (6.6) that the function V(t, x, w) is definite positive and decreasing. Assumption
2. There exist: V,,(t,x), Vzz(t, w) given in the assumption
l(1) and functions V,l(t.x,
(1) Functions
VZl(L x7 w). (2) Such constants
w) =
&, i = 1, 2; k = 1,. . . , 8, functions [, = ~,(jlxl~)and cr = cz(\jwl\) of the
K-class [12] that: (a) 0: VU (f, x) + 0: V,,(r, ~)Ix s PM C:; (b) 0:
V,,(t, x>I,, c Piz C-t +
(c)
D:V,,(f,
(4
D;:V,,(r>
41g2
(e)
0:
x,
(f)
J’E(~,
w)
+
D;V22(f, s
P13 51;2; w)Iw
6
P2,Tt;
P22C?1 + P23f1i2;
W) +
0:
D:Vl2(t,x,
W)(L
“Pz&
+ ,&si,Cz;
(g)
0:
w&l
s
PlbG
+
Pl’c-It;2
+
Pl&
(h)
D,:V,,(bx,
w&z
s
P26 c:
+ P27 Cl c,
+
P2s G.
Vlz(t>x?
Vlz(t,
LEMMA 6.2. If all conditions
x,
w)Ix
s
P14CI
+
Plj
Cl 52;
of the assumption 2 are fulfilled and the matrix c =
c11 c12 ) Cl2 = czl
c C21
c22
1
with the elements [14] Cl1 =
ml1
+ PE)
+ 21717I2(Pll
+ PI6
c22 =
m21
+
+ 2771772(Pla
+
CIZ = i(qiP13
P22) +
qip23)
+
77172@15
+ P26)i
P24 + Pzs); + p25
+
P17 +
827)
is definite negative, then a derivative D’V(t,
x, w) = $D+%(t,
x, w)q
of the function V(t, x, w) is a definite negative function in virtue of the system X1. Proof.
In virtue of the estimations
(a)-(d) D+V(t,x,
holds, where pT = (~l(IcYII)~ ~dll41>). A definite negativeness of the derivative
of the assumption 2 and estimation w) cpTCp
follows from the condition of the lemma 6.2.
The Lyapunov matrix-function
1457
6.1. If a two-component system Zi is such that all conditions of lemmas 6.1 and 6.2 are fulfilled, then the state of equilibrium x = 0, w = 0 of system Z1 is uniform asymptotically stable. If in the assumptions 1 and 2 N, = Rk, N, = Q, functions vi, vi, c, belong to the KR-class [12] and conditions of the lemmas 6.1, 6.2 are fulfilled, then the state of equilibrium x = 0, w = 0 of the system Xi is uniform asymptotically stable in the whole. THEOREM
Proof Having fulfilled the enumerated conditions the function V and its full derivative satisfies all conditions of the theorems 5.3, 5.4. It proves the statement of the theorem 6.1. Remark. If in estimations (a)-(d) of assumption 2 we change the sign of the inequality for the opposite one and leave in inequalities (3), (5) o f assumption 1 only estimations below, then it isn’t difficult to define conditions of instability of the state x = 0, w = 0 of the system Xi. 7. CONCLUDING
REMARKS
An application of the Lyapunov matrix-function to analysis of stability of the hybrid systems formalizes methods of construction of the suitable Lyapunov functions (functionals) and leads to a situation of applicability of the classical results obtained for nonstationary systems. Let us pay attention to the fact that in conditions (a), (b), of the assumption 2 it is not required that pii < 0 and /& < 0 should be presented simultaneously, i.e. the state of equilibrium x = 0, w = 0 of one of the subsystems crl, crz can be unstable, whereas the whole system Zi is stable. This fact can be significant while solving the applied problems. Acknowledgemenr--I am cordially thankful to Professor D. D. Siljak, who has kindly enabled me to study the work [16] which gave inspiration for this paper. REFERENCES 1. MARTYNYUKA. A., The Lyapunov matrix-function, Nonlinear Analysis 8, 1223-1226 (1984). 