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2 Exponential Stability for the C0-Norm: Analysis in the Frequency Domain

# 2 Exponential Stability for the C0-Norm: Analysis in the Frequency Domain

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90

3 Systems of Linear Conservation Laws

As we have already emphasized in Chapter 2, the system (3.3) can be regarded

as a set of scalar delay systems

Ri .t; L/ D Ri .t

Rj .t; 0/ D Rj .t

i ; 0/

j ; L/

i D 1; : : : ; m;

,

k

j D m C 1; : : : ; n;

L

; k D 1; : : : ; n;

k

which are interconnected by the boundary conditions (3.4). Taking the Laplace

transform, it follows that the characteristic function of the system (3.3), (3.4) is:

det In

s

diagfe

1

;:::;e

s

n

gK ;

(3.13)

where In is the identity matrix of Mn;n .R/. The roots of this function are called the

poles of the system.

Definition 3.4. The poles of the system (3.3), (3.4) are stable if there exists ı > 0

such that the poles are located in the half plane . 1; ı R.

A fundamental property is given in the following theorem.

Theorem 3.5. The system (3.3), (3.4) is exponentially stable for the C0 -norm if and

only if the poles of the system are stable.

t

u

Proof. See (Hale and Verduyn-Lunel 1993, Chapter 9, Theorem 3.5).

Remark 3.6. Theorem 3.5 deals with the C0 -norm. However, it must be pointed out

that the proof, as it is given by Hale and Verduyn-Lunel, also works for the Lp -norm

for every p 2 Œ1; C1.

Hence the stability analysis does not require to know the actual location of the

poles. It is sufficient to know that they have a negative real part which is bounded

away from zero. From the viewpoint of boundary control design, it is obviously of

major interest to predict the stability, and therefore the sign of the real parts of the

poles, directly from the coefficients of the matrix K. Two stability conditions are

presented below. The first one is the same as in the previous section.

Theorem 3.7. The poles of the system (3.3), (3.4) are stable if

Proof. If

2 .K/

2 .K/

< 1.

< 1 there exists Á 2 .0; 1/ and  2 Dn such that

kK 1 k 6 Á:

(3.14)

Let us assume that s is a pole of the system. Then

det In

D det .In

D det In

D0;

diagfe

s

1

diagfe

diagfe

s

1

;:::;e

s

1

s

n

;:::;e

;:::;e

s

n

g K

s

n

1

/Kg 

g K/

1

3.2 Exponential Stability for the C0 -Norm: Analysis in the Frequency Domain

91

which implies that

kdiagfe

s

1

;:::;e

s

n

g K 1 k > 1:

(3.15)

Since

kdiagfe

s

1

;:::;e

s

n

g K 1 k 6 kdiagfe

s

1

;:::;e

s

n

g kkK 1 k

6 exp. minf 1 <.s/; : : : ;

n <.s/g/kK

1

k

where <.s/ denotes the real part of the pole s, we have, using also (3.14) and (3.15),

exp. minf 1 <.s/; : : : ;

n <.s/g/Á

> 1:

(3.16)

Inequality (3.16) implies that

<.s/ 6 ı ,

ln.Á/

< 0:

maxf 1 ; : : : ; n g

t

u

Another stability condition is stated in the following theorem by Silkowski

(1976) which relies on the Kronecker density theorem (e.g., Bridges and Schuster

(2006)).

Theorem 3.8. Let

˚

N.K/ , maxf .diag e

iÂ1

;:::;e

iÂn

«

K/I .Â1 ; : : : ; Ân /T 2 Rn g

(3.17)

where .M/ denotes the spectral radius of the matrix M. If the time delays

. 1 ; : : : ; n / are rationally independent, the poles of the system (3.3), (3.4) are stable

if and only if N.K/ < 1.

