ABSTRACT
Figure Nonlinearity N
graph
N
sign
N
sgn
j
j or
j
j
j
j
or
j
j
N
sgn
j
j
j
j sgn
j
j
j
j
it sin
t
In this example
X
A
X X
jx
j
jx
j
jx
j
The set X
A
can be expressed in the form for which we set
b
b
b
so that
X
A
X X
max
jx
i
j
i
Figure Set X
A
The function H is found as
HX i
N
N
sin
t
N
N
N
N
jx
j
N
jx
j
T
so that the matrix A is determined by
A
%
B
B
B
&
'
C
C
C
A
In this case
c Ab
%
B
B
B
&
'
C
C
C
A
%
B
B
B
&
'
C
C
C
A
%
B
B
B
&
'
C
C
C
A
Hence c
i
for any The condition c
i
i
b
i
i of
Theorem is satised for any The result is that the set S X
A
is
itself the maximal estimate of D
ps
X
A
i ie
D
ps
X
A
i X
A
The preceding examples show that although necessity of the conditions of Theo
rem through Theorem is not proved these theorems can in a certain case
ps
A
A
A
set
Estimate of the domain of practical stability
with settling time
s
In the analysis of the domain D
p
s
X
A
X
F
I of system practical stability with
settling time
s
we use both the set X
A
of permitted states at any time t
and
the set X
F
of allowed states since the settling time
s
has passed
Theorem Let the system have generalised motions XtX
i C
t
X
a
I In order for a compact set S S X
A
to be estimate E
p
s
X
A
X
F
I
of the system practical stability with the settling time
s
with respect to fX
F
Ig
S E
p
s
X
A
X
F
I it is sucient that
S is connected with nonempty interior
S and S X
A
there are a connected compact subset E
F
of X
F
with nonempty connected
interior at most nite subset Z of S E
F
and a function v
n
fg such that
a vtX C
X
A
Z
b vtX v
mE
t vt Y tX Y
E
F
X
A
E
F
c there is a function
which is integrable over
and obeys
i D
vtX i t for all tX i
X
A
Z I
ii
Z
t
d v
mA
v
MS
for all t
and
iii
Z
t
d v
mE
F
v
MS
for all t T
s
rrr
Proof Let all the conditions of the theorem statement hold Hence all the condi
tions of Theorem also hold so that XtX
i X
A
tX
i
S I
ie S E
ps
X
A
I
From the conditions under c after integrating D
vtX we derive
vtXtX
i v X
v
mE
F
v
MS
tX
i
s
S I
This and v X
v
MS
X
S imply
vtXtX
i v
mE
F
tX
i
s
S I
or
XtX
i E
F
tX
i
s
S I
due to the condition b This result E
F
X
F
and the condition verify all require
ments of b of Denition S E
pc
s
X
F
I This and S E
ps
X
A
I
s
with re
spect to fX
A
X
F
Ig so that D
p
s
X
A
X
F
I X
F
and D
p
s
X
A
X
F
I
X
F
Therefore we accept
E
F
X
F
S X
F
S
and
v
MAE
F
sup D
vtX i tX i
X
A
E
F
I
Theorem Let the system have generalised motions XtX
i C
t
S I and the sets E
F
S and X
F
be compact and connected with nonempty
interiors Let and hold In order for the set S to be the estimate
E
p
s
X
A
X
F
I of the domain of the system practical stability with settling time
s
with respect to fX
A
X
F
Ig S E
p
s
X
A
X
F
I it is sucient that
there exist a function v
n
and an at most nite subset Z of
S E
F
obeying
a vtX C
X
A
Z
and
b vtX v
mE
F
t vt Y tX Y
E
F
X
A
E
F
v
MS
v
mA
t t v
MAS
for all t
and
v
MS
v
mE
F
t t v
MAE
F
for all t
s
rrr
Proof Let all the conditions hold Hence all conditions of Theorem hold
which means that S E
ps
X
A
I and XtX
i S tX
i
S I
By replacing v
MAS
by v
MAE
in the proof of Theorem we show that
vXtX
i v X
t v
MAE
tX
i
S I
which together with the condition yields
vXtX
i v X
v
mE
t v
MS
v
mE
t tX
i
s
S I
The last inequality is due to v
MS
v X
X
S The preceding result
and the condition b imply