2. MARTYNYUK A. A., On application of the Lyapunov matrix-functions m the theory of stability, Nonlinear Analysti 9, 1495-1501 (1985). 3. ZUBOV V. I., A. M. Lyapunov’s methods and their application, Izd. Leningr. gosunio-ta 238 (1957). 4. MARTYNYUKA. A., Sfubiliry of Motion of Complex Systems, p. 351, Nauka Dumka, Kiev (1975). 5. GRUJI~ LJ. T., MARTYNYVKA. A. & RIBBENS-PAVELLA M., Stability of Large-Scale Systems Under rhe Structural and Singular Perturbations, p. 305, Nauka Dumka, Kiev (198-t). 6. LYAPUNOVA. M., General problem of stability of motion, Kharkov (1892). 7. MATROSOVV. M., Vector Lyapunov functions in the analysis of nonlinear interconnected systems. Symp. Insf. Naz. Alta Mat. Mathematics VI, 209-242 (1971). 8. GRUJ~CLJ. T., Novel development of Lyapunov stability of motion, Int. I. Control 22, 529-549 (1975). 9. LAKSHMIKANTHA.M V. & LEELA S., Differential and Integral Inequalities-Theory and Applications, Vol. I, Academic Press, New York (1969). 10. HALANAYA., Differential Equations: Stability, Oscillations, Time Delay, Academic Press, New York (196-t). 11. YOSHISAWAT., The Stability Theory by Lyupunou’s Second JIethod, Mathematical Society of Japan, Tokyo (1966). 12. HAHN W., Stability of Motion, Springer, Berlin (1967). 13. SI~ZETDINOV T. K., Stability of Systemswirh Distributed Parameters, p. 180, Kazan (1971). 14. DJORDJEVICM. Z., Stability analysis of large-scale systems whose subsystems may be unstable. Report No. 83-A, ETH, Zurich (1983). 15. BARTYSHEVA. V., The vector Lyapunov functions application for an analysis of the two-component systems, in The Vector L upunoo Functions and their Construction, pp. 237-257, Nauka, Novosibirsk (1980). 16. IKEDA M. & 8 ILJAK D. D., Hierarchical Liapunov functions, Proc. 22nd IEEE Conf. on Decision and Control. San Antonio, Texas, December 14-16 (1983). 17. RASMUSSENR. D. & MICHELA. N., Stability of interconnected dynamical systems described on Banach space. IEEE Trans. Automatic Control AC-21, 464-471 (1976).
Methods & Appli~mom,
Vol
-
THE LYAPUNOV
10. No.
12. pp. 151S1457,
1986
O362-M.Y.86 Pergtmon
_
MATRIX-FUNCTION AND STABILITY SYSTEMS
$3 00 - 00 Journals Ltd.
OF HYBRID
A. A. MARTYNYUK Institute
of Mechanics,
The Ukrainian
(Received
1 February
Academy
of Sciences,
Nesterov
Str., 3, Kiev-57.
1985; received for publication 20 January
Key words and phrases: Lyapunov matrix-function, whole), asymptotic stability, two-component system,
Ukr.
SSR. 252057
1986)
hybrid system. stability, uniform stability (in the property of having fixed sign of matrix-functions.
