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Multiparameter singular perturbations of linear systems with multiple time scales
0005 1098/83 $3.00 + 0.00 Pergamon Press Ltd. © 1983 International Federation of Automatic Control.
Autoraatica, Vol. 19, No. 4, pp. 385 394, 1983
Printed in Great Britain.
Multiparameter Singular Perturbations of Linear Systems with Multiple Time Scales* G. S. LADDEI- and D. D. SILJAK:~
A schemefor order-decomposition and hierarchial aggregation of small parameters may be usedfor multiparameter singular perturbation of linear systems, when it Is essential to realize a multiple time scales assumption. Key Words--Perturbation techniques; approximation theory; system-order reduction; stability; hierarchial systems.
Campbell (1978, 1979); Campbell and Rose (1979). The objective of this paper is to consider a joint multiparameter-multitime scale singular perturbation of linear systems, in which small parameters are grouped into equivalent classes according to their order. A linear transformation is formulated, which can be used in a hierarchial scheme to obtain a suitable form of the original system, that recognizes explicitly the order-decomposition of small parameters. The resulting multitime scale problem can then be solved effectively in the framework of a generalized D-stability (Khalil and Kokotovic, 1979a).
Abstract--A joint multitime scale-multiparameter singular perturbation is formulated and resolved in the context of linear time-varying systems. An interesting feature of the solution procedure is a hierarchial scheme of aggregating and arranging of the groups of small parameters according to their order. The scheme provides a suitable framework for establishing qualitative properties of multitime scale systems.
1. INTRODUCTION MULTIPARAMETER singular perturbation
models arise whenever there are more than one small parameter representing physical constants such as inertias, masses, time constants, etc. (Vasileva, 1963). A way to simplify a multiparameter problem is to consider the parameters to be of the same order allowing for the ignorance of their ratios using the concept of D-stability (Khalil and Kokotovic, 1979a, b; Khalil, 1981). The same order assumption enables one to recast the multiparameter system into a model resembling a single parameter perturbation problem. The assumption can be removed to consider the multiple time-scale problem provided the bounds on the ratios of the parameters are allowed to be arbitrary (Khalil, 1981). These simplifications, however, may not be acceptable in problems where an explicit multiple time-scale assumption is essential and restricted to bounded variations of system parameters. Such problems have been analyzed by Hoppenstead (1971); O'Malley (1974), and more recently by
2. MULTITIME SCALE PERTURBATIONS
Consider a linear system described by
fc = Ao(t)x + i Bok(t)yk k=l
+ ~ Cok(t)xk k=l
'~iYi= A~o(t)x + ~ B~ik(t)yk k=l
+ i C~k(t)z~, ie J k=l
#j~j = A~o(t)x + ~ B~k(t)yk k=l
+ i C~k(t)Zk, j ~
(1)
k=l
* Received 18 March 1982; revised 17 November 1982. The original version of this paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by associate editor M. Jamshidi under the director of editor A. Sage. ]'Department of Mathematics, University of Texas at Arlington, Arlington, TX 76019, U.S.A. :~School of Engineering, University of Santa Clara, Santa Clara, CA 95053, U.S.A.
w h e r e x ~ ~.o, Yl E/~m,, Zj • ,~lj, ~ -- { 1, 2..... r}, j = {1,2,...,s}, m=Xi~jmi, l=IF,j~yl~, and the dimension of the system (1) is n = no + m + I. In (1) all matrix functions are continuous on ~ and have appropriate dimensions. The parameters e~ and #i are positive real numbers, all e~s and #is being of the 385
386
G . S . LADDE and D. D. SILJAK
same order, separately. This means that the ratios of e~s and #~s can be bounded as ~2~<--~<~,
Vi, k ~ J
~k
(2)
~ -< #--~< p, #k
vj, k e J
where, for simplicity, we omit the argument of the matrices. System (7) is equivalent to x = (Ao - Ao2A221A20)Yc + (A01 - Ao2A22XA21)Y, ~(to) = Xo
8y = DI(Alo - A12A21A20)Yc + Dl{All - A12A21A21)Y, y(to) = Yo (8) = - A 2 1 A 2 0 Y c - A21A21~.
where ~ ~, ~, fi are some positive numbers. Our crucial assumption is that e~s and #is are not of the same order. We define e ~--- (/~1~2 . . . . .
The #-boundary layer system is d~
- - = D2A22(to)~(zu), ~(0) = Zo - Z(to)
# -~- ( # 1 # 2 , . . . , # s ) 1Is (3)
er) l/r,
(9)
dz u
and use the following assumption. where z~ = (t - to)/# is the '#-stretched' time scale. A s s u m p t i o n 2.1. #/e ~ 0 as ~ ~ 0 +. Thus (1) is a
three-time scale multiparameter system. With (3), we can simplify the representation (1) as k = A o ( t ) x + A o l ( t ) y + Ao2(t)z, X(to) = Xo ep = D i A l o ( t ) x + D1A11(t)y + D1A12(t)z, y(to) = Yo
Next, the following assumption is introduced. A s s u m p t i o n 2.3. A l l ( t ) nonsingular for all t ~> to.
Letting lim Di-l(e)e = 0, we can reduce (8) to e~0
(4)
= (Ao - Ao2Az~A20)Y¢
#~ = D2A20(t)x + D2A21(t)y + D2A22(t)z, Z(to) = Zo
where y = (y~, Y2T. . . . . Yr) T T E ~ ' n , z = (Zl, T Z2 T . . . . ,Zs) T T ~1, and the block-matrices of (4) are formed in an obvious way from the matrices of (1) with DI and D2 being defined as D1 = diag
Ii1,~I12 .... ,~I1,
+ (Aol - Ao2A2~A21)Y,
(#1
#2
.....
= LAo - A o 2 A 2 ~ A 2 0 - (A01 - A o 2 A z ~ A 2 1 ) ( a l l - A 1 2 A 2 ~ A 2 1 ) -1 (Alo - A 1 2 A 2 ~ A 2 0 ) ] ~ , 2 ( t o ) = Xo (11)
Z12A2~h21)-l(a10 - a12A21A20)~.
The e-boundary layer system is
d: d**
- D 1 [ A 11 (to) - A 12 ( t o ) A 2 # ( t o ) A 21 (to)
]f,
y(0) = Yo - y(to) I1~ and I2j are identity matrices of appropriate dimensions. In view of (2) and (3), the elements of the matrices D1 and DE are bounded as ~: ~< - ~
~,
/~ ~< - - ~
~i
(6)
#j
where the bounds of (6) depend on the corresponding bounds of (2). In order to perform a two-stage reduction of (4), we need the next assumption. A s s u m p t i o n 2.2. A22(t ) is nonsingular for all t t> t o. First, we reduce (4) with respect to # by letting
lim D21(#)# = 0 #~0
= Ao2 + AolY + Ao2Z, 2(to) = Xo ey -----D I A 1 0 x + DIA11 ~ + DIA12z, y(to) = Yo
(7) 0 = A2ox + A21Y + A22z
(10)
which is equivalent to
)7 = - - ( A l l --
~ssI2s}
YC(to) = Xo
0 = (Alo -- A 1 2 A 2 1 A 2 0 ) £ + (All - A12A221A21)Y;
(5) D2= diag(#I21,~122
is
Ax2(t)A2~(t)A21(t)
(12)
where z~ = (t - to)/e is the 'z-stretched' time scale. The presented hierarchial order-reduction scheme provides a suitable basis for incorporating multitime scale singular perturbation in a natural way. Our primary objective is to use the scheme in studying stability of the system (4). As a by-product of this study, we will obtain approximations of solutions to (4) in terms of solutions to the overall reduced system (11) and to the boundary layer systems (9) and (12).
3. TRIANGULARIZATION In order to consider the multitime scale perturbations, we derive a three-fold version of Chang's (1972) transformation, which is necessary for transforming the original system (4) into a triangular form. The transformed system is suitable for deducing the qualitative properties of systems with multitime scale singular perturbations.
Multi-parameter singular perturbations
387
We rewrite the system (4) as
Ii I
IAo(t) Aol(t) = [e-lDtAlo(t) e-tD1Atl(t) LP-lD2A20(t) #-tD2A2a(t)
A0(t) 1[i1 e-tDtA12(t) I
(13)
#- XD2A22(t)J
and use a similarity transformation
til o °oil:l =
1(0
(14)
11
[_L2(t) Lel(t) I
to represent (13) in a triangular form -- Ao2L2 - (Aox - Ao2L2t)L1
Aol -- Ao2L 1
© ©
e-tDI(Ali
- A12L2t) + La(Aol - Ao2L2I)
©
~:-ID1A12 + LxAo2 /.~-lD2A22 + e - l L 2 t D 1 A l 2 + L2Ao
(15) where we again drop the argument of the coefficient matrix functions. The submatrices LI, L2, and L 21 in (14) are determined by the equations e/~l = Dr(All
-
A12Lzl)L1
-
DIAlo
-
eLIAo
We plan to use 5¢{M} on many occasions, first being the D-stability concept (Khalil and Kokotovic, 1979a) which is crucial for stability analysis of the fastest subsystem of (15),
+ DIA12L2 + eLtAotL1 + eLtAo2 (L2 - L21L1)
#z=ID22Az2(t)+~F(t,e,#)lz
#L2 = D2A22L2 - D2A2o + ~-(L21D1At2L2
where
- L21D1Axo) + #(L2Ao2L2 - L2Ao) (16) e
Assumption 4.1. For all D 2 satisfying (6), there exists a positive number 0c22 such that
- L21D1Alt) + #(L2Ao2L21 - L2A01) with the initial conditions Ll(to) = [At t(to) - A12(to)A2~(to)A21(to)]- 1 [A t o(to) - A 12(to)A~ (to)Azo(to) ] (17)
L2(to) = A2~(to)A2o(to) L21(to) = A2~(to)A21(to).
