LYAPUNOV MATRICES FOR A CLASS OF NEUTRAL TYPE TIME DELAY SYSTEMS

LYAPUNOV MATRICES FOR A CLASS OF NEUTRAL TYPE TIME DELAY SYSTEMS Vladimir L. Kharitonov

Automatic Control Department CINVESTAV-IPN A. P. 14-740, Mexico, D.F.

Abstract: A class of neutral type time delay systems is presented for which computation of Lyapunov matrices is reduced to solution of an auxiliary two point boundary problem for a delay free system of matrix equations. Copyright c 2006 IFAC. Keywords: Neutral type time delay systems, Lyapunov matrices

1. INTRODUCTION The complete type Lyapunov quadratic functionals show to be useful in solution of such problems as estimation of robustness bounds and calculation of exponential estimates for time delay systems, see (Kharitonov, 2005). One of the principal di¢ culties in applications of the functionals for an analysis of time delay systems is a lack of constructive algorithms for computation of the corresponding Lyapunov matrices. It has been shown in (Louisell, 2001) that for the case of single delay systems the matrices satisfy an auxiliary system of delay free matrix equations with a special set of two point boundary conditions. This opens the door for application of standard procedures for computation of the Lyapunov matrices. To the best of our knowledge this is the only reported case when Lyapunov matrices can be computed in a semi-analytic manner. In this contribution a new class of neutral type time delay systems with distributed delay is presented for which Lyapunov matrices can be computed as solutions of a special two point boundary problem for a delay free system of linear matrix equations, see Theorems 1, 4. In the following Section some basic notations are given and Lyapunov matrices are introduced.

Main results are presented in Section 3. It is shown here that under some special assumptions on the delay distributed term the computation of Lyapunov matrices is reduced to a solution of special delay free system of linear di¤erential matrix equations. A corresponding set of boundary conditions is provided as well. An illustrative example is given in Section 4. Some concluding remarks end the contribution.

2. PRELIMINARIES We deal with the following time delay system d [x(t) + Cx(t h)] = A0 x(t) + A1 x(t dt Z 0 + G( )x(t + )d ; t 0:

h) (1)

h

Here A0 , A1 and C are given real constant n n matrices, delay h > 0, and G( ) is a continuous matrix valued function de…ned for 2 [ h; 0].

2.1 Fundamental matrix Let n n matrix K(t) satis…es the matrix equation

d [K(t) + K(t h)C] = K(t)A0 + K(t dt Z 0 K(t + )G( )d ; t 0; +

h)A1

h

with the following initial condition K( ) = 0, 2 [ h; 0), K(0) = I, and the sewing condition K(t) + K(t

the symmetry condition (3) compensates the de…ciency. In the following we show that under some additional assumptions on the matrix G( ) one may compute the Lyapunov matrix as a solution of a special two point boundary problem for an auxiliary system of delay free matrix equations.

h)C is continuous for t > 0:

Matrix K(t) is known as the fundamental matrix of the system (1), see (Bellman and Cooke, 1963).

3.1 A particular case We begin with the case when

2.2 Lyapunov matrices G( ) = A key role in construction of Lyapunov quadratic functionals with a prescribed time derivative for system (1) is played by Lyapunov matrices.

m X

j

Bj ;

(7)

j=0

where B0 ; :::; Bm are constant n

n matrices.

Let us de…ne matrices De…nition 1. Given a symmetric matrix W . Matrix valued function U ( ), 2 [ h; h], is called Lyapunov matrix for system (1) associated with W if it satis…es the following three conditions

Z( ) = U ( ); V ( ) = U ( and 2(m + 1) auxiliary matrices

dynamic property 0

+U (

Xj ( ) =

0

U ( )+U ( h)C = U ( )A0 Z 0 h)A1 + U ( + )G( )d ;

Yj ( ) =

0;

h

) = U T ( );

0;

Z

Z

0 j

U ( + )d ;

j = 0; 1; :::; m;

j

V(

j = 0; 1; :::; m:

h 0

)d ;

h

(2)

Then, the dynamic property (2) can be presented in the form

(3)

Z 0 ( )+V 0 ( )C = Z( )A0 +V ( )A1 +

symmetry property U(

h);

algebraic property U 0 (0) T

C [U (+0)