Proof. See (Hale and Verduyn-Lunel 1993, Chapter 9, Theorem 6.1).

t

u

The statement of this theorem includes the rather unexpected feature that the time

delays have to be ‘rationally independent’ which is a generic property. In fact, when

the i ’s are rationally dependent the condition N.K/ < 1 is no longer necessary

and can be violated while keeping the exponential stability as we shall illustrate

with a simple example below. In Michiels et al. (2001), it is explained how “when

approaching rational dependence of the delays, the supremum of the real parts of

the poles can have a discontinuity (. . . ) compatible with the continuous movement

of individual roots” in the complex plane.

92

3 Systems of Linear Conservation Laws

3.2.1 A Simple Illustrative Example

Let us now present an example that illustrates the conditions of Theorems 3.7

and 3.8. We consider the most simple case of a system of two linear conservation

laws with a full matrix K. More precisely, we have the system

@t

R1

R2

!

C

!

0

1

0

R1

@x

!

D 0;

R2

2

2

<0<

1;

(3.18)

with the boundary condition

R1 .t; 0/

R2 .t; L/

!

D

K

‚ …„ !

ƒ

!

R1 .t; L/

k0 k1

k2 k3

R2 .t; 0/

:

(3.19)

Taking the Laplace transform of system (3.18), (3.19), the characteristic equation is

.es

1

k0 /.es

2

k3 /

k1 k2 D 0:

(3.20)

Let us consider the very special case 1 D 1, 2 D 2 which allows a simple and

explicit computation of the poles. In this case, the characteristic equation is

e3s

k0 e2s

k3 es C k0 k3

k1 k2 D 0:

(3.21)

Defining z , es , we get the third-order polynomial equation

z3

k0 z2

k3 z C k0 k3

k1 k2 D 0:

(3.22)

Let z` (` D 1; 2; 3) denote the three roots of this polynomial. Then, for each z` Ô 0,

there is an infinity of system poles sn D n C j!n lying on a vertical line in the

complex plane:

n

D ln jzi j; !n D 2 n C arg.zi /; n D 0; ˙1; ˙2; : : : :

(3.23)

The poles are stable if and only if jz` j < 1, ` D 1; 2; 3.

For simplicity, let us now address the special case where k0 k3 = k1 k2 . In that case,

it can be shown after a few calculations that the stability condition of Theorems 3.7

and 3.8 is

N.K/ D

2 .K/

D jk0 j C jk3 j < 1:

(3.24)

The region of stability corresponding to this condition is thus the square represented

in Fig. 3.2. From Theorem 3.8 we know that the condition is necessary and sufficient

3.2 Exponential Stability for the C0 -Norm: Analysis in the Frequency Domain

93

when 1 = 2 is an irrational number. But, when 1 = 2 is rational, the stability region

may be larger as we shall now illustrate by computing the poles of the system.

Using the condition k0 k3 D k1 k2 the polynomial equation (3.22) becomes

z.z2

k3 / D 0

k0 z

(3.25)

and we can compute the roots explicitly

z1 D 0;

z2;3 D

k0 ˙

q

k02 C 4k3

2

:

(3.26)

Remark that es D z1 D 0 has no solution. The system poles corresponding to z2 and

z3 are stable if and only if the parameters k0 and k3 satisfy one of the following two

conditions:

q

k02 C 4k3 > 0 and jk0 ˙ k02 C 4k3 j < 2

(3.27)

or

k02 C 4k3 < 0 and

q

k02 C jk02 C 4k3 j < 2:

(3.28)

The region of stability in the .k0 ; k3 / plane is the triangular region shown in Fig. 3.2.