£/'{D2A22(t)} <~ --0~22
4. PRELIMINARY RESULTS Our main device in qualitative study ofmultitime scale problems is the logarithmic norm of a square matrix M lim sup III + hMl[ - 1 h
(18)
Vt
>/to.
(21)
We immediately note that Assumption (4.1) implies Re2{DzA2z(t)} <~ -2a,
The triangular form (15) of (13) is suitable for a hierarchial study of stability properties of the multitime scale problems such as (13). The verification of the transformation (14), which produced (15), is postponed until Section 5. In the next section, we establish certain preliminary results necessary for the verification procedure.
h--*0 +
F(t,e,#) -- L2x(t)DtAt2(t) + epLz(t)Ao2(t).(20) For this purpose, we introduce the following assumptions.
#/',21 = D2A22L2x - D2A21 + ~-(L2xD1At2L21
5e{M}
(19)
Vt >t to
(22)
where 2~ = :~22, but it is not implied by (22). The condition (22) means that A22(t) is D-stable, that is, D2A22(t) is a stable matrix for all D2. Our use of a more restricted condition (21) is motivated by the fact that it is verifiable in finite number of steps (Ladde, 1977). Furthermore, Assumption (4.1) provides a convenient setting for proving qualitative properties of equations such as (16). The use of the logarithmic norm in showing Dstability of a given matrix, is preceded by a proper choice of vector and matrix norms. For example, let us consider a square block-matrix A(t)= [Ais(t)] with s square diagonal blocks All(t), and assume that there exists a constant positive definite matrix P = diag {Px, P2 . . . . . Ps}
(23)
388
G . S . LADDE and D. D. SILJAK
such that the matrix Q(t) defined as
AT(t)P + PA(t) = -Q(t)
(24)
is positive definite for all t >/to. We choose as a vector norm Ilalle~, =(aTPD-la) ~/2 and as a matrix norm IlMllpo , = )~2 {DP- t MTpD- ~M} where 2u denotes the largest eigenvalue of the indicated matrix. We also assume that 2,,{Q(t)} >1 for all t/> to where K is a positive number. Then
11@2(t,r,e,p)[I ~< exp - - ~ 2 ( t - z-) , VI ~> c. P
f32)
Proof Again, we set m(t)= ]~2(t,v,~:,P)ll. By following the proof of Lemma 4.1 and using the properties of the logarithmic norm, we arrive at D+m(t) <, [~t/{~D2Azz(t)}
~{DA(t)} = ).M{DP-t [AT(t)P + PA(t)]}. (25) To conclude D-stability of A using (23) and (24) when D has the form (5), we derive the inequality
~{DA(t)} <~ -~,
Vt >>.to
(26)
where ~ - ~:~ccW{-P-~}, which is Assumption 4.1 We note, however, that the logarithmic norm cannot be used to establish D-stability for all the cases listed in Khalil and Kokotovic (1979a). Now, we are in a position to derive estimates of various state transition matrices associated with (15). First, we consider the system (27)
fl~ = DzA22(t)7.
+ 5¢
F(t, e, p
)}]
t
fl
)7
m(t) <. exp (l (t - z)[-~22
D+m(t) <~S{D2A22(t)}m(t), gt >~z
(29)
with re(r) = 1. By solving (29), we get
m(t)<~ e x p I - l ~ 2 2 ( t -
Vt ~> r.
z)], Vt>~r.
-Flimsup{t--2~¢ ' f ' IIr(O' e' ~)11dO
t~
+ ~
}
1
HF(z,e,P)lldr < + o c (31) 0
(35)
t]
~< :~22(36)
This means that one can find a positive number T such that
(30)
This proves Lemma 4.1. To derive the estimate of the state transition matrix ~2(t, z, e, #) for the full system (19), we assume that F(t, e, p) is continuous in t and satisfies the condition
,
At this point, we invoke our basic Assumption 2.1 and condition (31) to conclude that there exist two positive numbers 5" and p* such that e ~ e~', #/e ~ #*/e* and
L e J,
limsup
IIC(0, 5,~)11 dO
~k,-+~
which yields
(34)
Using Assumption 4.1 and the estimate (28), we get
+ --C,l--z
O(h)
~
+ 5fll~F(O,e,p)~JdOlm(t), Vt >~z.
11022(t,z)ll ~ exp - - ~ 2 2 ( t - r) , Vt ~> r. (28)
11¢22(t, r)ll +
1
m(t) <~exp ( f [~f'{~D2A22(O)}
Lemma 4.1. Under Assumption 4.1
m(t + h) - re(t) = ll(I)22(t + h, r)ll -l~22(t,z)l] ~< (11I + hO2A22(t)11- 1)
(33)
with re(z) = 1. Solving (33), we get
which is formally obtained from (19) by ignoring the term (p/e)F(t, 5, p). We denote (I)22(t , z) as the state transition matrix of (27) and prove the following:
Proof Set m(t) = 11~22(t, T)II. For small h > 0
re(t), Vt <~r
IIr(0, e, p)ll dO ~< exp E½~22(t -- "C)], Vt/> r + T. (37)
From continuity of F(t,~,/~), we can find a positive number v such that sup {[IF(0,~,#)t[} r<~O~+ T
~< v. Again, on the basis of Assumption 2.1, we can find positive numbers e~', #7 such that ~: < e~, P/~ < #2/e22 and 2(p/e)v ~< e22. Then
which will be justified later.
Lemma 4.2. Under Assumptions (2.1) and (4.1), and the condition (31), there exist positive numbers ct2, ~*, and/t* such that if e ~< e*, p/e ~< ~*/e*, then
expI fi
dO]
~< exp [½~2;(t - r)],
V t ~ [ r , z + T].
(38)
Multi-parameter singular perturbations Choose /3* = min {/31,ez}, be* = min {M', be*}, and az = ~22. From (35), (37), and (38), we get
m(t)~exp
[1
-~22(t-z)
]
, Vt~>z
with re(z)--0. Using the comparison principle (Siljak, 1978) we produce the estimate
m(t) <<.r(t;z,O), Vt ~ z
(39)
whenever /3 ~ r (40)
corresponding to the boundary-layer system (9) with the state transition matrix
389
where r(t;z,O) is the solution of the differential equation
r = I~{1D2A22(t)} + ~cP{~F(t,/3,#)}]r + s(t) (49) with
r(z) =
0, and
s(t)=l{ D2[A22(t)-A22(z)]+beF(t' }e, e'")
exp[1D2A22(z)(t- z)],t>/"c. (41)
~ 2 ( t , z , be) =
(48)
exp[-#1~x22(t - z)l.
(50)
For this we need the following assumption.
Assumption4.2.
The matrix A22(t) is Lipschitzian on ~ that is, there exists a positive number K such that
By applying the elementary variation of constants formula, we solve (49) to get
r(t;z,O)
=
£e ~D2A22(tl)
exp
IIA22(q) - A22(t2)ll ~< Kit1 - t21, Vtl, t2 e , ~ .
(42)
We state the following lemma.
Lemma 4.3.
Under Assumptions 2.1, 4.1, and 4.2, and the condition (31), there exist positive numbers /30, beo, (o such that if e ~
~_(o, Vt >/z. /3
With estimate (32) and definition of s(t) in (50), we get
r(t;z,O)<~lexp[-52(t - z)l ft { D2[A22(O)
(43) -
Proof Comparing (9) and (19), we conclude that W2(t, z,/3, be) satisfies a nonhomogeneous differential equation beP=[D2A22(t)+~F(t,/3,#)]P+q(t,/3,#)(44) with
p(r) =
1
{D 2 [A22(t) - AE2(Z)]
+ ~F(t,/3,be)}*2(t,z, be).
1
+ ½gllD211(t-
/
(45)
[
From Assumption 4.1 and Lemma 4.1 we have
[1
1
(52)
Since A22(t ) is Lipschitzian by Assumption 4.2 and F(0,e,#) satisfies (31), our basic Assumption 2.1 allows us to compute further (52) as
0, and the perturbation term
q(t,/3,be) =
}d0.
+
(t
z)] + KIID211V1
,
1
2
1~2(t, T, be)l[ ~< exp - ~ 2 2 ( t - z) , Vt I> z.(46) We use again a differential inequality solution approach to establish (43). We set raft)= IlqJe(t,z,/3,be)ll and form the inequality
D+m(t) <-N[~{1D2A22(t)} + ~c~'{1F(t,/3,be)}]rn(t) 1
+-IIq(t,/3,be)ll, Vt >1z. be
(47)
where the positive number ~u is obtained from (31) as ~u = max {~1,~2}. For any positive number there exist positive numbers T(~) and ~1(~) such that ~ IIF(0,e,#)lld0 ~< G(t - z), Vt/> z + T
390
G.S. LADDE and D. D. SILJAK
and ~1 = limsup~
1
-
5.1 there exist positive numbers e + and # + such that ife ~< e + and #/e ~< #+/e +, then the solutions L2(t) and L21(t) of the auxiliary system (16) with the initial conditions (17) are bounded for all t ~< to.
