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
kK 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 kkK 1 k
6 exp. minf 1 <.s/; : : : ;
n <.s/g/kK
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;
with adequately adapted boundary conditions:
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
Load
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
steady state
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`