1. INTRODUCTION
THE LYAPUNOVmatrix-function [l, 21 allows us to apply construction of the direct Lyapunov method [6] to analysis of hybrid systems, the free subsystems of which can be unstable. In this case the outside of diagonal elements of the matrix-function must supply the more complete account of the effect of interconnection functions, which in case of the vector-functions application can be considered sometimes as a stabilizing factor. Let us also remark that attempts to take into account the effect of interconnection functions as the stabilizing ones, were made even earlier [4] while consideration of the weakly connected subsystems, but effectivity of the criteria obtained in this very case, can be defined by existence of the general solution of independent subsystems. In a number of cases, particularly for the weakly connected oscillating systems of the second order, these criteria are effective [4,5]. The method of analysis proposed here is based on a knowledge of the Lyapunov matrix-function. The present paper is the further development of work [2]. 2. STATING
THE
PROBLELM
We consider a system 1, which consists of the m-interconnected subsystems tiJ!..i E [l, m]. System 2 can be called a hybrid one if subsystems 4 (or free oj) are described by different types of equations (finite-dimensional, integro-differential equations of parabolic type with a strongly elliptical operator, functional-differential ones, etc.). The couple (ai, go. i E [l, m] determines an i-interconnected subsystem in system Z. We shall not concretize yet the type of subsystems 4 (subsystems 0;) and interconnection functions g, between them, but suppose only that solutions of the correspondent Cauchy problems for each subsystem are defined correctly, and the state of equilibrium of system Z in the direct product of Banach spaces B, x B2 x . . . x B, = H, is the point 0. The aim of the present paper is a solution of the problem on stability of the state of equilibrium 0 of the system Z in case when there are both stable and unstable subsystems among oi, i E [l, m] and stability of the state of equilibrium of the whole system Z can be obtained by the stabilizing effect of the interconnection functions gi, i E [l, M].
It is widely (Oi,
giP
E
[I,
known ml)
3. THE LYAPUNOV MATRIX-FUNCTION [7] that the functioning of the hybrid system 2 (of the t%ality
can be described by a differential
equation in Banach space E = n Ei, i=l
1119
1350
A.
A.
L~~ARTYNYLK
where Ei are Banach spaces, corresponding to subsystems (oi, gJ i E [ 1, m] with the norms ]/ . Iii. For a hyper-vector xr = (XT, . . . , xi)’ this equation has the form
z: g = X(t, where X : T, X H+
x), (X(&j) =
x0
E Ho c
E), X(t, 0) = 0,
(3.1)
E, H = fi Hk (a compact in E set), Ho = fi Hok, X is in general a k=l
k=l
discontinuous, nonbounded operator), defined on a set S2 = T, x H C I’; I- = T, X E. Under a solution ~(t; to, x0) of the equation (3.1) on T, = [t, + x[, t E R for t0 E T, C R and x0 E Ho we understand an absolutely continuous on T, abstract vector-function, which satisfies this equation almost everywhere on T,. Let us recollect that system (3.1) consists of the isolated subsystems ui : dt
=
XJt, xi);
Xi
E
where Xi: T, x Hi+ Ei, which can be united by the interconnection gi: gi =
such that g,: T, x HI x . . . x H,+
gi(r>
Xl
9 *
.
.,x,),
(3.2)
Ei, functions
i E [l, M]
Ei. By a couple (ai, gJ an i-interconnected
(oi,gi):%=X;(t,Xi)
+gi(r,xr,.
(3.3) sybsystem
. *,X,)
is described for i = j respectively. Together with equations (3.4) for i E [l, m] we consider the real functionals j(z [l,m], vii: r + R and a “matrix-function” (matrix-functional)
B(f~x, = ivij(l,
x>l~j=l
(3.1)
Vij(r, x); i,
(3.5)
which maps r onto Rmx m. Elements Vij(t, Xi) for i = j can be constructed on the basis of equations of the free subsystems ai and Vii(t, x) for i # j can be constructed having taken into account the interconnection functions between i and j of subsystems ui and Uj. By means of a vector nT = (ql,. . ,, q,,J E R”, vi > 0, Vi E [l, m] we shall construct a scalar function (see [2, 141) V(t, x) = Il W, x)rl
(3.6)
for a hybrid system 2. Let us define the upper right derivative of the function V(t, x) in virtue of the system 2 by formula D+v(t,x)
= nrD+%(t,X)~,
D+93(r,x)
~{D+vii(‘,x)}~j_I
where
(3.7)
The Lyapunov
matrix-function
1151
and
D’ V,(t, X) = lim e-o+
SUP
0
f
{V,[t+
0,x+
&X(r,x)] - V,(t,x)},i,jE
[l,m].