[[F(O,e,#)IldO + ~.
The other positive number (2 is given as (2 =
max
Proof Let ~o(t, to) and ~1 l(t, to) denote the state transition matrices of the systems
]iF(t, e, #)ll-
r~
Now, we are in a position to choose the positive numbers e° and kt° such that whenever e ~< e° and p/e ~< ~t°/e°, we have 12(t-z) p
[
exp - 1 2 ( t - r ) p
1'
~
(55)
es: = D1All(t)x
(56)
and
respectively. From Assumption 5.1, Oo(t, t0) and O1 l(t, to) have the following estimates
I12,t--r)12exp[--~2(t--r'l<'--~
[~o(t, to)ll ~< exp [~o(t - to)], vt ~> to
where e is the Naperian base. With this choice of e° and #o, we can rewrite (53) as
KIID2III r(t; r, O) ~< #~-(~Ue~22+ ~e2~2Y ] Vt >~ r.
5: = Ao(t)x
(54)
From (48) and (54) we obtain the estimate (43) with ~o _ ((u/e~z) + e(KI[D2[I/2e2~2). This proves Lemma 4.3. We are now in a position to verify the validity of the basic transformation (14) which produces the triangular form (15) of the multitime scale system (13). The verification approach presented in the next section, are motivated by the work of Khalil and Kokotovic (1979a) on two-time scale singular perturbation problems.
(57)
and I~11(t, to)lJ ~< exp [~11(t - to)], vt ~> to. (58) where So and ~1 a are positive numbers. We note that • o(to, t) and ~ 1(to, t) are state transition matrices of the adjoint systems corresponding to (55) and (56), respectively. Then, ~22(t, to)Lz(to)Oo(to, t) and 022(t, to)L2x(to)Oll(to, t) are solutions of the initial value problems
#Z = D2Az2(t)Z - #ZAo(t),
Z(to) = L2(to) (59)
and
i.tZ = D2A22(t)Z - ~ZD1All(t),
Z(to) = L21(to) (60)
5. AUXILIARY SYSTEM
Our objective in this section is to establish the qualitative properties of the auxiliary system (16) which are necessary in establishing the validity of the triangular form (15) of the original system (4). From continuity of the system matrices in (4) and continuous differentiability of the right side of the auxiliary system (16), follows the existence and uniqueness of solutions of the initial value problem (16) and (17). For validity of the transformation (14), which produces the triangular form (15), we also need boundedness of solutions of the auxiliary system. This we show next. We immediately recognize the fact that the last two equations are decoupled from the first in (16). This allows us to establish boundedness in two stages. We first consider the boundedness of L2(t) and L21(t) for which we need:
respectively. Comparing (59) and (60) with their counterparts in (16), we conclude that the equations for L2(t) and L2 l(t) can be considered as perturbed systems by (59) and (60). This allows us to use the variation of constants formula and represent the solutions as follows:
Assumption 5.1. The matrix functions Ao(t), Aol(t), Ao2(t), Alo(t), All(t), Air(t), A2o(t), and A2~(t) are bounded on ~.
L21 (t) = (I) 22(t, to)L 21 (to)(I)l 1 (to, t)
We prove the following lemma.
Lemma 5.1. Under the Assumptions 2.1, 4.1, and
L2(t) = ~22(t, to)L(to)~o(to, t)
+ ~'oCb22(t,z)(-;D2A20(z) 1
+ - [L21 (z)D1A x2(z)L2 (z)
-- L21(z)D1Alo(T)] + L2(z)Ao2(z)Lz(z))(1)o(Z , t) dr
+ f'C~22(t'z)(
lt
1 + -/-,21 ( r ) D I A 12 (r)L21(z) c.
(61)
391
Multi-parameter singular perturbations + L2(T)AoE(T)L21(z)
- LE(Z)Aol(z))~ll(Z, t) dz.
/zA
LE1 =
IDEAEE(t)+#L21(t)D1All(t)IAL21/3
(62) - - / i A L 2 1 ( D a [All(t) /3
We choose positive numbers/3+ and #+ such that
/3
2~ + max {~o,~li} ~< ~t22 16~t+ l~(llAoz(t)llp
AtE(t)Lz1(t)])
+ # [AL21DIAIE(t)AL21 ]
+ IIAodt)ll) ~< 1
+ ~R21(t,/3,#), /3
(~22
ALEl(to)
=
0
(65)
+ 1
16#~+--(IIDAI2(t)IIP + IIDiAlo(t)ll) <<,1
where by Lemma 5.1, L2(t) and LEl(t) are bounded solutions of (16)
/3 0~22
where
/~,2(t) = p =
:2 (max
0~22
(66) and RE(t,e,/g) =
For/3 ~< e + and W/3 ~< #+//3+, we have IIL2(t)ll < P and IIL2dt)ll < p for all t>~ to, which can be established by standard techniques. The proof is complete. In order to go to the second stage of validation of the transformation (14), which is concerned with the solution Lt(t) in (16), we need to establish the convergence of L2(t) and L21(t). We introduce the following assumption.
Assumption 5.2. The matrix functions Aft(t) AEo(t), A22i(t)A21(t), and their first derivatives are bounded on With this assumption, we formulate the following convergence result. Lemma 5.2. Under the assumptions of Lemma 5.1 and Assumption 5.2, we have L2(t) =
LEl(t ) = A~(t)AEt(t)
{IIDEA2011,IID2AEIlI}
+ max {llLz(to)ll, ItLEdto)}).
LEI(t) =
A21(t)A2o(t),
a22i(t)A2o(t) + O(#/e) A2~(t)A2~(t) + O(p/e).
EE(t)DiA1E(t)LE(t) -- L21(t)DiAlo(t) + e[E2(t)Ao2(t)E2(t) - L2(t)Ao(t) - L2(t)]
R2i(t,/3, #) = F.21(t)D1Ax2(t)r.21(t) - LEl(t)D1Atl(t) +/3 [LE(t)Ao(t)LEI(t) -- LE(t)Aol(t) -/~El(t)].
(67)
From Assumptions 5.1, 5.2, and boundedness of and R21(t, e,/g), and the coefficient matrices of (65), are all bounded and continuous. This implies the existence of solutions to the initial-value problem (65). Further, we note that the right sides of (65) are continuously differentiable with respect to ALE and ALE1. Therefore, the solutions are also unique. Let ¢Pn(t,z), @~(t,z), and ~a(t,z) be state transition matrices of LE(t) and L E l ( t ) , w e conclude that R 2 ( t , e , # )
ILl = [Ao(t) --
AoE(t)A22i(t)AEo(t)]rI
(63) / l ~ = # D 1 [A 1 l(t) /3
Proof. We first write A221(t)AEo(t) + ALE(t), AL2(to) = 0 LEl(t) = AE~(t)AEl(t) + AL21(t), ALEl(to) = 0. LE(t) =
A1E(t)A•(t)AEI(t)]Z
I~=IDEAEE(t)+~A2~(t)AEI(t)DIAII(t)]~)
(64)
(68)
The terms ALE(t) and ALE l(t) satisfy the differential equations
respectively. We note that I~n(t, ~)ll and I~x(t, 011 have the estimates similar to (57) and (58) with rates an and ~x. We also observe that the f~-subsystem in (68) has the same form as (19) with F(t,e,/~) = A221(t)A21(t)D1All(t). By Lemma 4.2, I~n(t,x)ll satisfies an estimate as in (32) with 0~f~< ~22- Moreover, ¢I)n(t,to)OOn(t, to) and @n(t, t0)~-tl)x(t, to) are solutions of the equations
#A L2 =[D2A22(t) + ~L,2i(t)DIAll(t)]AL2 Ao:(t)LE(t)] + li [AL2AoE(t)ALE + LE(t)AoE(t)AL2 ] - #ALE [Ao(t) -
+ P/3 [ALExD1AI,(t)/~2(t)] + ~RE(t,/3,#), /3
AL2(to) = 0
IglVl = ID2A22(t) + # -- # M [Ao(t)
392
G.S. LADDE and D. D. -
By (Ihl(t,z), @l(t,z,~,#), ~l(t,'C,~:), and q~x(t, z, e, p) we denote the state transition matrices of the systems
M(to)=®.
Ao2(t)Lz(t)],
SILJAK
~IV = [DeAzz(t) + ~L,21(t)D1Al l(t)IN
e l~' ---- D 1 / [ 1 l(t)y
# N(D1 JAil(t)
~37'= D1 [A, l(t) + O(u/e)]y
-- Alz(t)F, zl(t)]),
N(to) = E.
(69)
Equations (69) can be considered as unperturbed system corresponding to (65). Therefore, the solutions of(65) can be represented in the context of (69) as ALz(t) =
i'
On(t, Q(AL2(z)Ao2(r)AL2(3)
(75)
system (12), and
Wl(t,r,e,#)=Ol(t,z,e,#)-+(t,z,c,#).
(76)
Lemma 5.2. Under Assumptions 2.1 and 5.2, the matrices ~l~(t, 3) and Oi(t, 3, e, p) have estimates similar to those of (28) and (32) with the rates ~11/e and ~l/e where 711 is defined in (74) and ~1 = ½~11.