Xj ( )Bj :

j=0

C T U 0 (0)C = [U 0 (+0) 0

m X

0

U ( 0)] C =

U 0 ( 0)] W:

At the same time,

(4)

If system (1) is exponentially stable the Lyapunov matrix can be presented as Z 1 U( ) = K T (t)W K(t + )dt: (5) 0

C T Z 0 ( ) + V 0 ( ) = AT0 U T (h m X AT1 V T (h ) BjT XjT (h

+

AT0 U (

Observe that

h)C

h)A1 +

AT1 U (0)C

) + U(

h)C]

+ U ( ) + C U (h + ) G( ) d :

(6)

):

j=0

In the explicit form the algebraic property (4) looks as W = AT0 U (0) + U (0)A0 +C T U (h)A0 + AT1 U (h) + U ( Z 0 T +C U (0)A1 + GT ( ) [U ( h T

)

U T (h

) = V ( ); V T (h

XjT (h

) = Yj ( );

) = Z( ); and

j = 0; 1; :::; m;

so, C T Z 0 ( ) + V 0 ( ) = AT0 V ( ) m X BjT Yj ( ):

AT1 Z( )

j=0

3. MAIN RESULTS In this section we present an algorithm for computation of the Lyapunov matrices for system (1). The lack of the initial conditions for the Lyapunov matrix U ( ) as a solution of the delay matrix equation (2) constitutes a serious di¢ culty for computation of the matrix. To some extend

The …rst derivative of the auxiliary matrices X0 ( ) and Y0 ( ) X00 ( ) = U ( ) Y00 (

U(

h) = Z( )

) = V ( ) + V ( + h) =

Now, for j = 1; :::; m:

V ( );

V ( ) + Z( ):

Xj0 ( ) =

j

=0

U( + )

= h

j

= ( h) V ( ) j

Yj0 ( ) =

jXj

Z j

0 j 1 h

1(

Z =0 ) = h+ j

V(

= ( h)j Z( ) + jYj

1(

U ( + )d

); 0 j 1

V(

)d

h

):

And we arrive at the following delay free system of 2 (m + 2) matrix di¤erential equations

Lyapunov matrix of the system associated with a symmetric matrix W . Then, there exists a solution Z( ), V ( ), Xj ( ), Yj ( ), j = 0; 1; :::; m, of the delay free system of matrix equations (8) such that U ( ) = Z( ), 2 [0; h]. The solution satis…es the following set of two point boundary conditions 8 Z(0) = V (h); > > > > Xj (0) = YjT (h); j = 0; 1; :::; m; > > > > Yj (0) = XjT (h); j = 0; 1; :::; m; > > > > < Z(0)A0 + AT Z(0) + C T Z(0)A1 + AT Z(0)C 0

8 0 Z ( ) + V 0 ( )C = Z( )A0 + V ( )A1 > > m > X > > > + Xj ( )Bj > > > > j=0 > > > > C T Z 0 ( ) + V 0 ( ) = AT1 Z( ) AT0 V ( ) > > > m > X > > < BjT Yj ( ) > > X00 ( > > > > Y00 ( > > > 0 > > > Xj ( > > > > > 0 > > > Yj ( :

j=0

(8)

) = Z( ) V ( ) ) = Z( ) V ( ) ) = ( h)j V ( ) jXj 1 ( ); j = 1; 2; :::; m; ) = ( h)j Z( ) + jYj 1 ( ); j = 1; 2; :::; m:

Remark 1. It is worth of mentioning that the spectrum of system (8) is symmetrical with respect to the imaginary axis. A solution of the system (8) which de…nes the Lyapunov matrix U ( ) satis…es the conditions:

There are some relations between the auxiliary matrices that are given in the following lemma. Lemma 2. The auxiliary matrices Xk ( ) and Yk ( ), k = 0; 1; :::; m, are not independent, they satisfy the relations: Pk 1. Xk ( ) = ( 1)k j=0 j!(kk! j)! hj Yk j ( ); Pk 2. Yk ( ) = ( 1)k j=0 j!(kk! j)! hj Xk j ( ). Proof. The …rst set of relations can be easily obtained as follows. By de…nition Xk ( ) =

Z(0) = V (h); =

and, for j = 0; 1; :::; m, Xj (0) = = Yj (0) = =

Z

Z

Z

=

0 j

U (h +

h)d

h T

0 j

V (h

h

Z

j

U(

Z

Z

Z

0

= XjT (h);

T

+ C Z(0)A1 +

AT1 Z(0)C

j=0

T

o

=

W:

k

)d

k X

k! 4( 1)k hj j!(k j)! h j=0 k X

hi

k! hj Yk j!(k j)!