It can be easily checked that the stability conditions (3.27), (3.28) can be in fact

formulated in the following simpler way:

k3 >

1; k0 C k3 < 1; k0

k3 >

1:

(3.29)

A numerical illustration is given in Fig. 3.3(a) for parameter values k0 D k1 D

k2 D k3 D 0:6 which satisfy inequalities (3.29) but not inequality (3.24). As

expected from the above analysis, the poles are located on a vertical line in the left

half complex plane.

k3

Fig. 3.2 Regions of stability

1

-2

-1

1

-1

2

k0

94

3 Systems of Linear Conservation Laws

a

b

40

40

20

20

-0.5

-0.5

0.5

0.5

Fig. 3.3 Pole configuration for the characteristic equation (3.20) with k0 D k1 D k2 D k3 D

0:6: (a) 1 D 1; 2 D 2, (b) 1 D 1; 2 D 2:1

3.2.2 Robust Stability

From this simple example, it appears that the strict negativity of the pole real parts is

not a robust stability condition as far as robustness with respect to small variations

of the characteristic velocities is concerned. A numerical illustration of this fact is

given in Fig. 3.3(b) where the 2 value of the previous example is slightly perturbed:

2 is set to 2.1 instead of 2. The consequence of this small variation of 2 is that half

of the poles are progressively shifted to the right up to instability.

In contrast, we may observe that the stability condition N.K/ < 1 is robust

with respect to small changes on K since N.K/ depends continuously on K.

We then introduce the following definition for the robustness with respect to the

characteristic velocities.

Definition 3.9. The system (3.3), (3.4) is robustly exponentially stable with respect

to the characteristic velocities if there exists " > 0 such that the perturbed system

@t

RC

R

!

C

QC

ƒ

0

0

Q

ƒ

!

@x

Q such that j Q i

is exponentially stable for every ƒ

RC

!

D 0:

R

ij

6 " 8i 2 1; : : : ; n.

3.2 Exponential Stability for the C0 -Norm: Analysis in the Frequency Domain

95

It is then evident that the following robust stability condition follows as a simple

corollary of Theorem 3.8.

Corollary 3.10. The system (3.3), (3.4) is robustly exponentially stable with

respect to the characteristic velocities if and only if N.K/ < 1.

t

u

3.2.3 Comparison of the Two Stability Conditions

Another interesting observation is that we have N.K/ D 2 .K/ in the above simple

example. This observation suggests to further investigate the comparison between

N and 2 and to determine to which extent they can be equal. This is done in the

following theorems.

Theorem 3.11. For every integer n and for every real n

2 .K/.

n matrix K, N.K/ 6

Proof. For every .Â1 ; : : : ; Ân /T 2 Rn and for every D 2 Dn

«

˚

«

˚

.diag eÃÂ1 ; : : : ; eÃÂn K/ D .D diag eÃÂ1 ; : : : ; eÃÂn KD 1 /

«

˚

D .diag eÃÂ1 ; : : : ; eÃÂn DKD 1 /

«

˚

6 kdiag eÃÂ1 ; : : : ; eÃÂn DKD 1 k

˚

«

6 kdiag eÃÂ1 ; : : : ; eÃÂn kkDKD 1 k D kDKD 1 k:

t

u

Theorem 3.12.

(a) For every n 2 f1; 2; 3; 4; 5g and for every real n n matrix K, N.K/ D 2 .K/.

(b) For every integer n > 5, there exist a real n n matrix K such that N.K/ <

2 .K/.

The proof of this theorem can be found in Appendix C. The following corollary

follows trivially.

e =

Corollary 3.13. If there exist a permutation matrix P such that the matrix K

PKP 1 is a block diagonal matrix

«

˚

e2 ; : : : ; K

ep

e D diag K

e1 ; K

K

ei is a real ni

where each block K

N.K/ D 2 .K/.

ni matrix with ni 2 f1; 2; 3; 4; 5g, then

t

u

96

3 Systems of Linear Conservation Laws

3.3 The Rate of Convergence

In the previous two sections, we have given explicit conditions on the matrix

K that guarantee the exponential convergence to zero of the solutions of the

system (3.3), (3.4). From a control design viewpoint it is also of major interest to be

able to quantify the rate of convergence. For that purpose, we define the following

change of state variables:

S.t; x/ , e P. x/R.t; x/; 0 <

2 R; P. x/ ,

t

PC . x/

0

0

P . x/

!

