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Lyapunov Matrices for Time Delay System with Commensurate Delays
Copyright © IFAC System, Structure and Control Oaxaca, Mexico, USA, 8-10 December 2004
ElSEVIER
IFAC PUBLICATIONS
www_elsevier_comllocate/ifac
LYAPUNOV MATRICES FOR TIME DELAY SYSTEM WITH COMMENSURATE DELAYS Hiram Garcia-Lozano t , VIadimir L. Kharitonov
Departartament of Automatic Cont1'01
CINVESTAV-IPN Afe:rico . D.F.
MEXICO
t
E-mail: hgarcia(!.;i ctrl.cinvestav.mx
Abst.ract: In this paper, two procedures are proposed for construction of the Lyapunov matri ces for time delay systems with commensurable delays. The first one is semianalytical, while the second one provides continuous piece-wise linear approximations. Copyright © 2004 IFAC Keywords: Time delay systems, Lyapunov-Krasovskii Functional. Lyapullov Matrix, Krollecker product Numerical algorithms.
matrices . This procedure remains practical in the case of systems with non commensurate delays.
1. INTRODUCTION
The stability of time delay systems has been intensively studied in the last years. Some results are based on the reduce type Lyapunov-Krasovskii functionals, see survey papers (Kharitonov. 1999) • (Niculescu, 2001). Recently. new advances in construction of the complete type LyapunovKrasovskii functionals were reported in (Kharitonov, 2(03). It has been show there that the key component in such a construction is played by Lyapunov matrices which were introduced there. On the other hand, there were no algorithms proposed for const.ructive computation of the matrices . In this note we propose two numerical procedures for computation of the matrices. The first one is based on a special system of linear mat.rix equations without delay which has been reported for the case of systems with single delay in (Infante, 1978). Here we extend this procedure to the case of systems with multiple delays. The second procedure consists in computation of linear piece-wise approximations of the Lyapunov
1
In Section 1 some basic notations are introduced and useful auxiliary results are given. In Section 2 we introduce complete type LyapunovKrasovskii functionals and give a formal definition of Lyapunov matrices for time delay systems. Section 3 is dedicated to the semi-analyctic approach for computation of the Lyapunov matrices for time delay systems. \Ve introduce here a set of matrices which define the Lyapunov matrix and show that this set of matrices satisfies a system of matrix equations without of delay. To identify the solution which corresponds to the Lyapunov matrix a special set of bowldary conditions are given. In Section 4 we propose the second procedure which consists in computation of piece-wise linear approximations of the Lyapunov matrices. Some concluding remarks close the note.
The work was partially supported by CO.\"ACyT
Sll
2.
PRELllvIINARY
K(t) is the fundamental matrix of system (1) . Matrix U( T) satisfies the equation, see (Infante, 1978) ,
Consider a time delay system of the form rn
/Tt
t
~
O.
(1)
U'(T)
=
k=O ~
0 is a given
where U'(T)
= -:f;U(T) .
The following properties of U( T) can be found in (Infante, 1978).
Given a piece-wise continuous vector fUllction
Lemma 1. Matrix: U(T) satisfies the following condi tions: The symmetry condition for T ~ O.
U( - T) = UT(T).
is the segment of the solution. exist T ~ 1 and (7 > 0, such that every solution, l' (t ,.,:», of system (1) sat.isfies the inequality
')' 11
(5 )
The algebraic condition
If syst.em (1) is exponent.ially stable, then t.here
~
(4)
k=O
Here Ab are given n x n matrices. h time delay and x( t) = 'L~/~tl.
11 1: (t.
L U(T - kh)Ak.
-w =
A~U(O)
+ U(O)Ao+
(6)
",
for t ~ 0,
+
L AJU(kh) + U T (kh)Ak. k=O
here
1I
=
max
HE [- mh .O]
1I
As Cl. consequence. having const.ruct.ed the matrix U(T) we have in our ha nd s the fuuctionall'(l·t ).
Define a quadratic functional of III
W(.Tt) = x T (t)H"o :r (t)+
L
.1:
T
(t - kh)IVk·.T(t - kh) +
3. A SE:-'II-ANALYTIC CONSTRUCTION OF MATrnX U(T)
k= 1
+
f
fO T(t + O)H"II/H .T(t + O)rlO. X
In this sect ion we consider a semi -analitic procedure for const.1'1lction of Lyapunov mat rices for the system (1).