(3.8)
If V,(t, x) are continuously differential in all their arguments, then D’B(t,
X) ’ {DVij(t, x)}Y,=~
and in detail V(i==j): DVii(fyX) = aVii/at+
VVii(t,Xi)r(X~(t,X1)
V(i#j):
m C, [VV,(t,~)r(X<(r,xi)
DV,(t,X)
= dV,/dt+
+ gi(ttXlj * . .,Xm))i (3.9) + gi(f,xl,.
. ..~m))].
i=l
Definition 3.1. Matrix-function B(t, x) can be called the Lyapunov one on T, X G. G c N if: (1) Function (3.6) is absolutely semi-continuous over any interval [to, rl] C T,. (2) D+V(r,x) 6 0 on T, x G. On the basis of function (3.6) and its derivative (3.7) for a hybrid system ): the theorems on stability are stated according to the scheme of the direct Lyapunov method. At first let us give some definitions. Definition 3.2. The function V(t, x): r + R can be called: (1) Positive semi-definite on T, if there exists a connected time-invariant neighbourhood N of the point x = 0, NC H such that: (a) V(t, x) is absolutely semi-continuous over any internal [to, ti] C T,; (b) V(t, x) is nonnegative on T, x N, i.e. V(t, x) > 0 V(r. x) E T, x N; (c) V(t, X) vanishes for x = 0 (Vij(t, 0) = 0 Vt E Tr). (2) Positive semi-definite on T, x cp if the conditions of the definition 3.2(l) are fulfilled for N = S. (3) Semi-definite in the whole on T, if the conditions of the definition 3.2(l) are fulfilled for N = H. (4) Negative semi-definite (in the whole) on T, (on T, x N) if (- V(r, x)) is positive semidefinite (in the whole on T, (on T, x N)) respectively. In definitions 3.2 an expression “on T,” can be omitted if and only if all their conditions are fulfilled for t E R [8]. Definition 3.3. The function V(t, x): r-, R is to be called: (1) Positive definite on T,, t E R if there exists a connected time-invariant neighbourhood N of the point x = 0, NC H such that V(t, x) is positive semi-definite on T, X A’ and if there exists a positive definite function W(x) on N, W: H-, R satisfying the condition W(x) s V(t, x) V(r, x) E T, x N. (2) Positive definite on T, x S if all the conditions of definition 3.3(l) are fulfilled for N= S. (3) Positive definite in the whole on T, if all the conditions of definition 3.3(l) are fulfilled for N = H.
(-I) Negative definite in the whole on 7, (on vvhole) on T, Con T, x .L’) respectively. In definitions 3.3 an expression “on T,” can fulfilled for T E R [8]. Thus. function (3.6) and its derivative (3.7) signs of which allows us to solve problems on
1. DEFIN[TIOSS
OF
Tr x ,V) if (- V(r. x)) is positive be omitted
OF
(in the
if and only if a11 their conditions
are the scalar ones. the property stabilitv of the hybrid systems.