0
Proof Similar to the proofs of Lemmas 4.1 and
+ L2(3)Aoz(3)AL2(3)
4.2.
1
+ - [AL2i(z)D1A 11('1~)/~2('/~)
Assumption 5.4. The matrix ~ztI 1 (t) is Lipschitzian
e
on ~.
+ R2(r, e, It) ])@n(3, t) dr AL21(t) =
On(t,z)
[AL21(z)D1A12(z)AL21(z)
0
+ R21(z, e, #)])~z(z, t) dz.
(70)
Lemma 5.3. Under Assumptions 2.1, 5.3, and 5.4, there exist positive numbers e~, #¢, (* such that if e -%
/
From Assumption 5.2 and equations (64), it is obvious that ALz(t) and AL2x(t) constitute a bounded solution of (65). Then, from (70), after simple calculations, we conclude (63). This completes the proof of Lemma 5.2. In order to complete the analysis of the auxiliary system (14), it remains to establish the boundedness and convergence properties of L~(t). For this purpose, assuming Lemma 5.2 holds, we rewrite the equation for L~(t) of (14) as e/'~1 = O1 JAil(t) -- A~2(t)A2~(t)A2~(t)
Proof Similar to that of Lemma 4.3. From Lemmas 4.3, Assumptions 5.3 and 5.4, and noting the fact that the coefficient matrices in Assumption 5.3 are bounded, it follows that the solution Ll(t) of (71) is bounded. Finally, to establish convergence property of Ll(t), we introduce the following assumption.
Assumption 5.5. The matrix function ,4111(t) [Alo(t) - Alz(t)Af2 l(t)A2o(t)] and its derivative are bounded.
+ O(#/e)]L1 - Dx [Axo(t) - A12(t)A2z~(t)A20(t)
+ o(#/e)] -- eL i [Ao(t ) -
Aoz(t)A2~(t)Azo(t) + O(#/e) + L~ ]Aox(t)
- Aoz(t)A~(t)Aza(t) + O(#/e)]L1
(71)
with the initial condition Ll(to) - [A1 ~(to) -- Al:(to)A~2X(to)A21(to)]-I [A~o(to) -- A12(to)A2~(to)A2o(to)].
(72)
We define ,'tab(t) = All(t) - Ai2(t)A22~(t)A21(t)
(73)
Ll(t) = Ai-xl(t) [Alo(t) - Alz(t)A2~(t)Azo(t)] + O(e). (77)
Proof Similar to that of Lemma 5.2. By establishing boundedness and convergence of the solutions L l(t), Lz(t), and Lzl(t) of the auxiliary system (16), we have verified the validity of the transformation (14). Therefore, the transformed system (15) can be rewritten in terms of the original matrices of (13). We define the coefficient matrices of the p-reduced system (8) as
Ao(t) = Ao(t) - Ao2(t)A2~(t)A2o(t),
and introduce the following assumption.
Assumption 5.3. For Dx satisfying (6), there exists
/lot(t) = Aol(t) - Aoz(t)Az1(t)A21(t) /llo(t) = Alo(t) - A12(t)A221(t)Azo(t),
a positive number ~ll such that L~'{D1All(t)} ~< -~11,
Lemma 5.4. Under the assumptions of Lemma 5.2 and Assumptions 5.3, 5.4, and 5.5, we have
Vt i> to.
(74)
/lll(t)=A1~(t)-Alz(t)Af~(t)Azl(t)
(78)
Multi-parameter singular perturbations
393
and the e-reduced system (11) as AR(t ) --- ,4o(t) -- Aot(t)7t~tl(t).~to(t)
(79)
and rewrite (15) as Aol -- Ao2(t),4[at(t)Alo(t)Dl,'~xo(t) D1Alx(t ) + O(/~/e) + O(e)
(3
+ O(e)
~,4;11(t)/{loAo2(t) + OaAlz(t ) + eO(e D2A22(t) + O(#/e)
(80) with initial conditions where ~t(t, to, e,#) is the state transition matric corresponding to the decoupled e-subsystem with the matrix D1Att(t) + O(p/e) + O(e),
U(to) = x o
V(to) = -4;d(to)/{lo(to)Xo + Yo + O(e) W(to) = A~l(to)Azo(to)Xo + A~(to)A2t(to)Yo + z0 + O(#/e).
B23(t, e, p) = -~[~(t)Ato(t)Ao2(t) + 1Dt,412(t) (81)
On the basis of the transformed system (80), we establish in the next section our main result concerning the qualitative properties of the original system (13). 6. MAIN RESULT
We are now in a position to use the triangular form (30) and establish stability of the original multitime scale singularly perturbed system (13). For this, we need our last assumption concerning stability of the reduced subsystem (11) with the system matrix AR of (79).
Assumption 6.1. There exists a positive number
+ O(e).
and w(t) is the solution of the p-subsystem (19)
w(t) = ~2(t, to, e, p)W(to)
t~
to)1
IIv(t)[I ~ e x p I - ~ x ( t -
( llv(t° )ll + ~2 -
)
(85)
where (p/e)F(t, e, p) = O(p/e). Because of Assumption 5.3, ~)l(t, to, e, #) has an estimate similar to (32) with the rate ~1 < ~x 1. Moreover, by Assumptions 5.1 and 5.5 with (78), we have boundedness of B23(t,e,t~) with respect to t. Therefore, by (32) we can derive from (84) the following inequality
~R such that limsup
(84)
-po ~ 1
llB2 3(t' e' P )ll llw(t° )ll) (86)
~{A~(z)}dz ~< --~a. (82)
+ cX~
0
Assumption 6.1 implies that the reduced subsystem (11) is globally exponentially stable.
Theorem 6.1. Under Assumptions 2.1, 2.2, 2.3, 4.1, 4.2, 5.1, 5.2, 5.3, 5.4, 5.5, and 6.1, there exist positive number g and /i such that if e ~< g and #/e ~< ki/g then the equilibrium of the system (13) is globally exponentially stable.
Proof Stability of (80) implies stability of (13). To show stability of (80), we should verify stability of each decoupled subsystem and smallness of their interconnections considered as perturbation terms. We recall Lemma (4.2) and conclude immediately stability of the p-subsystem of (80). To establish stability of the e-subsystem, we use (80) and write the corresponding solution as v(t) = ~ t ( t , to, e, p)v(to) +
~o
+t(t,z,e,#)Bz3(Z)w(z)dz
(83)
By choosing gl and/it sufficiently small, so that fi~-~l < ~z% et
(87)
and noting the fact that ~lllBz3(t, gt,/Jt)ll is bounded, we conclude from (85) exponential stability of the e-subsystem. Finally, using Assumption 6.1 and repeating the similar arguments, we establish exponential stability of the reduced subsystem with perturbation terms, which is defined in (80). This results in the values gR and/~R of the upper bounds of e and p for stability of the reduced subsystem. By choosing g = min {gl, gR} and ~i = min {/it,/~a} exponential stability of (86) and, thus, (13) follows. This completes the proof. We remark that under the assumptions of Theorem 6.1 (except Assumption 6.1), we can conclude that the solutions of the original system (13) can be approximated by the solutions of the overall reduced system, and e and p boundary layer system. The inverse of the transformation (14) is
394
G.S. LADDE and D. D. SILJAK
,o
o Ool[!
-Ll(t)
I1
- L 2 ( t ) + L2~(t)La(t)
This together with Lemmas 4.3 and 5.3 gives the desired result
-L21(t)
.
t88)
I
Energy, Electric Energy Systems Division, under contract DEAC03-77ET29138, and in part by The National Science Foundation under grant ECS-8011210.
x(t) = ~(t) + 0 ( 0 y(t) = --A?ll(t)A1o(t)Yc(t) + ~(t) + O(g)
z(t) = [ - A ; ~ ( t ) A 2 o ( t )
+ A ~ (t)A z 1(t)d;11 (t)al o(t)])~(t) -
-
A£¢(t)A2~(t)y(t) + ~(t) + 0 ( # / 0 + O(e). (89)
Equations (89) are valid on any compact time interval. This result parallels that of Theorem 2 in Khalil and Kokotovic (1979a). 7. CONCLUSION
A hierarchial scheme was developed for aggregation of small parameters in singular perturbation analysis of linear systems having multiple time scales. The scheme is effected by a linear transformation which produces an explicit orderdecomposition of the system. The resulting multiparameter-multitime-scale problem is solved in the context of generalized D-stability. Acknowledgements--The authors are grateful to Professor S. L. Campbell, Department of Mathematics, North Carolina State University at Raleigh, for his valuable comments on this paper. This research was supported in part by The Department of
REFERENCES Campbell, S. L. (1978). Singular perturbation of autonomous linear systems II. J. Diff. Eqn., 29, 362. Campbell, S. L. (1979). On a singularly perturbed autonomous linear control problem. IEEE Trans A ut. Control, AC-24, 115. Campbell, S. L. and N. J. Rose (1979). Singular perturbation of autonomous linear systems. SlAM J. Math. Anal., 10, 542. Chang, K. W. (1972). Singular perturbations of a general boundary value problem. SIAM J. Math. Anal., 3, 520. Hoppenstead, F. (1971). Properties of solution of ordinary differential equations with small parameters. Commun. Pure Appl. Math., 24, 807. Khalil, H. K. (1981). Asymptotic stability of nonlinear multiparameter singularly perturbed systems. Automatica, 17, 797. Khalil, H. K. and P. V. Kokotovic (1979a). D-stability and multiparameter singular perturbations. SIAM J. Control Optimiz., 17, 56. Khalil, H. K. and P. V. Kokotovic (1979b). Control of linear systems with multiparameter singular perturbations. Automatica, 15, 197. Ladde, G. S. (1977). Logarithmic norm and stability of linear systems with random parameters. Int. J. Systems Sci., 8, 1057. O'Malley, Jr, R. E. (1974). Introduction to Singular Perturbations, Academic Press, New York. Siljak, D. D (1978). Large-scale Dynamic Systems: Stabili O' and Structure. North-Holland, New York. Vasileva, A. B. (1963). Asymptotic behavior of solutions to certain problems involving nonlinear differential equations containing a small parameter multiplying the highest derivatives. Russian Math. Surveys, lg, 13.