- kh where t.he IT'; are definite positive matrices. The corresponding complete type Lyapunov-Krasovskii functional is a quadratic fun ctional which satisfies the equation d -V(.Tt) = -w(:rtl , t ~ O. 1.:=1
dt
Let us introduce 2m auxiliary matrices
Xj(T)=U(T+jh) r E[O. hJ, { j = -m , -m + 1. .... 0,1. ... , m - 1.
.
Let j
The functional has the form, see (Kharitonov. 200:3) , uo(x,) = x T (t)U(O)l:(t)+
~
(7)
0, then 11/
Xj(r)
= U'(r + jh) = L
U(r
+ (j
- k)h).4 k·.
k = ll
+
?;2XT(t) ikhU(-kh-O)AkX(t+O)dO+
+
fffo.
k=} j=)
X
Note that - rn ~ j - k < rn, for j ~ O. therefore the last equation can be written as
0
m
(10
XT(t+O})AJ
-kh
x
U(O) _ O2 + kh _ jh)Aj.-r(t
for r E [0. hJ.
+ (2)d0 2 )
dOl
+ When j < 0, applying
properly (5) we obtain Xj(T)=U(T+jh) = UT( - T - jh).
-jh
f
m
+L k=)
x x(t
(8)
k= O
0 X T
(t
+ 0)
[l-Vk
+ (kh + O)IFrn+k J X
So ,
-kh
+ O)dO,
(2)
Xj(T) = -U'(-T - jh),
where the matrix
U(T) =
1'' '
KT (t) [1'Vo
x K(t
+ T)dt.
=
+ ~(Wk + khW,n+k)] x
r"l'" -lE
U( -T - (j
+ k)h)Ak ] T
m
= -
(3)
L k=O
92
AJU(r + (j
+ h:)h) .
,
Observe that for j < 0 we have -m::; j + k < m, therefore the last equation can be written in the form
Here matrix
(1)
Ao x At =
(
m
L A[ Xj+dT),
Xj(T) = -
for T E [0, h],
(1)
al
(I)
(9)
k=O
(I»)
Ao a21 Ao ... ani Ao ai;) Ao a~;) Ao .. . a~,lJ Aa . . . ... .. ..
all
n
,
(I)
(I)
Ao a2n Ao ... allllAa
is the Kronecker product of matrices Ao
( al~») n
Equations (8) and (9) allow us to write the system of linear matrix equations without delay
and Al =
(al;») n
.)=1
•
=
and E is t.he
.)=1
identity n x n matrix.
rn
L Xj-k(T)Ak,
Xj(T) =
In a similar way, the vectorial form of the boundary conditions is:
k=U
j=O,I, ... ,m-1. TE [O.h].
( 10)
,n
X .; (T) = -
L AJ Xj+d T ),
+ 1. .... -1
T
711+ 1. .. .. 0, 1, .. ., In
j = -rn, -
k=O
) = -m. -m
( 14)
Xj+l(O) = .1:j(h).
E [0. h] .
-l·V = (Ati' x E) Xo
+ (E
2:
-
x Ao)xu+
(15)
U1
Theorem 1. The matrices (7) sat.isfy system (10) and the following boundary condit.ions
+
L [(AT x E) xk - I(11) + (Eo A.,jXk- l(hl]. k=1
Here matrix Xj+l
(0) = Xj (h) ,
(11)
( aiO) (ail») Ta~O) (a;I») T... a;O)(a~I»)T a~O)(a~I») T ...
j= - m,-m+ 1. .... 0,1. ... ,m-2:
- ~V = Ati' Xo(O)
+ Xo(O)Ao+
l
AooAI =
'/I
+
L [A~ Xk-dh) + X!_I (h)AkJ
(12)
k=1 (0)
1
(0 )
.1:'_1 (T) .T.~( T)
l <, -:1 r-Lo
the vector
(T)
«T)
= -
L
(AJ
-
L; E [0, h]
x
E)
(I)
, 0." arecolull1ns
(T) XO(T)
l Xm - :I(T)
(16)
J
- Lm 0 - Lo -L1
1
-Lm
Lo
L", L m -
0
0
1
0
Lu
and
xj-d T),
T
J
L", L", -1
0
Now the equation (10) can be "'.'fitten as
j = 0, 1. ... , m-I;
... .