STABILITY
definite
,A HYBRID
of having
are fixed
S\.STEXI
For Banach spaces Ek and their direct product E we detine a norm \jxli = jj.\.,]i + ]/,y2]i+ + I/x,,,//and formulate definitions of stability of the state of equilibrium x = 0 of system IX. attraction and asymptotic stability takin 9 into account results of the work [5]. Definitiorz -1.1. The state .Y= 0 of the system Z is: (a) stable with respect to Ti if for every f. E T, and every E > 0 there exists 6(t,,. E) > 0 such that for ,Y~E I-2,. /]_~,,]l < 8( lo, E) an inequality ]l~(t; rU. s,,)/l < t‘ Vr E T,, holds; (b) uniformly stable with respect to T, if the conditions of the definition 4.1(a) are fulfilled and for every e > 0 the corresponding maximal value d satisfies the condition inf[b w(~, E): t E T,] > 0: (c) stable in the whole with respect to T, if the conditions of the definition 4.1(a) are fulfilled and a.Ll(f, E)- + x for E- + 2, Vt E T,; (d) uniformly stable in the whole with respect to T, if the conditions of the definition 1.1(b) and (c) are fulfilled; (e) unstable with respect to T, if there exist f. E T,, E E 10. i x[ and r’ E T!.. r > co such that for every b E IO, + x[ there exists x0 E H,. /IsUjj< b for which jl~(r*: to, x,,) 3 E. An expression %th respect to T,‘. in definitions (a)-(e) can be omitted if and only if T, = R. Dejkirion 4.7. The state s = 0 of system X is: (a) attracting with respect to T, if for any t, E T, there exist A(r,) > 0 and any c > 0 there is a T(t,, x0, <) E [O. + x[ such that for .yo E Ho, I/X,,]/< A(r,) an inequality (IX@;lo, x0)]] < ;
Vt E ]to + T(r,l. s,j, ;). + y-[
holds; (b) so-uniform attracting with respect to T, if the conditions of definition 4.2(a) are fulfilled and for any I,, E T, and any c E [0, + x[ there exist A(r,) > 0 and T,,(t,. A(t,). f) E [O. + x[ such that sup[T,,(to.xo,
c): x0 E B,(r,)
C H,] = T,(t,.
A(t,).
;);
c lo uni orm attracting with respect to T, if the conditions of definition 4.7(a) an’d !for-an,’,’ ; E 10. t x[ there exist A > 0 and T,(.r,. CT)E [O. i x[ such that SUP[T,(~~.XO.
(d) uniformly attracting fulfilled. i.e. (a) is correct
f):
are fulfilled
(to,Xg) E T, X Bl C T, X HO] = T,,(A. <);
with respect to T, if the conditions of definitions 4.2(b) and (c) are and for every < E IO, + =[ there exist A > 0 and T,(A. I) E [O. + x[
The Lyapunov matrix-function
l-153
-
such that sup[T,(t,.xO.
;): (fg.~~g) E T, x B, C T, x H,] = T,,(Ls. i”);
(e) properties (a)-(d) hold “in the whole” if the conditions of definition 4.2(a) are fulfilled for any A(r,) E 10, + =[ and f. E T,. An expression “with respect to T,” in definitions 3.5(a)-(e) can be omitted if
Defirzirion 4.3. The state x = 0 of system 2 is: (a) asymptotically stable with respect to T, if definitions 3.4(a) and 4.2(a) hold: (b) equi-asymptotically stable with respect to T, if definitions 4.1(a) and 4.2(b) hold; (c) quasi-uniform asymptotically stable with respect to T, if definitions 4.1(b) and 4.2(c) hold; (d) uniform asymptotically stable with respect to T, if definitions 4.1(b) and 4.2(d) hold; (e) the properties of (a)-(d) hold in the whole if the corresponding type of stability x = 0 and attraction of x = 0 hold in the whole. In definitions (a)-(e) an expression “with respect to T,” can be omitted if T, = R.
5. THEOREMS
OS STABILITY
According to the definitions accepted in Section 1 we consider stability of the state of equilibrium of a hybrid system with respect to one measure p = ]jx](.The presence of a matrixfunction B(f, X) and, consequently, of the function V(r, x) (formula (3.6)) with the derivative D’V(t, x) (formula (3.7)), allows us to formulate all basic theorems on stability and instability with respect to a measure. similarly as it has been done in the monograph [5] for the nonstationary systems. Moreover, numerous classical results, illustrated in monographs [6. 913] can also be extended onto the hybrid systems on the basis of a suitable matrix-function. 5.1. Let a hybrid system Z be such that there exist: (a) an open connected neighbourhood N of the point x = 0; (b) the matrix-function ‘Z(t. X) with the element
THEOREM
V,: T, x A’+
R,
V(i.j)
E [I, m];
(c) a vector q* > 0 with the element
‘7i> 0 Vi E [l. M] such that: (d) a defined by the expression (3.6) function V(r, x) which is continuously decreasing and positive definite with respect to a measure p on N(T, X N); (e) D+V(t, x) 6 0 V(r, x) E R x N [V(t, x) E T, x N]. Then the state of equilibrium x = 0 of system E is uniformly stable with respect to a measure p [on T,] respectively. Proof. Having fulfilled conditions (b)-(d) of the theorem we obtain the function V(t.x): T, x A’--, R which is a continuous. scalar, decreasing. positive definite on N (on T, x iv) one.