Autoraatica, Vol. 19, No. 4, pp. 385 394, 1983
Printed in Great Britain.
Multiparameter Singular Perturbations of Linear Systems with Multiple Time Scales* G. S. LADDEI- and D. D. SILJAK:~
A schemefor order-decomposition and hierarchial aggregation of small parameters may be usedfor multiparameter singular perturbation of linear systems, when it Is essential to realize a multiple time scales assumption. Key Words--Perturbation techniques; approximation theory; system-order reduction; stability; hierarchial systems.
Campbell (1978, 1979); Campbell and Rose (1979). The objective of this paper is to consider a joint multiparameter-multitime scale singular perturbation of linear systems, in which small parameters are grouped into equivalent classes according to their order. A linear transformation is formulated, which can be used in a hierarchial scheme to obtain a suitable form of the original system, that recognizes explicitly the order-decomposition of small parameters. The resulting multitime scale problem can then be solved effectively in the framework of a generalized D-stability (Khalil and Kokotovic, 1979a).
Abstract--A joint multitime scale-multiparameter singular perturbation is formulated and resolved in the context of linear time-varying systems. An interesting feature of the solution procedure is a hierarchial scheme of aggregating and arranging of the groups of small parameters according to their order. The scheme provides a suitable framework for establishing qualitative properties of multitime scale systems.
1. INTRODUCTION MULTIPARAMETER singular perturbation
models arise whenever there are more than one small parameter representing physical constants such as inertias, masses, time constants, etc. (Vasileva, 1963). A way to simplify a multiparameter problem is to consider the parameters to be of the same order allowing for the ignorance of their ratios using the concept of D-stability (Khalil and Kokotovic, 1979a, b; Khalil, 1981). The same order assumption enables one to recast the multiparameter system into a model resembling a single parameter perturbation problem. The assumption can be removed to consider the multiple time-scale problem provided the bounds on the ratios of the parameters are allowed to be arbitrary (Khalil, 1981). These simplifications, however, may not be acceptable in problems where an explicit multiple time-scale assumption is essential and restricted to bounded variations of system parameters. Such problems have been analyzed by Hoppenstead (1971); O'Malley (1974), and more recently by
2. MULTITIME SCALE PERTURBATIONS
Consider a linear system described by
fc = Ao(t)x + i Bok(t)yk k=l
+ ~ Cok(t)xk k=l
'~iYi= A~o(t)x + ~ B~ik(t)yk k=l
+ i C~k(t)z~, ie J k=l
#j~j = A~o(t)x + ~ B~k(t)yk k=l
+ i C~k(t)Zk, j ~
(1)
k=l
* Received 18 March 1982; revised 17 November 1982. The original version of this paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by associate editor M. Jamshidi under the director of editor A. Sage. ]'Department of Mathematics, University of Texas at Arlington, Arlington, TX 76019, U.S.A. :~School of Engineering, University of Santa Clara, Santa Clara, CA 95053, U.S.A.
w h e r e x ~ ~.o, Yl E/~m,, Zj • ,~lj, ~ -- { 1, 2..... r}, j = {1,2,...,s}, m=Xi~jmi, l=IF,j~yl~, and the dimension of the system (1) is n = no + m + I. In (1) all matrix functions are continuous on ~ and have appropriate dimensions. The parameters e~ and #i are positive real numbers, all e~s and #is being of the 385
386
G . S . LADDE and D. D. SILJAK
same order, separately. This means that the ratios of e~s and #~s can be bounded as ~2~<--~<~,
Vi, k ~ J
~k
(2)
~ -< #--~< p, #k
vj, k e J
where, for simplicity, we omit the argument of the matrices. System (7) is equivalent to x = (Ao - Ao2A221A20)Yc + (A01 - Ao2A22XA21)Y, ~(to) = Xo
8y = DI(Alo - A12A21A20)Yc + Dl{All - A12A21A21)Y, y(to) = Yo (8) = - A 2 1 A 2 0 Y c - A21A21~.
where ~ ~, ~, fi are some positive numbers. Our crucial assumption is that e~s and #is are not of the same order. We define e ~--- (/~1~2 . . . . .
The #-boundary layer system is d~
- - = D2A22(to)~(zu), ~(0) = Zo - Z(to)
# -~- ( # 1 # 2 , . . . , # s ) 1Is (3)
er) l/r,
(9)
dz u
and use the following assumption. where z~ = (t - to)/# is the '#-stretched' time scale. A s s u m p t i o n 2.1. #/e ~ 0 as ~ ~ 0 +. Thus (1) is a
three-time scale multiparameter system. With (3), we can simplify the representation (1) as k = A o ( t ) x + A o l ( t ) y + Ao2(t)z, X(to) = Xo ep = D i A l o ( t ) x + D1A11(t)y + D1A12(t)z, y(to) = Yo
Next, the following assumption is introduced. A s s u m p t i o n 2.3. A l l ( t ) nonsingular for all t ~> to.
Letting lim Di-l(e)e = 0, we can reduce (8) to e~0
(4)
= (Ao - Ao2Az~A20)Y¢
#~ = D2A20(t)x + D2A21(t)y + D2A22(t)z, Z(to) = Zo
where y = (y~, Y2T. . . . . Yr) T T E ~ ' n , z = (Zl, T Z2 T . . . . ,Zs) T T ~1, and the block-matrices of (4) are formed in an obvious way from the matrices of (1) with DI and D2 being defined as D1 = diag
Ii1,~I12 .... ,~I1,
+ (Aol - Ao2A2~A21)Y,
(#1
#2
.....
= LAo - A o 2 A 2 ~ A 2 0 - (A01 - A o 2 A z ~ A 2 1 ) ( a l l - A 1 2 A 2 ~ A 2 1 ) -1 (Alo - A 1 2 A 2 ~ A 2 0 ) ] ~ , 2 ( t o ) = Xo (11)
Z12A2~h21)-l(a10 - a12A21A20)~.
The e-boundary layer system is
d: d**
- D 1 [ A 11 (to) - A 12 ( t o ) A 2 # ( t o ) A 21 (to)
]f,
y(0) = Yo - y(to) I1~ and I2j are identity matrices of appropriate dimensions. In view of (2) and (3), the elements of the matrices D1 and DE are bounded as ~: ~< - ~
~,
/~ ~< - - ~
~i
(6)
#j
where the bounds of (6) depend on the corresponding bounds of (2). In order to perform a two-stage reduction of (4), we need the next assumption. A s s u m p t i o n 2.2. A22(t ) is nonsingular for all t t> t o. First, we reduce (4) with respect to # by letting
lim D21(#)# = 0 #~0
= Ao2 + AolY + Ao2Z, 2(to) = Xo ey -----D I A 1 0 x + DIA11 ~ + DIA12z, y(to) = Yo
(7) 0 = A2ox + A21Y + A22z
(10)
which is equivalent to
)7 = - - ( A l l --
~ssI2s}
YC(to) = Xo
0 = (Alo -- A 1 2 A 2 1 A 2 0 ) £ + (All - A12A221A21)Y;
(5) D2= diag(#I21,~122
is
Ax2(t)A2~(t)A21(t)
(12)
where z~ = (t - to)/e is the 'z-stretched' time scale. The presented hierarchial order-reduction scheme provides a suitable basis for incorporating multitime scale singular perturbation in a natural way. Our primary objective is to use the scheme in studying stability of the system (4). As a by-product of this study, we will obtain approximations of solutions to (4) in terms of solutions to the overall reduced system (11) and to the boundary layer systems (9) and (12).
3. TRIANGULARIZATION In order to consider the multitime scale perturbations, we derive a three-fold version of Chang's (1972) transformation, which is necessary for transforming the original system (4) into a triangular form. The transformed system is suitable for deducing the qualitative properties of systems with multitime scale singular perturbations.