Ll
=M
-LI
0
M=
k)
( I)
tL
where
X;"')(T) t.he columns of matrix Xi(T) , and define
L (E x A
( I)
H
71.
We arrive at the following system
Let. us present the matrL'( equat.ion (10) and the bounda.ry conditions (11), (12), in the vector form. To t.his end, we denote by x;I )(T), x;2)(T), ....
k=O
1
H
( 0)
.1.1 Vector form presentation
Xj(T) =
a~:l)(ail»)Tj
(~(I») T alU) (~(I») T ... a(O) (atl») T
anda l , 0. 2 , ,,,,0.,, .0. 1 , a 2 of Ao and AI respectively.
Proof. The fact that matrices (7) satis fy system (10) follows directly form the previous computations. The first set of boundary condit.ions is a consequence of the property of matrix (7) . The last boundary conditions is consequence of (G) . •
(a;ll)T\
.
.
0.(0)
a~;l)
(13)
= AiT
X
E.
L;=ExA;,
xJ+d T),
i
= 0,1, .. , m,
i=O,l , .. ,m .
Remark 1. If there exists only one solution of (16) which satisfies the boundary conditions of
k=O
)=-711,-rn+l , .... -2.- 1; TE[O,h].
93
the Theorem 1, then the solution defines the Lyapunov matrix in accordance with (7): XO(T) ,
rical and contain only (~) . So, one has to reduce the dimension of vector c and the vector W, respectively.
for T E [0, h]; h), for T E [h ,2h];
!
~I(T -
U(T) =
tions. on the other hand matrix U (0) is symmet-
:
X",_I(T - (m -1)h), for T E [(m - l)h , mhJ,
4. PIECE-WISE LINEAR APPROXIMATION
if there are several solution:; of (16), then one has to select the one which satisfies equation (4) and the conditions (5)-(6). This :;olution defines the Lyapunov matrix.
In this section we propose a numerical scheme to compute continuous piece-wise linear approximation of the Lyapunov matri.x U( T) . It follows directly from equation (4) that U(T)e - AOT = U(O)
The general solution of (16) is of the form y = e.'VlT c.
f: 11'
U(~ - kh)Ake- An!d~. 1.0=1 We will use this expression in order to derive the desired approximation. Let lIS divide the interval [0. h] into N equal segments
(17)
+
Here c=
[k1'. (k where the vector c i:; unknown.
-
t) c
+ 1)1']
From (19) we have
.,=1
( 18)
= Ul .
Comparing matrices U(j1') and U(jr'
U(j1' 0
0
0
0
t li U(~- kh)Ake- A;'!d~. r
U(j1')e AOJ " = U(O)+
Here
E ...
k = O.1, .... N-1.
where
The vector c can be fount! if we substitute expression (17) in conditions (14) , (1.5). and solve then the obtained equations with respect to the vectors. To do this we rewrite the conditions (14) and (15) in a matrix block form . These conditions takes I.he form
(Te·Mh
(19)
j
+ r')e - AoU+lJr
(j+ I J ,'
+ r)
we get
= U(j1')+
111
L
U(~ - kh)Ake - A' o~ d~ . 1.0 =1 Changing the integ ration variable we finally arrive at the equality
+ ,
• J"
T=
0 0
E
0
0
0
E
°
0 0
U(j1'
0 . ..
E 0 0 ... KI K2 ... I<""m - I A'",
° o°E .. · ° °° °° E° E° ... 0
T=
l oo °°
IV = [0 ... 0 0 vec( - W) T
=
-(A~ x E
Ki = AJ x E
U(O
+ j1' - k(N1· )) Ake - AOH dO].
Now. we denote by U(T) the approximation of the Lyapunov matrix and introduce the following matrices
0 0
U; = O(jr), On each segment tion is defined as
1T
+E x
(20)
k=1
0
U(s) =
where E is the n x n identity block. and
Ko
= U(j1')+
r
+
E 0J
0 0 0
Ko
f1
+ r)e - AOr
(1 +
j = 0,1. ,.. . m.