3

k j5

V(

j(

)d

):

The second set of relations can be veri…ed in a similar way. Lemma 2 provides a substantial reduction of the system (8). Observe, that the sum Lemma 2 provides a substantial reduction of the system (8). Observe, that the sum m m X X BjT Yj ( ) = BjT ( 1)j hj X0 ( ) j=0

+jhj

We may formulate now the following statement. ::: + Theorem 1. Given a time delay system (1) where matrix G( ) is of the form (7). Let U ( ) be the

h)d

) V(

j=0

h

Xj (0) + C T Xj (h)

V ( + + h)d = h =

0

= ( 1)

U (h + )d

AT0 Z(0)

k

( h h 2

T j

U( + + h

h 0

h 0

k h 0

k

h)d

+V (0)A1 + AT1 V T (0) + AT0 V (0)C m n X +C T V T (0)A0 + Xj (0) + C T Xj (h) Bj +BjT

Z

0

as well as, the algebraic property Z(0)A0 +

=

= YjT (h);

)d

1

+V (0)A1 + AT1 V T (0) + AT0 V (0)C > m n > X > > T T > +C V (0)A + Xj (0) + C T Xj (h) Bj > 0 > > > j=0 > o > > : +B T X (0) + C T X (h) T = W: j j j (9)

j(j

j=0

1

1) 2!

X1 ( ) + h2 X j

2(

j(j

1)

2!

hj

) + jhXj

so, if we de…ne the matrix

2

X2 ( )+

1(

) + Xj ( ) ;

B(h) =

m X

f (0) (s) =

( h)j BjT ;

j=0

f (k) (s) =

then m X j=0

BjT Yj ( ) =

m X

k=0

1 (k) B (h) Xk ( ); k!

and the second equation of the system (8) takes the form C T Z 0 ( ) + V 0 ( ) = AT1 Z( ) AT0 V ( ) m X 1 (k) B (h) Xk ( ): k! k=0

Therefore, the system (8) is reduced to the following system of (m + 3) matrix equations 8 0 Z ( ) + V 0 ( )C = Z( )A0 + V ( )A1 > > > m > X > > > + Xj ( )Bj > > > > j=0 > > > < V 0 ( ) + C T Z 0 ( ) = AT1 Z( ) AT0 V ( ) m X 1 (k) > B (h) Xk ( ) > > k! > > k=0 > > > X00 ( ) = Z( ) V ( ) > > > > X 0 ( ) = ( h)j V ( ) jXj 1 ( ); > > : j j = 1; 2; :::; m: (10) In a similar way one may reduce the set of boundary conditions (9) as follows 8 Z(0) = V (h); > > > k > X > k! > k > X (0) = ( 1) hj XkT j (h); k > > j!(k j)! > > j=0 > > > k = 0; 1; :::; m; > < Z(0)A0 + AT0 Z(0) + C T Z(0)A1 + AT1 Z(0)C > > +V (0)A1 + AT1 V T (0) + AT0 V (0)C > > m n > X > > > +C T V T (0)A0 + Xj (0) + C T Xj (h) Bj > > > > j=0 > o > > : +B T X (0) + C T X (h) T = W: j j j (11) There is a certain connection between the spectrum of the time delay system (1) and that of the delay free system (8). Theorem 3. Given a time delay system (1) where matrix G( ) is of the form (7). Let s0 be an eigenvalue of the system such that s0 be also an eigenvalue of the system. Then s0 belongs to the spectrum of the delay free system (8).

where

; and

dk f (0) (s) ; dsk

k = 1; :::; m:

On the other hand, a complex number s belongs to the spectrum of the delay free system (8) if and only if there exist a non trivial set of n (0) (0) n constant matrices Z (0) ; V (0) ; Xj ; Yj ; j = 0; 1; :::; m, such that 8 (0) > sZ + sV (0) C = Z (0) A0 + V (0) A1 > > m > X > > (0) > + Xj B j > > > > j=0 > > > T (0) (0) > sC Z + sV = AT1 Z (0) AT0 V (0) > > > m < X (0) BjT Yj > > j=0 > > (0) > (0) (0) > sX = Z V > 0 > > (0) > > sY = Z (0) V (0) > > 0(0) > (0) j (0) > jXj 1 ; j = 1; 2; :::; m > > sXj = ( h) V > (0) : (0) sYj = ( h)j Z (0) + jYj 1 ; j = 1; 2; :::; m: (13) Multiplying the …rst equality in (12) by from the left, and the second equality by e hs0 T from the right we obtain T

s0

hs0

+ s0 e

m X

T

T

C

f (k) (s0 )

T

A0

e

hs0

T

A1

Bk = 0;

k=0

and s0 e

hs0

+AT1

T

T

+ s0 C T T + AT0 e hs0 T m X + e hs0 f (k) ( s0 )BkT T = 0: k=0

If we de…ne matrices Z (0) = (0) Xj (0) Yj

T

=f =e

(j)

; V (0) = e T

(s0 )

hs0 (j)

f

;

hs0

T

;

j = 0; 1; :::; m;

( s0 )

T

;

j = 0; 1; :::; m;

then the previous two equalities take the form s0 Z (0) + s0 V (0) C Z (0) A0 m X (0) V (0) A1 Xk Bk = 0; k=0

+ s0 V (0) + AT0 V (0) m X (0) +AT1 Z (0) + BkT Yk = 0:

s0 C Z

k=0

hs

e s

As s0 and s0 are eigenvalues of the system then there exist two non trivial vectors and such that T G(s0 ) = 0; GT ( s0 ) = 0: (12)

T

Proof. The characteristic matrix of system (1) is m X G(s) = sI +se hs C A0 e hs A1 f (k) (s)Bk ;

1

(0)

k=0

That is, for s = s0 the matrices satisfy the …rst two equations of system (13). It is just a matter of simple calculations to show that for s = s0

they satisfy also the rest of 2(m + 1) matrix equations in (13). It is evident that the de…ned set (0) (0) of matrices Z (0) ; V (0) ; Xj ; Yj ; j = 0; 1; :::; m, is not trivial. Therefore, the complex value s0 belongs to the spectrum of the delay free system of matrix equations (8). The same is true also for s0 . Remark 2. The statement remains true if we replace in Theorem 3 the delay free system (8) by the reduced system (10). 3.2 A general case Now we consider the case when matrix G( ) is of the form m X G( ) = (14) j ( )Bj ; j=0

where B0 ; :::; Bm are given n n matrices, and the scalar functions 0 ( ); :::; m ( ) are such that m X 0 j = 0; 1; :::; m: (15) jk k ( ); j( ) = k=0

Remark 3. In the previous subsection we have j ; j = 0; 1; :::; m. These functions satisfy j( ) = the equations 0 0(

0 j(

) = 0;

)=j

j 1(

);

j = 1; :::; m:

The time delay matrix equation for U ( ) is now of the form U 0( ) + U 0( h)C = U ( )A0 + U ( h)A1 Z m 0 X + 0: (16) j ( )U ( + )Bj d ;

=

j (0)Z( )

j ( h)V ( )

= j (0)Z( ) m X jk Xk ( );

Z

j(

0 0 j(

)U ( + )d

h

h)V ( )

j = 0; 1; :::; m;

k=0

while

=

Z

0

)V 0 ( h Z (0)V ( )+ ( h)Z( )+ j j Yj0 ( ) =

j(

)d 0 0 j(

)V (

)d

h

= j ( h)Z( ) j (0)V ( ) m X + j = 0; 1; :::; m: jk Yk ( ); k=0

And we arrive at the following system of delay free matrix equations 8 0 Z ( ) + V 0 ( )C = Z( )A0 + V ( )A1 > > > m > X > > > + Xj ( )Bj > > > > j=0 > T 0 > > C Z ( ) + V 0 ( ) = AT0 V ( ) AT1 Z( ) > > > m > X > > > BjT Yj ( ) > < j=0

> Xj0 ( ) = j (0)Z( ) j ( h)V ( ) > > m > X > > > j = 0; 1; :::; m; > jk Xk ( ); > > > k=0 > > 0 > > j (0)V ( ) > Yj ( ) = j ( h)Z( ) > m > X > > > j = 0; 1; :::; m: + > jk Yk ( ); : k=0