Multi-parameter singular perturbations
387
We rewrite the system (4) as
Ii I
IAo(t) Aol(t) = [e-lDtAlo(t) e-tD1Atl(t) LP-lD2A20(t) #-tD2A2a(t)
A0(t) 1[i1 e-tDtA12(t) I
(13)
#- XD2A22(t)J
and use a similarity transformation
til o °oil:l =
1(0
(14)
11
[_L2(t) Lel(t) I
to represent (13) in a triangular form -- Ao2L2 - (Aox - Ao2L2t)L1
Aol -- Ao2L 1
© ©
e-tDI(Ali
- A12L2t) + La(Aol - Ao2L2I)
©
~:-ID1A12 + LxAo2 /.~-lD2A22 + e - l L 2 t D 1 A l 2 + L2Ao
(15) where we again drop the argument of the coefficient matrix functions. The submatrices LI, L2, and L 21 in (14) are determined by the equations e/~l = Dr(All
-
A12Lzl)L1
-
DIAlo
-
eLIAo
We plan to use 5¢{M} on many occasions, first being the D-stability concept (Khalil and Kokotovic, 1979a) which is crucial for stability analysis of the fastest subsystem of (15),
+ DIA12L2 + eLtAotL1 + eLtAo2 (L2 - L21L1)
#z=ID22Az2(t)+~F(t,e,#)lz
#L2 = D2A22L2 - D2A2o + ~-(L21D1At2L2
where
- L21D1Axo) + #(L2Ao2L2 - L2Ao) (16) e
Assumption 4.1. For all D 2 satisfying (6), there exists a positive number 0c22 such that
- L21D1Alt) + #(L2Ao2L21 - L2A01) with the initial conditions Ll(to) = [At t(to) - A12(to)A2~(to)A21(to)]- 1 [A t o(to) - A 12(to)A~ (to)Azo(to) ] (17)
L2(to) = A2~(to)A2o(to) L21(to) = A2~(to)A21(to).
£/'{D2A22(t)} <~ --0~22
4. PRELIMINARY RESULTS Our main device in qualitative study ofmultitime scale problems is the logarithmic norm of a square matrix M lim sup III + hMl[ - 1 h
(18)
Vt
>/to.
(21)
We immediately note that Assumption (4.1) implies Re2{DzA2z(t)} <~ -2a,
The triangular form (15) of (13) is suitable for a hierarchial study of stability properties of the multitime scale problems such as (13). The verification of the transformation (14), which produced (15), is postponed until Section 5. In the next section, we establish certain preliminary results necessary for the verification procedure.
h--*0 +
F(t,e,#) -- L2x(t)DtAt2(t) + epLz(t)Ao2(t).(20) For this purpose, we introduce the following assumptions.
#/',21 = D2A22L2x - D2A21 + ~-(L2xD1At2L21
5e{M}
(19)
Vt >t to
(22)
where 2~ = :~22, but it is not implied by (22). The condition (22) means that A22(t) is D-stable, that is, D2A22(t) is a stable matrix for all D2. Our use of a more restricted condition (21) is motivated by the fact that it is verifiable in finite number of steps (Ladde, 1977). Furthermore, Assumption (4.1) provides a convenient setting for proving qualitative properties of equations such as (16). The use of the logarithmic norm in showing Dstability of a given matrix, is preceded by a proper choice of vector and matrix norms. For example, let us consider a square block-matrix A(t)= [Ais(t)] with s square diagonal blocks All(t), and assume that there exists a constant positive definite matrix P = diag {Px, P2 . . . . . Ps}
(23)
388
G . S . LADDE and D. D. SILJAK
such that the matrix Q(t) defined as
AT(t)P + PA(t) = -Q(t)
(24)
is positive definite for all t >/to. We choose as a vector norm Ilalle~, =(aTPD-la) ~/2 and as a matrix norm IlMllpo , = )~2 {DP- t MTpD- ~M} where 2u denotes the largest eigenvalue of the indicated matrix. We also assume that 2,,{Q(t)} >1 for all t/> to where K is a positive number. Then
11@2(t,r,e,p)[I ~< exp - - ~ 2 ( t - z-) , VI ~> c. P
f32)
Proof Again, we set m(t)= ]~2(t,v,~:,P)ll. By following the proof of Lemma 4.1 and using the properties of the logarithmic norm, we arrive at D+m(t) <, [~t/{~D2Azz(t)}
~{DA(t)} = ).M{DP-t [AT(t)P + PA(t)]}. (25) To conclude D-stability of A using (23) and (24) when D has the form (5), we derive the inequality
~{DA(t)} <~ -~,
Vt >>.to
(26)
where ~ - ~:~ccW{-P-~}, which is Assumption 4.1 We note, however, that the logarithmic norm cannot be used to establish D-stability for all the cases listed in Khalil and Kokotovic (1979a). Now, we are in a position to derive estimates of various state transition matrices associated with (15). First, we consider the system (27)
fl~ = DzA22(t)7.
+ 5¢
F(t, e, p
)}]
t
fl
)7
m(t) <. exp (l (t - z)[-~22
D+m(t) <~S{D2A22(t)}m(t), gt >~z
(29)
with re(r) = 1. By solving (29), we get
m(t)<~ e x p I - l ~ 2 2 ( t -
Vt ~> r.
z)], Vt>~r.
-Flimsup{t--2~¢ ' f ' IIr(O' e' ~)11dO
t~
+ ~
}
1
HF(z,e,P)lldr < + o c (31) 0
(35)
t]
~< :~22(36)
This means that one can find a positive number T such that
(30)
This proves Lemma 4.1. To derive the estimate of the state transition matrix ~2(t, z, e, #) for the full system (19), we assume that F(t, e, p) is continuous in t and satisfies the condition
,
At this point, we invoke our basic Assumption 2.1 and condition (31) to conclude that there exist two positive numbers 5" and p* such that e ~ e~', #/e ~ #*/e* and
L e J,
limsup
IIC(0, 5,~)11 dO
~k,-+~
which yields
(34)
Using Assumption 4.1 and the estimate (28), we get
+ --C,l--z
O(h)
~
+ 5fll~F(O,e,p)~JdOlm(t), Vt >~z.
11022(t,z)ll ~ exp - - ~ 2 2 ( t - r) , Vt ~> r. (28)
11¢22(t, r)ll +
1
m(t) <~exp ( f [~f'{~D2A22(O)}
Lemma 4.1. Under Assumption 4.1
m(t + h) - re(t) = ll(I)22(t + h, r)ll -l~22(t,z)l] ~< (11I + hO2A22(t)11- 1)
(33)
with re(z) = 1. Solving (33), we get
which is formally obtained from (19) by ignoring the term (p/e)F(t, 5, p). We denote (I)22(t , z) as the state transition matrix of (27) and prove the following:
Proof Set m(t) = 11~22(t, T)II. For small h > 0
re(t), Vt <~r
IIr(0, e, p)ll dO ~< exp E½~22(t -- "C)], Vt/> r + T. (37)
From continuity of F(t,~,/~), we can find a positive number v such that sup {[IF(0,~,#)t[} r<~O~+ T
~< v. Again, on the basis of Assumption 2.1, we can find positive numbers e~', #7 such that ~: < e~, P/~ < #2/e22 and 2(p/e)v ~< e22. Then
which will be justified later.
Lemma 4.2. Under Assumptions (2.1) and (4.1), and the condition (31), there exist positive numbers ct2, ~*, and/t* such that if e ~< e*, p/e ~< ~*/e*, then
expI fi
dO]
~< exp [½~2;(t - r)],
V t ~ [ r , z + T].
(38)
Multi-parameter singular perturbations Choose /3* = min {/31,ez}, be* = min {M', be*}, and az = ~22. From (35), (37), and (38), we get
m(t)~exp
[1
-~22(t-z)
]
, Vt~>z
with re(z)--0. Using the comparison principle (Siljak, 1978) we produce the estimate
m(t) <<.r(t;z,O), Vt ~ z
(39)
whenever /3 ~ r (40)
corresponding to the boundary-layer system (9) with the state transition matrix
389
where r(t;z,O) is the solution of the differential equation
r = I~{1D2A22(t)} + ~cP{~F(t,/3,#)}]r + s(t) (49) with
r(z) =
0, and
s(t)=l{ D2[A22(t)-A22(z)]+beF(t' }e, e'")
exp[1D2A22(z)(t- z)],t>/"c. (41)
~ 2 ( t , z , be) =
(48)
exp[-#1~x22(t - z)l.
(50)
For this we need the following assumption.
Assumption4.2.
The matrix A22(t) is Lipschitzian on ~ that is, there exists a positive number K such that
By applying the elementary variation of constants formula, we solve (49) to get
r(t;z,O)
=
£e ~D2A22(tl)
exp
IIA22(q) - A22(t2)ll ~< Kit1 - t21, Vtl, t2 e , ~ .
(42)
We state the following lemma.
Lemma 4.3.
Under Assumptions 2.1, 4.1, and 4.2, and the condition (31), there exist positive numbers /30, beo, (o such that if e ~
~_(o, Vt >/z. /3
With estimate (32) and definition of s(t) in (50), we get
r(t;z,O)<~lexp[-52(t - z)l ft { D2[A22(O)
(43) -
Proof Comparing (9) and (19), we conclude that W2(t, z,/3, be) satisfies a nonhomogeneous differential equation beP=[D2A22(t)+~F(t,/3,#)]P+q(t,/3,#)(44) with
p(r) =
1
{D 2 [A22(t) - AE2(Z)]
+ ~F(t,/3,be)}*2(t,z, be).
1
+ ½gllD211(t-
/
(45)
[
From Assumption 4.1 and Lemma 4.1 we have
[1
1
(52)
Since A22(t ) is Lipschitzian by Assumption 4.2 and F(0,e,#) satisfies (31), our basic Assumption 2.1 allows us to compute further (52) as
0, and the perturbation term
q(t,/3,be) =
}d0.