[j1', (j
j1'; s) Uj
+ 1)1'). +
the approxima-
C~
jr') Uj+I. (21)
Replacing (21) in the equation (20) and using the property (5) we obtain
Aa),
+ Eo Ai· Uj+IP = Uj
Remark 2. On one hand matrix equation ( 15) provides only (~) independent scalar equa-
+
In
In
k=1
k=1
L U'~k-jQk + L Uftk _j_1Rk ,
j = 0,1, .. ., Nm - 1.
94
(22)
where
If there exists only one solution of (24) then, tllis solution defines Lyapunov mat.rix.
5. CONCLUSIONS In this note we propose two different computational schemes to compute the Lyapunov matrix, this matrix is t.he key in the construction of Lyapunov-Krasovskii functiona!. It is worth to point out that, the second scheme can be extended to more general cases of time delay systems like 1l0n-conIllensurate time delay systems .
In the cases when j > kN the subscripts in UJ.k - j , UJ.k _ j _ lare negatives numbers, which introduces news variables not desired, in this case we can use the property (5), changing U,J .k - j by Uj-N.k. We obtain mN matrix equations for mN + 1 unknov.'11 matrices. The condition (6) add additional matrix equation
'"
+L
-14' = AJUo + UuAo
A"[.U.·. N
REFERENCES
+ U;r.NA k .
k= 1
Finally, we arrive at t.he system of rnN + 1 matrix equations for mN + 1 unknown matrices: In
+
UH1P = U j
L U~. k_jQk + L U'~k_j _ IRk
k=l j = 0, 1, .. .. mN - 1,
k= 1
(23) m
-w =
AJUo + UoAo
L [A J UN .•. + U~kA..].
+
k=1
If the system admits a unique solution. then the solution defines t.he desired contlllUoo piece-wise linear approxinlation of U(r).
4.1 Vector form In this subsection we write the equation (23) in the vector form. In order to do this, we first associate with a n x n mat.ri.'( X the corresponding n 2 - dimensional column vector .1:. In this case, t.he column vector which corresponds to the matrix product AoX Al has the form (Ao x Adx, while (Ao oA I)X represents the column vector associated with matrix the product AOXT Al . In thESe notations form
the equation (23) takes the m
(E
X P)Uj+1
= Uj
+L
(E
0
QdUNk-j+
k= 1
k=l
j = 0, 1, ... , m - 1.
-w
+L
=
[(.16
x E)
+ (E
[(A~ x EUNk)
x Ao)]
(24) 1£0+
+ (E OAk)U
NkJ
k=l
Here 1£0, 1£1, •.• U m , are n 2 -diInensional column vectors associat.ed with matrices Uo, UI , •• . , U"'.
95
R . nellman and K 1. Cooke. (1963) . DifferentialDifference Equations , Academic Press. Gu , K (1999). Discretized Lyapunov functional for uncertain systems with multiple time delay, International JOllrnal of control, pp. 1436-1445 . Hale, .Jack K, Sjoerd 1\1. VerduYll Lune!. (1993) . Introduction to Functional Differential Equations, Applied :\lat.hematical Sciences, Springer- Verlag. Huang W. (1989) . Generalization of Liapunov 's Theorem in a linear delay systems, Journal of Mathematical Analysis and Applications, pp. 83-94. Infante E. F. and W. n. Castelan. (1978). A Lyapunov functional for a matri.'( differncedifferential equations, .Journal DijJerential Equations, pp. 439-451. Kharitonov V. 1., and A . P. Zhahko. (200:3). Lyapunov-Krasovskii approach to the robust stability analysis of time Delay systems, Automatica, pp 15-20. Kharitonov V. L. (1999). Robust stabilit.y analysis if time delay systems: A survey, Annual Reviews in control, pp 185-196 Kolmanovskii, V. and A. Myshkis. (1999) . Introduction to the theon) and apl-ica.tions of the functional differ'ential equations, Academic Press. Kolmanovskii, V. (1999) . St.ability of some linear syst.ems whit. delays IEEE tmsactions on Automatic Contol, Academic Press. Niculescu, S (200 1). Delay Effec ts on Stability. S pringer-Verlag.