Together with condition (e). stability of the state x = 0 with respect to one measure follows from the theorem 2 of [l3]. The fact that the function V(t, x) is decreasing ensures uniformity of asymptotic stability on T, (see [j]).
THEOREM 5.2. Let a hybrid system5 be such that a vector-function on R x H (on I, x H). If there exist: (a) a matrix-function 9(t, x):
V,j: R X H*
R,V(i,j)
Xbe definite and continuous
E [I., m]:
(b) a vector
qr > 0 with the element 17,> 0 Vi E [l. m] such that: (c) a defined by the expression (3.6) function V(r, x) which is radially unbounded. decreasing and positive definite with respect to a measure p in the whole (on T,): (d) D+V(r,x) 2 q’D’%(r,x)q S 0, ‘v’(f,x) E R x H, [(t-x) E Y, x H]. Then the state of equilibrium x = 0 of system 1 is uniformly stable with respect to a measure p in the whole (on T,). THEOREM 5.3. Let a hybrid system X be such that: (a) conditions (a)-(d) of the theorem 5.1 are fulfilled; (b) a positive definite function v exists on N such that D’V(t, x) s -i+(x)
V(~,X) E R x N(V(r,x)
E T, x A’).
Then the state of equilibrium x = 0 of the system Z is uniform asymptotically respect to a measure p (on T,).
stable with
Proof. Having fulfilled condition (a) of the theorem 5.3 we obtain the function V(t. x) which is decreasing, positive definite with respect to a measure p on N (on T, X N). In virtue of the condition (b) its derivative D-V(t,x) is negative definite on iv’ (on T, X A’). Hence. V(r) s V(t,) - y(t - r,). Further on the standard arguments [13] lead to the statement of the theorem. THEOREM 5.4. Let a hybrid (a) conditions (b) a positive
(a)-(c) definite
system Z be such that: of the theorem 5.2 are fulfilled; with respect to a measure p in the whole function
D+V(t, x) s - I$(x)
w exists such that
V(r, x) E R x H(V(t, X) E T, x H).
Then the state of equilibrium x = 0 of the system whole with respect to a measure p.
C is uniform
asymptotically
stable
in the
Proof. Taking into account the said above in the proofs of the theorems of the theorem
is obtained
similarly
to the known
proofs
5.1.5.3 the statement (see, for example, [5, pp. 40-421).
6. STABILITY OF THE TWO-COMPONENT SYSTEMS ANALYSIS We consider a two-component system 2, (see, for example, [15, 171)
(6.1) (6.2)
The Lyapunov
matrix-function
where w(to, z) =
wO(z).,M(t,
d/az) wldR = ~‘(1. s),s
z.
E
aS?.R C R’.
X:ToxH--+R”;L:B,-+BI;~M:B,+B3.woEB,, L, M are some differential operators and B1,. . ., B, are Banach A hybrid
system
Z, consists
of the independent u,:
(72:
and interconnection
functions
dx z
X(r.x(r)):
=
Jg=L(t.
between
spaces.
subsystems (6.3)
7. a/a:)
bt
(6.4)
them
g,=g,(t.z.x,~):T~xRxHxQ+R”, g2 =g?(t.z.x. Let us introduce between them.
two assumptions
w): To x R x H x Q-R”. on subsystems
o,.
cr2 and
interconnection
functions
Assumption
1. There exist: (1) Functions VII(t, x) E %(R x N,), VJ2(t, w) E %(R x dV,,). ‘V, C C2,N,,. C Q are open connected neighbourhoods of the points x = 0, w = 0 respectively. (2) Functions qc:,([jx][), yi(][w]j) of the K-class [12]. i = 1. 2. (3) Positive values -cu,,. Cut;,i = 1, 2 and such estimations are fulfilled (uii&i]X]>
c V,,(t. x) s E,, gW).