;

(3.30)

C

where P and P are defined in (3.8) and (3.9) respectively. The dynamics of these

new coordinates can be shown to be governed by the same hyperbolic system as

is R:

St C ƒSx D 0;

SC .t; 0/

!

e /

D K.

S .t; L/

e /,

K.

D20 K00 PC .

SC .t; L/

S .t; 0/

L/D0 4

P . L/K10 PC .

!

;

D20 K01 D1 2

L/D0 4 P . L/K11 D1 2

!

:

It follows that, for a given K such that N.K/ < 1, the robust convergence is

guaranteed with any such that

e // < 1:

N.K.

Let us define

c

It follows that, for any

of (3.3), (3.4),

e // < 1g:

, supf W N.K.

2 .0;

c /,

there exists C > 0 such that, for every solution

kR.t; :/kL2 ..0;L/IRn / Ä Ce

t

kRo kL2 ..0;L/IRn / ; 8t 2 Œ0; C1/:

3.4 Differential Linear Boundary Conditions

97

3.3.1 Application to a System of Two Conservation Laws

Let us consider again the system of two conservation laws

!

!

!

R1

R1

1 0

@t

C

@x

D 0;

1 > 0;

0

R2

R2

2

2

> 0;

(3.31)

under the boundary condition

R1 .t; 0/

!

D

R2 .t; L/

k0 k1

!

k2 k3

R1 .t; L/

R2 .t; 0/

!

:

(3.32)

In this case, the change of coordinates (3.30) is

S1 D p1 e t e

x=

e / is

and the matrix K.

0

B

e /,B

K.

B

@

k0 e

k2 e

1

2

. 1 C 2 / D1

D20

1

R1 ;

S2 D p2 e t e

1

D20

k1 2 C

D1 C

C;

A

2

k3 e

with

i

x=

,

2

R2 ;

L

; i D 1; 2:

i

In the special case of local boundary conditions where k0 D k3 D 0 and jk1 k2 j < 1,

the value of c is explicitly given by

Ã

Â

1

1

e

ln

c D supf W N .K. // D 1g D

jk1 k2 j

which is, as expected, identical to the convergence rate that we have found in

Chapter 2.

3.4 Differential Linear Boundary Conditions

Up to now, in this chapter, we have discussed the stability of linear hyperbolic

systems under static linear boundary conditions. In this section, we examine how

the previous results can be generalized to the case of boundary conditions that are

dynamic and represented by linear differential equations. More precisely, we consider the linear hyperbolic system of conservation laws in Riemann coordinates (3.3)

under linear differential boundary conditions of the following form:

˘

X D AX C BRout .t/;

Rin .t/ D CX C KRout .t/;

(3.33)

98

3 Systems of Linear Conservation Laws

where A 2 M`;` .R/, B 2 M`;n .R/, C 2 Mn;` .R/, K 2 Mn;n .R/, X 2 R` , ` 6 n.

The notations Rin and Rout were introduced in Section 1.1 and stand for

Rin .t/ ,

RC .t; 0/

!

R .t; L/

;

RC .t; L/

Rout .t/ ,

!

R .t; 0/

:

The well-posedness of the Cauchy problem associated with this system is addressed

in Appendix A, see Theorem A.6.

3.4.1 Frequency Domain

Using the Laplace transform, the system (3.3), (3.33) is written in the frequency

domain as

Rout .s/ D D.s/Rin .s/ with D.s/ , diagfe

.sI

A/X.s/ D BRout .s/;

s

1

;:::;e

s

n

g;

i

D L= i ;

Rin .s/ D CX.s/ C KRout .s/:

Hence the poles of the system are the roots of the characteristic equation

i

h

det I D.s/ C.sI A/ 1 B C K D 0:

Theorem 3.14. The steady state R.t; x/ Á 0 of the system (3.3), (3.4) is exponentially stable for the L1 -norm if and only if the poles of the system are stable (i.e.,

have strictly negative real parts and are bounded away from zero).