(17) One is looking for a solution of the system which satis…es the two point boundary conditions (9).

h

j=0

Let once again introduce matrices Z( ) = U ( ), V ( ) = U( h), and Xj ( ) = Yj ( ) =

Z Z

0 j(

)U ( + )d ;

j = 0; :::; m;

j(

)V (

j = 0; :::; m:

h 0

)d ;

h

Theorem 4. Given a time delay system (1) where matrix G( ) is of the form (14). Let U ( ) be the Lyapunov matrix of the delay system associated with a symmetric matrix W . Then, there exists a solution Z( ), V ( ), Xj ( ), Yj ( ), j = 0; 1; :::; m, of the delay free system of matrix equations (17) such that U ( ) = Z( ), 2 [0; h]. The solution satis…es the boundary conditions (9).

Then equation (16) looks as Z 0 ( ) + V 0 ( )C = Z( )A0 + V ( )A1 m X + Xj ( )Bj ; 2 [0; h]; j=0

and,

C T Z 0 ( ) + V 0 ( ) = AT1 Z( ) m X BjT Yj ( ):

AT0 V ( )

For the general case the statement of Theorem 3 remains true. Theorem 5. Given a time delay system (1) where matrix G( ) is of the form (14). Let s0 be an eigenvalue of the time delay system such that s0 be also an eigenvalue of the system. Then s0 belongs to the spectrum of the delay free system (17).

j=0

Now,

Xj0 (

)=

Z

0 j( h

0

)U ( + )d

Sometimes the delay free system (17) allows a reduction. It happens when functions j ( ), j = 0; 1; :::; m, satisfy the conditions

j(

h) =

m X

jk k (

);

k=0

2 [ h; 0]; where

jk

Yj ( ) =

Z

= = = =

Z Z

j = 0; 1; :::; m;

are constant coe¢ cients. In this case 0 j(

)V (

j(

)U (

)d

h 0

h)d = h =

h 0 j( h

m X

k=0 m X

jk

hi

h)U ( + )d Z

Figure 1: Matrix U ( ) components for = 0:3:

0 k(

)U ( + )d

5. CONCLUSIONS

h

jk Xk (

);

k=0

and one can exclude matrices Yj ( ) of the system (17), and of the boundary conditions (9) as well.

4. EXAMPLE

In this contribution a class of time delay systems is introduced for which the problem of computation of the Lyapunov matrices can be reduced to a solution of a two point boundary problem for an auxiliary delay free system of matrix equations. Presented results can be also applied to the system Z 0 d x(t) + Cx(t h) + P ( )x(t + )d dt h Z 0 e0 x(t) + A e1 x(t h) + e )x(t + )d ; =A G( h

Consider the time delay system x(t) + J x(t 1) = x(t) + Jx(t 1) Z 0 + [sin(2 )I + cos(2 )J] x(t + )d : 1

where J =

0 1 . 10

e0 A0 = A

e1 + P ( h); and A1 = A e ) + P 0 ( ): G( ) = G( P (0);

6. ACKNOWLEDGMENT

The reduced system of delay free matrix equations is now of the form 8 0 Z + V 0 J = Z + V J + X0 + X1 J; > > < 0 V JZ 0 = JZ + V + X0 + JX1 ; 0 X = 2 X1 ; > > : 00 X 1 = Z V + 2 X0 :

The corresponding set of boundary conditions is the following 8 Z(0) = V (1); X0 (0) = X0T (1); > > > > < X1 (0) = X1T (1); 2Z(0) 2 2 JZ(0)J + ( I + 2 J)X0 (0) > > > +X0 (1)( I + 2 J) JX1 (0)(I + J) > : + (I J)X1 (1)J = W:

Let us select matrix

W =

because it can be written in the form (1) with

20 02

;

the four components of the corresponding Lyapunov matrix are depicted on the Figure 1.

The work was partly supported by the CONACyT, Mexico.

REFERENCES Bellman, R. and K.L. Cooke (1963). Di¤ erential Di¤ erence Equations. Academic Press. New York. Kharitonov, V.L. (2005). Lyapunov functionals and Lyapunov matrices for neutral type time delay systems: a single delay case. International Journal of Control 78, 783–800. Louisell, J. (2001). A matrix method for determining the imaginary axis eigenvalues of a delay system. IEEE Transactions on Automatic Control 46, 2008–2012.