ElSEVIER
IFAC PUBLICATIONS
www_elsevier_comllocate/ifac
LYAPUNOV MATRICES FOR TIME DELAY SYSTEM WITH COMMENSURATE DELAYS Hiram Garcia-Lozano t , VIadimir L. Kharitonov
Departartament of Automatic Cont1'01
CINVESTAV-IPN Afe:rico . D.F.
MEXICO
t
E-mail: hgarcia(!.;i ctrl.cinvestav.mx
Abst.ract: In this paper, two procedures are proposed for construction of the Lyapunov matri ces for time delay systems with commensurable delays. The first one is semianalytical, while the second one provides continuous piece-wise linear approximations. Copyright © 2004 IFAC Keywords: Time delay systems, Lyapunov-Krasovskii Functional. Lyapullov Matrix, Krollecker product Numerical algorithms.
matrices . This procedure remains practical in the case of systems with non commensurate delays.
1. INTRODUCTION
The stability of time delay systems has been intensively studied in the last years. Some results are based on the reduce type Lyapunov-Krasovskii functionals, see survey papers (Kharitonov. 1999) • (Niculescu, 2001). Recently. new advances in construction of the complete type LyapunovKrasovskii functionals were reported in (Kharitonov, 2(03). It has been show there that the key component in such a construction is played by Lyapunov matrices which were introduced there. On the other hand, there were no algorithms proposed for const.ructive computation of the matrices . In this note we propose two numerical procedures for computation of the matrices. The first one is based on a special system of linear mat.rix equations without delay which has been reported for the case of systems with single delay in (Infante, 1978). Here we extend this procedure to the case of systems with multiple delays. The second procedure consists in computation of linear piece-wise approximations of the Lyapunov
1
In Section 1 some basic notations are introduced and useful auxiliary results are given. In Section 2 we introduce complete type LyapunovKrasovskii functionals and give a formal definition of Lyapunov matrices for time delay systems. Section 3 is dedicated to the semi-analyctic approach for computation of the Lyapunov matrices for time delay systems. \Ve introduce here a set of matrices which define the Lyapunov matrix and show that this set of matrices satisfies a system of matrix equations without of delay. To identify the solution which corresponds to the Lyapunov matrix a special set of bowldary conditions are given. In Section 4 we propose the second procedure which consists in computation of piece-wise linear approximations of the Lyapunov matrices. Some concluding remarks close the note.
The work was partially supported by CO.\"ACyT
Sll
2.
PRELllvIINARY
K(t) is the fundamental matrix of system (1) . Matrix U( T) satisfies the equation, see (Infante, 1978) ,
Consider a time delay system of the form rn
/Tt
t
~
O.
(1)
U'(T)
=
k=O ~
0 is a given
where U'(T)
= -:f;U(T) .
The following properties of U( T) can be found in (Infante, 1978).
Given a piece-wise continuous vector fUllction
Lemma 1. Matrix: U(T) satisfies the following condi tions: The symmetry condition for T ~ O.
U( - T) = UT(T).
is the segment of the solution. exist T ~ 1 and (7 > 0, such that every solution, l' (t ,.,:», of system (1) sat.isfies the inequality
')' 11
(5 )
The algebraic condition
If syst.em (1) is exponent.ially stable, then t.here
~
(4)
k=O
Here Ab are given n x n matrices. h time delay and x( t) = 'L~/~tl.
11 1: (t.
L U(T - kh)Ak.
-w =
A~U(O)
+ U(O)Ao+
(6)
",
for t ~ 0,
+
L AJU(kh) + U T (kh)Ak. k=O
here
1I
=
max
HE [- mh .O]
1I
As Cl. consequence. having const.ruct.ed the matrix U(T) we have in our ha nd s the fuuctionall'(l·t ).
Define a quadratic functional of III
W(.Tt) = x T (t)H"o :r (t)+
L
.1:
T
(t - kh)IVk·.T(t - kh) +
3. A SE:-'II-ANALYTIC CONSTRUCTION OF MATrnX U(T)
k= 1
+
f
fO T(t + O)H"II/H .T(t + O)rlO. X
In this sect ion we consider a semi -analitic procedure for const.1'1lction of Lyapunov mat rices for the system (1).