(yT7~+u;(~~w~~) s v2& _-(4) Functions
w) s 6:: w3llwi)
V,,(t. s. w) = Vz,(t, X, w) V,Z(t, X,
(5) Such positive
values
W)
E
%(R X iv., X Lv,,).
-cy,?, Cu,? that
~,?~,(ll~ll)v/,(IIWII> G V,z(f,,Y, )V)s
LEMMA 6.1. If all the conditions
are definite
positive,
of assumption
~,~47I~ll-~!l~w~~ll~ll~.
1 are fulfilled
and matrices
then the function V(t, x, w) = qT3(t, x, w)q,
where
qT = (vi, qz) is a vector
with positive
components,
is definite
(6.5) positive
and decreasing.
1456
MARTYNYUK
A. A.
Proof.
Similarly to [14] we introduce the notations rT = (f7~l(ll-~ll>~ V~l(l/wlI))-qT = (vJ2(llxll)~ v2(IIwll))~ B = (Z’
“,,I.
Having fulfilled the conditions of assumption 1 for the function (6.5) such a bilateral estimation rTBrAIBrS
qTa(t,x,
~qTBTA~Bq
w)~
(6.6)
holds. In virtue of conditions of the lemma 6.1 it follows from the estimation (6.6) that the function V(t, x, w) is definite positive and decreasing. Assumption
2. There exist: V,,(t,x), Vzz(t, w) given in the assumption
l(1) and functions V,l(t.x,
(1) Functions
VZl(L x7 w). (2) Such constants
w) =
&, i = 1, 2; k = 1,. . . , 8, functions [, = ~,(jlxl~)and cr = cz(\jwl\) of the
K-class [12] that: (a) 0: VU (f, x) + 0: V,,(r, ~)Ix s PM C:; (b) 0:
V,,(t, x>I,, c Piz C-t +
(c)
D:V,,(f,
(4
D;:V,,(r>
41g2
(e)
0:
x,
(f)
J’E(~,
w)
+
D;V22(f, s
P13 51;2; w)Iw
6
P2,Tt;
P22C?1 + P23f1i2;
W) +
0:
D:Vl2(t,x,
W)(L
“Pz&
+ ,&si,Cz;
(g)
0:
w&l
s
PlbG
+
Pl’c-It;2
+
Pl&
(h)
D,:V,,(bx,
w&z
s
P26 c:
+ P27 Cl c,
+
P2s G.
Vlz(t>x?
Vlz(t,
LEMMA 6.2. If all conditions
x,
w)Ix
s
P14CI
+
Plj
Cl 52;
of the assumption 2 are fulfilled and the matrix c =
c11 c12 ) Cl2 = czl
c C21
c22
1
with the elements [14] Cl1 =
ml1
+ PE)
+ 21717I2(Pll
+ PI6
c22 =
m21
+
+ 2771772(Pla
+
CIZ = i(qiP13
P22) +
qip23)
+
77172@15
+ P26)i
P24 + Pzs); + p25
+
P17 +
827)
is definite negative, then a derivative D’V(t,
x, w) = $D+%(t,
x, w)q
of the function V(t, x, w) is a definite negative function in virtue of the system X1. Proof.
In virtue of the estimations
(a)-(d) D+V(t,x,
holds, where pT = (~l(IcYII)~ ~dll41>). A definite negativeness of the derivative
of the assumption 2 and estimation w) cpTCp
follows from the condition of the lemma 6.2.