+
(t
z)] + KIID211V1
,
1
2
1~2(t, T, be)l[ ~< exp - ~ 2 2 ( t - z) , Vt I> z.(46) We use again a differential inequality solution approach to establish (43). We set raft)= IlqJe(t,z,/3,be)ll and form the inequality
D+m(t) <-N[~{1D2A22(t)} + ~c~'{1F(t,/3,be)}]rn(t) 1
+-IIq(t,/3,be)ll, Vt >1z. be
(47)
where the positive number ~u is obtained from (31) as ~u = max {~1,~2}. For any positive number there exist positive numbers T(~) and ~1(~) such that ~ IIF(0,e,#)lld0 ~< G(t - z), Vt/> z + T
390
G.S. LADDE and D. D. SILJAK
and ~1 = limsup~
1
-
5.1 there exist positive numbers e + and # + such that ife ~< e + and #/e ~< #+/e +, then the solutions L2(t) and L21(t) of the auxiliary system (16) with the initial conditions (17) are bounded for all t ~< to.
[[F(O,e,#)IldO + ~.
The other positive number (2 is given as (2 =
max
Proof Let ~o(t, to) and ~1 l(t, to) denote the state transition matrices of the systems
]iF(t, e, #)ll-
r~
Now, we are in a position to choose the positive numbers e° and kt° such that whenever e ~< e° and p/e ~< ~t°/e°, we have 12(t-z) p
[
exp - 1 2 ( t - r ) p
1'
~
(55)
es: = D1All(t)x
(56)
and
respectively. From Assumption 5.1, Oo(t, t0) and O1 l(t, to) have the following estimates
I12,t--r)12exp[--~2(t--r'l<'--~
[~o(t, to)ll ~< exp [~o(t - to)], vt ~> to
where e is the Naperian base. With this choice of e° and #o, we can rewrite (53) as
KIID2III r(t; r, O) ~< #~-(~Ue~22+ ~e2~2Y ] Vt >~ r.
5: = Ao(t)x
(54)
From (48) and (54) we obtain the estimate (43) with ~o _ ((u/e~z) + e(KI[D2[I/2e2~2). This proves Lemma 4.3. We are now in a position to verify the validity of the basic transformation (14) which produces the triangular form (15) of the multitime scale system (13). The verification approach presented in the next section, are motivated by the work of Khalil and Kokotovic (1979a) on two-time scale singular perturbation problems.
(57)
and I~11(t, to)lJ ~< exp [~11(t - to)], vt ~> to. (58) where So and ~1 a are positive numbers. We note that • o(to, t) and ~ 1(to, t) are state transition matrices of the adjoint systems corresponding to (55) and (56), respectively. Then, ~22(t, to)Lz(to)Oo(to, t) and 022(t, to)L2x(to)Oll(to, t) are solutions of the initial value problems
#Z = D2Az2(t)Z - #ZAo(t),
Z(to) = L2(to) (59)
and
i.tZ = D2A22(t)Z - ~ZD1All(t),
Z(to) = L21(to) (60)
5. AUXILIARY SYSTEM
Our objective in this section is to establish the qualitative properties of the auxiliary system (16) which are necessary in establishing the validity of the triangular form (15) of the original system (4). From continuity of the system matrices in (4) and continuous differentiability of the right side of the auxiliary system (16), follows the existence and uniqueness of solutions of the initial value problem (16) and (17). For validity of the transformation (14), which produces the triangular form (15), we also need boundedness of solutions of the auxiliary system. This we show next. We immediately recognize the fact that the last two equations are decoupled from the first in (16). This allows us to establish boundedness in two stages. We first consider the boundedness of L2(t) and L21(t) for which we need:
respectively. Comparing (59) and (60) with their counterparts in (16), we conclude that the equations for L2(t) and L2 l(t) can be considered as perturbed systems by (59) and (60). This allows us to use the variation of constants formula and represent the solutions as follows:
Assumption 5.1. The matrix functions Ao(t), Aol(t), Ao2(t), Alo(t), All(t), Air(t), A2o(t), and A2~(t) are bounded on ~.
L21 (t) = (I) 22(t, to)L 21 (to)(I)l 1 (to, t)
We prove the following lemma.
Lemma 5.1. Under the Assumptions 2.1, 4.1, and
L2(t) = ~22(t, to)L(to)~o(to, t)
+ ~'oCb22(t,z)(-;D2A20(z) 1
+ - [L21 (z)D1A x2(z)L2 (z)
-- L21(z)D1Alo(T)] + L2(z)Ao2(z)Lz(z))(1)o(Z , t) dr
+ f'C~22(t'z)(
lt
1 + -/-,21 ( r ) D I A 12 (r)L21(z) c.
(61)
391
Multi-parameter singular perturbations + L2(T)AoE(T)L21(z)
- LE(Z)Aol(z))~ll(Z, t) dz.
/zA
LE1 =
IDEAEE(t)+#L21(t)D1All(t)IAL21/3
(62) - - / i A L 2 1 ( D a [All(t) /3
We choose positive numbers/3+ and #+ such that
/3
2~ + max {~o,~li} ~< ~t22 16~t+ l~(llAoz(t)llp
AtE(t)Lz1(t)])
+ # [AL21DIAIE(t)AL21 ]
+ IIAodt)ll) ~< 1
+ ~R21(t,/3,#), /3
(~22
ALEl(to)
=
0
(65)
+ 1
16#~+--(IIDAI2(t)IIP + IIDiAlo(t)ll) <<,1
where by Lemma 5.1, L2(t) and LEl(t) are bounded solutions of (16)
/3 0~22
where
/~,2(t) = p =
:2 (max
0~22
(66) and RE(t,e,/g) =
For/3 ~< e + and W/3 ~< #+//3+, we have IIL2(t)ll < P and IIL2dt)ll < p for all t>~ to, which can be established by standard techniques. The proof is complete. In order to go to the second stage of validation of the transformation (14), which is concerned with the solution Lt(t) in (16), we need to establish the convergence of L2(t) and L21(t). We introduce the following assumption.
Assumption 5.2. The matrix functions Aft(t) AEo(t), A22i(t)A21(t), and their first derivatives are bounded on With this assumption, we formulate the following convergence result. Lemma 5.2. Under the assumptions of Lemma 5.1 and Assumption 5.2, we have L2(t) =
LEl(t ) = A~(t)AEt(t)
{IIDEA2011,IID2AEIlI}
+ max {llLz(to)ll, ItLEdto)}).
LEI(t) =
A21(t)A2o(t),
a22i(t)A2o(t) + O(#/e) A2~(t)A2~(t) + O(p/e).
EE(t)DiA1E(t)LE(t) -- L21(t)DiAlo(t) + e[E2(t)Ao2(t)E2(t) - L2(t)Ao(t) - L2(t)]
R2i(t,/3, #) = F.21(t)D1Ax2(t)r.21(t) - LEl(t)D1Atl(t) +/3 [LE(t)Ao(t)LEI(t) -- LE(t)Aol(t) -/~El(t)].
(67)
From Assumptions 5.1, 5.2, and boundedness of and R21(t, e,/g), and the coefficient matrices of (65), are all bounded and continuous. This implies the existence of solutions to the initial-value problem (65). Further, we note that the right sides of (65) are continuously differentiable with respect to ALE and ALE1. Therefore, the solutions are also unique. Let ¢Pn(t,z), @~(t,z), and ~a(t,z) be state transition matrices of LE(t) and L E l ( t ) , w e conclude that R 2 ( t , e , # )
ILl = [Ao(t) --
AoE(t)A22i(t)AEo(t)]rI
(63) / l ~ = # D 1 [A 1 l(t) /3
Proof. We first write A221(t)AEo(t) + ALE(t), AL2(to) = 0 LEl(t) = AE~(t)AEl(t) + AL21(t), ALEl(to) = 0. LE(t) =
A1E(t)A•(t)AEI(t)]Z
I~=IDEAEE(t)+~A2~(t)AEI(t)DIAII(t)]~)
(64)
(68)
The terms ALE(t) and ALE l(t) satisfy the differential equations
respectively. We note that I~n(t, ~)ll and I~x(t, 011 have the estimates similar to (57) and (58) with rates an and ~x. We also observe that the f~-subsystem in (68) has the same form as (19) with F(t,e,/~) = A221(t)A21(t)D1All(t). By Lemma 4.2, I~n(t,x)ll satisfies an estimate as in (32) with 0~f~< ~22- Moreover, ¢I)n(t,to)OOn(t, to) and @n(t, t0)~-tl)x(t, to) are solutions of the equations
#A L2 =[D2A22(t) + ~L,2i(t)DIAll(t)]AL2 Ao:(t)LE(t)] + li [AL2AoE(t)ALE + LE(t)AoE(t)AL2 ] - #ALE [Ao(t) -
+ P/3 [ALExD1AI,(t)/~2(t)] + ~RE(t,/3,#), /3
AL2(to) = 0
IglVl = ID2A22(t) + # -- # M [Ao(t)
392
G.S. LADDE and D. D. -
By (Ihl(t,z), @l(t,z,~,#), ~l(t,'C,~:), and q~x(t, z, e, p) we denote the state transition matrices of the systems
M(to)=®.
Ao2(t)Lz(t)],
SILJAK
~IV = [DeAzz(t) + ~L,21(t)D1Al l(t)IN
e l~' ---- D 1 / [ 1 l(t)y
# N(D1 JAil(t)
~37'= D1 [A, l(t) + O(u/e)]y
-- Alz(t)F, zl(t)]),
N(to) = E.