- kh where t.he IT'; are definite positive matrices. The corresponding complete type Lyapunov-Krasovskii functional is a quadratic fun ctional which satisfies the equation d -V(.Tt) = -w(:rtl , t ~ O. 1.:=1
dt
Let us introduce 2m auxiliary matrices
Xj(T)=U(T+jh) r E[O. hJ, { j = -m , -m + 1. .... 0,1. ... , m - 1.
.
Let j
The functional has the form, see (Kharitonov. 200:3) , uo(x,) = x T (t)U(O)l:(t)+
~
(7)
0, then 11/
Xj(r)
= U'(r + jh) = L
U(r
+ (j
- k)h).4 k·.
k = ll
+
?;2XT(t) ikhU(-kh-O)AkX(t+O)dO+
+
fffo.
k=} j=)
X
Note that - rn ~ j - k < rn, for j ~ O. therefore the last equation can be written as
0
m
(10
XT(t+O})AJ
-kh
x
U(O) _ O2 + kh _ jh)Aj.-r(t
for r E [0. hJ.
+ (2)d0 2 )
dOl
+ When j < 0, applying
properly (5) we obtain Xj(T)=U(T+jh) = UT( - T - jh).
-jh
f
m
+L k=)
x x(t
(8)
k= O
0 X T
(t
+ 0)
[l-Vk
+ (kh + O)IFrn+k J X
So ,
-kh
+ O)dO,
(2)
Xj(T) = -U'(-T - jh),
where the matrix
U(T) =
1'' '
KT (t) [1'Vo
x K(t
+ T)dt.
=
+ ~(Wk + khW,n+k)] x
r"l'" -lE
U( -T - (j
+ k)h)Ak ] T
m
= -
(3)
L k=O
92
AJU(r + (j
+ h:)h) .
,
Observe that for j < 0 we have -m::; j + k < m, therefore the last equation can be written in the form
Here matrix
(1)
Ao x At =
(
m
L A[ Xj+dT),
Xj(T) = -
for T E [0, h],
(1)
al
(I)
(9)
k=O
(I»)
Ao a21 Ao ... ani Ao ai;) Ao a~;) Ao .. . a~,lJ Aa . . . ... .. ..
all
n
,
(I)
(I)
Ao a2n Ao ... allllAa
is the Kronecker product of matrices Ao
( al~») n
Equations (8) and (9) allow us to write the system of linear matrix equations without delay
and Al =
(al;») n
.)=1
•
=
and E is t.he
.)=1
identity n x n matrix.
rn
L Xj-k(T)Ak,
Xj(T) =
In a similar way, the vectorial form of the boundary conditions is:
k=U
j=O,I, ... ,m-1. TE [O.h].
( 10)
,n
X .; (T) = -
L AJ Xj+d T ),
+ 1. .... -1
T
711+ 1. .. .. 0, 1, .. ., In
j = -rn, -
k=O
) = -m. -m
( 14)
Xj+l(O) = .1:j(h).
E [0. h] .
-l·V = (Ati' x E) Xo
+ (E
2:
-
x Ao)xu+
(15)
U1
Theorem 1. The matrices (7) sat.isfy system (10) and the following boundary condit.ions
+
L [(AT x E) xk - I(11) + (Eo A.,jXk- l(hl]. k=1
Here matrix Xj+l
(0) = Xj (h) ,
(11)
( aiO) (ail») Ta~O) (a;I») T... a;O)(a~I»)T a~O)(a~I») T ...
j= - m,-m+ 1. .... 0,1. ... ,m-2:
- ~V = Ati' Xo(O)
+ Xo(O)Ao+
l
AooAI =
'/I
+
L [A~ Xk-dh) + X!_I (h)AkJ
(12)
k=1 (0)
1
(0 )
.1:'_1 (T) .T.~( T)
l <, -:1 r-Lo
the vector
(T)
«T)
= -
L
(AJ
-
L; E [0, h]
x
E)
(I)
, 0." arecolull1ns
(T) XO(T)
l Xm - :I(T)
(16)
J
- Lm 0 - Lo -L1
1
-Lm
Lo
L", L m -
0
0
1
0
Lu
and
xj-d T),
T
J
L", L", -1
0
Now the equation (10) can be "'.'fitten as
j = 0, 1. ... , m-I;
... .