Proof. See (Hale and Verduyn-Lunel 2002, Section 3) and (Michiels and Niculescu

2007, Section 1.2).

t

u

3.4.2 Lyapunov Approach

In the line of the previous results of this chapter, we may also introduce the following

Lyapunov function candidate:

Z

VD

0

L

2

m

X

pi

4

iD1

i

R2i .t; x/ exp.

x

i

/C

n

X

pi

iDmC1

i

3

`

X

x

R2i .t; x/ exp.C /5 dx C

qj Xj2

i

jD1

(3.34)

with X , .X1 ; : : : ; X` /T , pi > 0 (i D 1; : : : ; n), qj > 0 (j D 1; : : : ; `).

The time derivative of this function along the C1 -solutions of (3.3), (3.33) is

˘

VD

Ã

Rout

;

/

X

Â

VC

.RTout ; XT /M.

3.4 Differential Linear Boundary Conditions

99

with the matrix

M. / ,

KT P1 . /K

P2 . /

CT P1 . /K C QB

!

KT P1 . /C C BT Q

Q C CT P1 . /C C .AT Q C QA/

;

and

P1 . / , diag p1 ; : : : ; pm ; pmC1 exp.

P2 . / , diag p1 exp.

L

1

L

/; : : : ; pn exp.

mC1

/

n

L

/; : : : ; pm exp.

L

/; pmC1 ; : : : ; pn

m

Q , diagfq1 ; : : : ; q` g :

Exponential stability holds if there exist pi > 0 and qj > 0 such that the matrix

M.0/ is negative definite (see Castillo et al. (2012) for a related reference). A simple

example of this Lyapunov approach can be found in Theorem 2.9 for the stability

analysis of a density-flow system under Proportional-Integral control.

3.4.3 Example: A Lossless Electrical Line Connecting

an Inductive Power Supply to a Capacitive Load

Let us come back to the example of a lossless electrical line that we have presented

in Section 2.1. We now consider the case where the line connects an inductive power

supply to a capacitive load as shown in Fig. 3.4.

The dynamics of the line are described by the following system of two conservation laws:

@t I C

Power

supply

1

@x V D 0;

L`

@t V C

1

@x I D 0;

C`

(3.35)

R0

Transmission line

U (t)

x

L0

0

RL

L

Fig. 3.4 Transmission line connecting an inductive power supply to a capacitive load

CL

100

3 Systems of Linear Conservation Laws

with the differential boundary conditions:

L0

dI.t; 0/

C R0 I.t; 0/ C V.t; 0/ D U.t/;

dt

CL

dV.t; L/

V.t; L/

D I.t; L/:

C

dt

RL

(3.36)

For a given constant input voltage U.t/ D U , the system has a unique constant

U

;

R0 C R`

I D

R` U

:

R0 C R`

V D

The Riemann coordinates are defined as

R1 , V

V

C I

R2 , V

V

I

p

L` =C` ;

p

I

L` =C` ;

I

with the inverse coordinates

IDI C

R2 p

R1

L` =C` ;

2

R1 C R2

VDV C

:

2

Then, expressing the dynamics (3.35) and the boundary conditions (3.36) in

Riemann coordinates, we have:

@t R1 C

1 @x R1

˘

D 0;

!

X1

˘

X2

D

@t R2

˛1 0

0

2 @x R2

!

˛2

X1

D 0;

!

0

C

1

ˇ1

!

D

2

,p

R1 .t; L/

!

R2 .t; 0/

ˇ2 0

„ ƒ‚ …

„ ƒ‚ …

A

B

!

!

!

!

!

0 1

10

R1 .t; 0/

R1 .t; L/

X1

C

D

;

R2 .t; L/

R2 .t; 0/

10

X2

01

„ ƒ‚ …

„ƒ‚…

C

K

X2

with

1

˛1 D

L0

2

ˇ1 D

L0

s

C`

R0

C ;

L`

L0

s

C`

;

L`

1

˛2 D

CL

2

ˇ2 D

CL

s

s

L`

:

C`

L`

1

C

;

C`

RL CL

;

1

;

L` C`

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