(69)
Equations (69) can be considered as unperturbed system corresponding to (65). Therefore, the solutions of(65) can be represented in the context of (69) as ALz(t) =
i'
On(t, Q(AL2(z)Ao2(r)AL2(3)
(75)
system (12), and
Wl(t,r,e,#)=Ol(t,z,e,#)-+(t,z,c,#).
(76)
Lemma 5.2. Under Assumptions 2.1 and 5.2, the matrices ~l~(t, 3) and Oi(t, 3, e, p) have estimates similar to those of (28) and (32) with the rates ~11/e and ~l/e where 711 is defined in (74) and ~1 = ½~11.
0
Proof Similar to the proofs of Lemmas 4.1 and
+ L2(3)Aoz(3)AL2(3)
4.2.
1
+ - [AL2i(z)D1A 11('1~)/~2('/~)
Assumption 5.4. The matrix ~ztI 1 (t) is Lipschitzian
e
on ~.
+ R2(r, e, It) ])@n(3, t) dr AL21(t) =
On(t,z)
[AL21(z)D1A12(z)AL21(z)
0
+ R21(z, e, #)])~z(z, t) dz.
(70)
Lemma 5.3. Under Assumptions 2.1, 5.3, and 5.4, there exist positive numbers e~, #¢, (* such that if e -%
/
From Assumption 5.2 and equations (64), it is obvious that ALz(t) and AL2x(t) constitute a bounded solution of (65). Then, from (70), after simple calculations, we conclude (63). This completes the proof of Lemma 5.2. In order to complete the analysis of the auxiliary system (14), it remains to establish the boundedness and convergence properties of L~(t). For this purpose, assuming Lemma 5.2 holds, we rewrite the equation for L~(t) of (14) as e/'~1 = O1 JAil(t) -- A~2(t)A2~(t)A2~(t)
Proof Similar to that of Lemma 4.3. From Lemmas 4.3, Assumptions 5.3 and 5.4, and noting the fact that the coefficient matrices in Assumption 5.3 are bounded, it follows that the solution Ll(t) of (71) is bounded. Finally, to establish convergence property of Ll(t), we introduce the following assumption.
Assumption 5.5. The matrix function ,4111(t) [Alo(t) - Alz(t)Af2 l(t)A2o(t)] and its derivative are bounded.
+ O(#/e)]L1 - Dx [Axo(t) - A12(t)A2z~(t)A20(t)
+ o(#/e)] -- eL i [Ao(t ) -
Aoz(t)A2~(t)Azo(t) + O(#/e) + L~ ]Aox(t)
- Aoz(t)A~(t)Aza(t) + O(#/e)]L1
(71)
with the initial condition Ll(to) - [A1 ~(to) -- Al:(to)A~2X(to)A21(to)]-I [A~o(to) -- A12(to)A2~(to)A2o(to)].
(72)
We define ,'tab(t) = All(t) - Ai2(t)A22~(t)A21(t)
(73)
Ll(t) = Ai-xl(t) [Alo(t) - Alz(t)A2~(t)Azo(t)] + O(e). (77)
Proof Similar to that of Lemma 5.2. By establishing boundedness and convergence of the solutions L l(t), Lz(t), and Lzl(t) of the auxiliary system (16), we have verified the validity of the transformation (14). Therefore, the transformed system (15) can be rewritten in terms of the original matrices of (13). We define the coefficient matrices of the p-reduced system (8) as
Ao(t) = Ao(t) - Ao2(t)A2~(t)A2o(t),
and introduce the following assumption.
Assumption 5.3. For Dx satisfying (6), there exists
/lot(t) = Aol(t) - Aoz(t)Az1(t)A21(t) /llo(t) = Alo(t) - A12(t)A221(t)Azo(t),
a positive number ~ll such that L~'{D1All(t)} ~< -~11,
Lemma 5.4. Under the assumptions of Lemma 5.2 and Assumptions 5.3, 5.4, and 5.5, we have
Vt i> to.
(74)
/lll(t)=A1~(t)-Alz(t)Af~(t)Azl(t)
(78)
Multi-parameter singular perturbations
393
and the e-reduced system (11) as AR(t ) --- ,4o(t) -- Aot(t)7t~tl(t).~to(t)
(79)
and rewrite (15) as Aol -- Ao2(t),4[at(t)Alo(t)Dl,'~xo(t) D1Alx(t ) + O(/~/e) + O(e)
(3
+ O(e)
~,4;11(t)/{loAo2(t) + OaAlz(t ) + eO(e D2A22(t) + O(#/e)
(80) with initial conditions where ~t(t, to, e,#) is the state transition matric corresponding to the decoupled e-subsystem with the matrix D1Att(t) + O(p/e) + O(e),
U(to) = x o
V(to) = -4;d(to)/{lo(to)Xo + Yo + O(e) W(to) = A~l(to)Azo(to)Xo + A~(to)A2t(to)Yo + z0 + O(#/e).
B23(t, e, p) = -~[~(t)Ato(t)Ao2(t) + 1Dt,412(t) (81)
On the basis of the transformed system (80), we establish in the next section our main result concerning the qualitative properties of the original system (13). 6. MAIN RESULT
We are now in a position to use the triangular form (30) and establish stability of the original multitime scale singularly perturbed system (13). For this, we need our last assumption concerning stability of the reduced subsystem (11) with the system matrix AR of (79).
Assumption 6.1. There exists a positive number
+ O(e).
and w(t) is the solution of the p-subsystem (19)
w(t) = ~2(t, to, e, p)W(to)
t~
to)1
IIv(t)[I ~ e x p I - ~ x ( t -
( llv(t° )ll + ~2 -
)
(85)
where (p/e)F(t, e, p) = O(p/e). Because of Assumption 5.3, ~)l(t, to, e, #) has an estimate similar to (32) with the rate ~1 < ~x 1. Moreover, by Assumptions 5.1 and 5.5 with (78), we have boundedness of B23(t,e,t~) with respect to t. Therefore, by (32) we can derive from (84) the following inequality
~R such that limsup
(84)
-po ~ 1
llB2 3(t' e' P )ll llw(t° )ll) (86)
~{A~(z)}dz ~< --~a. (82)
+ cX~
0
Assumption 6.1 implies that the reduced subsystem (11) is globally exponentially stable.
Theorem 6.1. Under Assumptions 2.1, 2.2, 2.3, 4.1, 4.2, 5.1, 5.2, 5.3, 5.4, 5.5, and 6.1, there exist positive number g and /i such that if e ~< g and #/e ~< ki/g then the equilibrium of the system (13) is globally exponentially stable.
Proof Stability of (80) implies stability of (13). To show stability of (80), we should verify stability of each decoupled subsystem and smallness of their interconnections considered as perturbation terms. We recall Lemma (4.2) and conclude immediately stability of the p-subsystem of (80). To establish stability of the e-subsystem, we use (80) and write the corresponding solution as v(t) = ~ t ( t , to, e, p)v(to) +
~o
+t(t,z,e,#)Bz3(Z)w(z)dz
(83)
By choosing gl and/it sufficiently small, so that fi~-~l < ~z% et
(87)
and noting the fact that ~lllBz3(t, gt,/Jt)ll is bounded, we conclude from (85) exponential stability of the e-subsystem. Finally, using Assumption 6.1 and repeating the similar arguments, we establish exponential stability of the reduced subsystem with perturbation terms, which is defined in (80). This results in the values gR and/~R of the upper bounds of e and p for stability of the reduced subsystem. By choosing g = min {gl, gR} and ~i = min {/it,/~a} exponential stability of (86) and, thus, (13) follows. This completes the proof. We remark that under the assumptions of Theorem 6.1 (except Assumption 6.1), we can conclude that the solutions of the original system (13) can be approximated by the solutions of the overall reduced system, and e and p boundary layer system. The inverse of the transformation (14) is
394
G.S. LADDE and D. D. SILJAK
,o
o Ool[!
-Ll(t)
I1
- L 2 ( t ) + L2~(t)La(t)
This together with Lemmas 4.3 and 5.3 gives the desired result
-L21(t)
.
t88)
I
Energy, Electric Energy Systems Division, under contract DEAC03-77ET29138, and in part by The National Science Foundation under grant ECS-8011210.
x(t) = ~(t) + 0 ( 0 y(t) = --A?ll(t)A1o(t)Yc(t) + ~(t) + O(g)
z(t) = [ - A ; ~ ( t ) A 2 o ( t )
+ A ~ (t)A z 1(t)d;11 (t)al o(t)])~(t) -
-
A£¢(t)A2~(t)y(t) + ~(t) + 0 ( # / 0 + O(e). (89)
Equations (89) are valid on any compact time interval. This result parallels that of Theorem 2 in Khalil and Kokotovic (1979a). 7. CONCLUSION
A hierarchial scheme was developed for aggregation of small parameters in singular perturbation analysis of linear systems having multiple time scales. The scheme is effected by a linear transformation which produces an explicit orderdecomposition of the system. The resulting multiparameter-multitime-scale problem is solved in the context of generalized D-stability. Acknowledgements--The authors are grateful to Professor S. L. Campbell, Department of Mathematics, North Carolina State University at Raleigh, for his valuable comments on this paper. This research was supported in part by The Department of
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