Ll
=M
-LI
0
M=
k)
( I)
tL
where
X;"')(T) t.he columns of matrix Xi(T) , and define
L (E x A
( I)
H
71.
We arrive at the following system
Let. us present the matrL'( equat.ion (10) and the bounda.ry conditions (11), (12), in the vector form. To t.his end, we denote by x;I )(T), x;2)(T), ....
k=O
1
H
( 0)
.1.1 Vector form presentation
Xj(T) =
a~:l)(ail»)Tj
(~(I») T alU) (~(I») T ... a(O) (atl») T
anda l , 0. 2 , ,,,,0.,, .0. 1 , a 2 of Ao and AI respectively.
Proof. The fact that matrices (7) satis fy system (10) follows directly form the previous computations. The first set of boundary condit.ions is a consequence of the property of matrix (7) . The last boundary conditions is consequence of (G) . •
(a;ll)T\
.
.
0.(0)
a~;l)
(13)
= AiT
X
E.
L;=ExA;,
xJ+d T),
i
= 0,1, .. , m,
i=O,l , .. ,m .
Remark 1. If there exists only one solution of (16) which satisfies the boundary conditions of
k=O
)=-711,-rn+l , .... -2.- 1; TE[O,h].
93
the Theorem 1, then the solution defines the Lyapunov matrix in accordance with (7): XO(T) ,
rical and contain only (~) . So, one has to reduce the dimension of vector c and the vector W, respectively.
for T E [0, h]; h), for T E [h ,2h];
!
~I(T -
U(T) =
tions. on the other hand matrix U (0) is symmet-
:
X",_I(T - (m -1)h), for T E [(m - l)h , mhJ,
4. PIECE-WISE LINEAR APPROXIMATION
if there are several solution:; of (16), then one has to select the one which satisfies equation (4) and the conditions (5)-(6). This :;olution defines the Lyapunov matrix.
In this section we propose a numerical scheme to compute continuous piece-wise linear approximation of the Lyapunov matri.x U( T) . It follows directly from equation (4) that U(T)e - AOT = U(O)
The general solution of (16) is of the form y = e.'VlT c.
f: 11'
U(~ - kh)Ake- An!d~. 1.0=1 We will use this expression in order to derive the desired approximation. Let lIS divide the interval [0. h] into N equal segments
(17)
+
Here c=
[k1'. (k where the vector c i:; unknown.
-
t) c
+ 1)1']
From (19) we have
.,=1
( 18)
= Ul .
Comparing matrices U(j1') and U(jr'
U(j1' 0
0
0
0
t li U(~- kh)Ake- A;'!d~. r
U(j1')e AOJ " = U(O)+
Here
E ...
k = O.1, .... N-1.
where
The vector c can be fount! if we substitute expression (17) in conditions (14) , (1.5). and solve then the obtained equations with respect to the vectors. To do this we rewrite the conditions (14) and (15) in a matrix block form . These conditions takes I.he form
(Te·Mh
(19)
j
+ r')e - AoU+lJr
(j+ I J ,'
+ r)
we get
= U(j1')+
111
L
U(~ - kh)Ake - A' o~ d~ . 1.0 =1 Changing the integ ration variable we finally arrive at the equality
+ ,
• J"
T=
0 0
E
0
0
0
E
°
0 0
U(j1'
0 . ..
E 0 0 ... KI K2 ... I<""m - I A'",
° o°E .. · ° °° °° E° E° ... 0
T=
l oo °°
IV = [0 ... 0 0 vec( - W) T
=
-(A~ x E
Ki = AJ x E
U(O
+ j1' - k(N1· )) Ake - AOH dO].
Now. we denote by U(T) the approximation of the Lyapunov matrix and introduce the following matrices
0 0
U; = O(jr), On each segment tion is defined as
1T
+E x
(20)
k=1
0
U(s) =
where E is the n x n identity block. and
Ko
= U(j1')+
r
+
E 0J
0 0 0
Ko
f1
+ r)e - AOr
(1 +
j = 0,1. ,.. . m.