The Lyapunov matrix-function
1457
6.1. If a two-component system Zi is such that all conditions of lemmas 6.1 and 6.2 are fulfilled, then the state of equilibrium x = 0, w = 0 of system Z1 is uniform asymptotically stable. If in the assumptions 1 and 2 N, = Rk, N, = Q, functions vi, vi, c, belong to the KR-class [12] and conditions of the lemmas 6.1, 6.2 are fulfilled, then the state of equilibrium x = 0, w = 0 of the system Xi is uniform asymptotically stable in the whole. THEOREM
Proof Having fulfilled the enumerated conditions the function V and its full derivative satisfies all conditions of the theorems 5.3, 5.4. It proves the statement of the theorem 6.1. Remark. If in estimations (a)-(d) of assumption 2 we change the sign of the inequality for the opposite one and leave in inequalities (3), (5) o f assumption 1 only estimations below, then it isn’t difficult to define conditions of instability of the state x = 0, w = 0 of the system Xi. 7. CONCLUDING
REMARKS
An application of the Lyapunov matrix-function to analysis of stability of the hybrid systems formalizes methods of construction of the suitable Lyapunov functions (functionals) and leads to a situation of applicability of the classical results obtained for nonstationary systems. Let us pay attention to the fact that in conditions (a), (b), of the assumption 2 it is not required that pii < 0 and /& < 0 should be presented simultaneously, i.e. the state of equilibrium x = 0, w = 0 of one of the subsystems crl, crz can be unstable, whereas the whole system Zi is stable. This fact can be significant while solving the applied problems. Acknowledgemenr--I am cordially thankful to Professor D. D. Siljak, who has kindly enabled me to study the work [16] which gave inspiration for this paper. REFERENCES 1. MARTYNYUKA. A., The Lyapunov matrix-function, Nonlinear Analysis 8, 1223-1226 (1984). 2. MARTYNYUK A. A., On application of the Lyapunov matrix-functions m the theory of stability, Nonlinear Analysti 9, 1495-1501 (1985). 3. ZUBOV V. I., A. M. Lyapunov’s methods and their application, Izd. Leningr. gosunio-ta 238 (1957). 4. MARTYNYUKA. A., Sfubiliry of Motion of Complex Systems, p. 351, Nauka Dumka, Kiev (1975). 5. GRUJI~ LJ. T., MARTYNYVKA. A. & RIBBENS-PAVELLA M., Stability of Large-Scale Systems Under rhe Structural and Singular Perturbations, p. 305, Nauka Dumka, Kiev (198-t). 6. LYAPUNOVA. M., General problem of stability of motion, Kharkov (1892). 7. MATROSOVV. M., Vector Lyapunov functions in the analysis of nonlinear interconnected systems. Symp. Insf. Naz. Alta Mat. Mathematics VI, 209-242 (1971). 8. GRUJ~CLJ. T., Novel development of Lyapunov stability of motion, Int. I. Control 22, 529-549 (1975). 9. LAKSHMIKANTHA.M V. & LEELA S., Differential and Integral Inequalities-Theory and Applications, Vol. I, Academic Press, New York (1969). 10. HALANAYA., Differential Equations: Stability, Oscillations, Time Delay, Academic Press, New York (196-t). 11. YOSHISAWAT., The Stability Theory by Lyupunou’s Second JIethod, Mathematical Society of Japan, Tokyo (1966). 12. HAHN W., Stability of Motion, Springer, Berlin (1967). 13. SI~ZETDINOV T. K., Stability of Systemswirh Distributed Parameters, p. 180, Kazan (1971). 14. DJORDJEVICM. Z., Stability analysis of large-scale systems whose subsystems may be unstable. Report No. 83-A, ETH, Zurich (1983). 15. BARTYSHEVA. V., The vector Lyapunov functions application for an analysis of the two-component systems, in The Vector L upunoo Functions and their Construction, pp. 237-257, Nauka, Novosibirsk (1980). 16. IKEDA M. & 8 ILJAK D. D., Hierarchical Liapunov functions, Proc. 22nd IEEE Conf. on Decision and Control. San Antonio, Texas, December 14-16 (1983). 17. RASMUSSENR. D. & MICHELA. N., Stability of interconnected dynamical systems described on Banach space. IEEE Trans. Automatic Control AC-21, 464-471 (1976).