[j1', (j
j1'; s) Uj
+ 1)1'). +
the approxima-
C~
jr') Uj+I. (21)
Replacing (21) in the equation (20) and using the property (5) we obtain
Aa),
+ Eo Ai· Uj+IP = Uj
Remark 2. On one hand matrix equation ( 15) provides only (~) independent scalar equa-
+
In
In
k=1
k=1
L U'~k-jQk + L Uftk _j_1Rk ,
j = 0,1, .. ., Nm - 1.
94
(22)
where
If there exists only one solution of (24) then, tllis solution defines Lyapunov mat.rix.
5. CONCLUSIONS In this note we propose two different computational schemes to compute the Lyapunov matrix, this matrix is t.he key in the construction of Lyapunov-Krasovskii functiona!. It is worth to point out that, the second scheme can be extended to more general cases of time delay systems like 1l0n-conIllensurate time delay systems .
In the cases when j > kN the subscripts in UJ.k - j , UJ.k _ j _ lare negatives numbers, which introduces news variables not desired, in this case we can use the property (5), changing U,J .k - j by Uj-N.k. We obtain mN matrix equations for mN + 1 unknov.'11 matrices. The condition (6) add additional matrix equation
'"
+L
-14' = AJUo + UuAo
A"[.U.·. N
REFERENCES
+ U;r.NA k .
k= 1
Finally, we arrive at t.he system of rnN + 1 matrix equations for mN + 1 unknown matrices: In
+
UH1P = U j
L U~. k_jQk + L U'~k_j _ IRk
k=l j = 0, 1, .. .. mN - 1,
k= 1
(23) m
-w =
AJUo + UoAo
L [A J UN .•. + U~kA..].
+
k=1
If the system admits a unique solution. then the solution defines t.he desired contlllUoo piece-wise linear approxinlation of U(r).
4.1 Vector form In this subsection we write the equation (23) in the vector form. In order to do this, we first associate with a n x n mat.ri.'( X the corresponding n 2 - dimensional column vector .1:. In this case, t.he column vector which corresponds to the matrix product AoX Al has the form (Ao x Adx, while (Ao oA I)X represents the column vector associated with matrix the product AOXT Al . In thESe notations form
the equation (23) takes the m
(E
X P)Uj+1
= Uj
+L
(E
0
QdUNk-j+
k= 1
k=l
j = 0, 1, ... , m - 1.
-w
+L
=
[(.16
x E)
+ (E
[(A~ x EUNk)
x Ao)]
(24) 1£0+
+ (E OAk)U
NkJ
k=l
Here 1£0, 1£1, •.• U m , are n 2 -diInensional column vectors associat.ed with matrices Uo, UI , •• . , U"'.
95
R . nellman and K 1. Cooke. (1963) . DifferentialDifference Equations , Academic Press. Gu , K (1999). Discretized Lyapunov functional for uncertain systems with multiple time delay, International JOllrnal of control, pp. 1436-1445 . Hale, .Jack K, Sjoerd 1\1. VerduYll Lune!. (1993) . Introduction to Functional Differential Equations, Applied :\lat.hematical Sciences, Springer- Verlag. Huang W. (1989) . Generalization of Liapunov 's Theorem in a linear delay systems, Journal of Mathematical Analysis and Applications, pp. 83-94. Infante E. F. and W. n. Castelan. (1978). A Lyapunov functional for a matri.'( differncedifferential equations, .Journal DijJerential Equations, pp. 439-451. Kharitonov V. 1., and A . P. Zhahko. (200:3). Lyapunov-Krasovskii approach to the robust stability analysis of time Delay systems, Automatica, pp 15-20. Kharitonov V. L. (1999). Robust stabilit.y analysis if time delay systems: A survey, Annual Reviews in control, pp 185-196 Kolmanovskii, V. and A. Myshkis. (1999) . Introduction to the theon) and apl-ica.tions of the functional differ'ential equations, Academic Press. Kolmanovskii, V. (1999) . St.ability of some linear syst.ems whit. delays IEEE tmsactions on Automatic Contol, Academic Press. Niculescu, S (200 1). Delay Effec ts on Stability. S pringer-Verlag.