XtX
i E
F
tX
i
s
S I
and
XtX
i X
F
tX
i
s
S I
due to E
F
X
F
This and the properties of S to be compact connected with non
empty interior show that all conditions of b of Denition are satised so that
S
s
X
A
X
Q Q
ij
nn
a
Q
ij
sup h
ii
X i X i X
A
X
F
I
ij
max f sup h
ij
X i sign x
j
x i X
A
X
F
Ig b
X
F
fX b
T
jXj g b
and
S fX b
T
jXj g
Fig
Theorem Let the system have generalised motions XtX
i C
t
S I Let the matrix Q and sets X
F
and S be dened by and
respectively In order for the set S to be the estimate E
p
s
X
A
X
F
I of the
domain D
p
s
X
A
X
F
I S E
p
s
X
A
X
F
I it is sucient that
all the conditions of the Theorem hold and
t t
s
where
n
X
i
max
q
i
b
i
q
i
b
i
a
and
q q
q
q
n
T
Q
T
b b
rrr
Proof Let all the conditions hold Therefore S D
ps
X
A
I Theorem
and XtX
i X
A
tX
i
S I We accept E
F
X
F
and use S
to nd
v
MS
v
MS
v
mE
t v
mE
v
mF
a
and
v
MAF
b
due to and From and follows that the condition of
Theorem is satised All other conditions of the same theorem are also satised
because the sets S and X
F
are Ouniquely bounded with the same generating func
tion u v Altogether S E
pc
s
X
F
I that together with S E
ps
X
A
I
X
F
X X
T
PX
P P
T
p
ij
S
X X
T
PX
and
M m
ij
H
T
P PH a
m
ij
ij
sup w
ii
X i X i X
A
X
F
I
ij
max f sup w
ij
X i sign x
i
x
j
x i X
A
X
F
Ig
b
M m
ij
nn
c
Theorem Let the system have generalised motions XtX
i C
t
S I and Theorem hold Let the sets X
F
and S and the
matrix M obey the following
t t
s
a
with
M
P
if M
M
P
if M
b
Then the set S is estimate E
p
s
X
A
X
F
I of D
p
s
X
F
I
S E
p
s
X
A
X
F
I
rrr
Proof Under the conditions of the theorem statement we calculate for vX
X
T
PX
v
MS
v
mF
v
MAF
which together with verify the condition of Theorem Since the sets S
and X
F
are Ouniquely bounded then they satisfy all other conditions of the same
theorem too Therefore S E
p
s
X
A
X
F
I
If Fig
X
F
X max
jx
i
j
b
i
i n
b
S
X max
jx
i
j
i n
condi
tions under which the set S is the estimate of D
p
s
X
A
X
F
I
Let
R r
ij
nn
a
r
ij
ij
max Q
ij
Q
ij
ij
Q
ij
b
p p
p
p
n
T
R
T
b c
Theorem Let the system have generalised motions XtX
i C
t
S I Theorem hold the sets X
F
and S and the vector p
obey
t max
p
i
b
i
i n
t
s
Then the set S is estimate E
p
s
X
A
X
F
I of D
p
s
X
A
X
F
I
S E
p
s
X
A
X
F
I
rrr
Proof Let all the condition be valid Theorem guarantees S E
ps
X
A
I
and XtX
i X
A
tX
i
S I In this case
v
MS
v
MF
v
MAF
max
p
i
b
i
i n
These results and show that the condition of Theorem holds Since
the sets X
A
X
F
and S are Ouniquely bounded with the same generating func
tion u v vX max
jx
i
j
b
i
i n
and numbers and are
positive then the sets satisfy all the conditions of Theorem that implies S
E
p
s
X
A
X
F
I
The preceding results are based on properties of Ouniquely bounded sets and are
generalised as follows
Theorem Let the system have generalised motions XtX
i C
t
X
A
I and all the conditions of Theorem hold Let the sets X
A
S and X
F
be uniquely bounded with the same generating function u and be determined
by and respectively
X
F
fX uX g
In order for the set S to be estimate E
p
s
X
A
X
F
I of the domain D
p
s
X
A
X
F
I of the system practical stability with the settling time
s
with respect to
fX
A
X
F
Ig it is sucient that
t u
MAF
t
are
satised because
unique boundedness of X
A
X
F
and S and
and
and guarantee that these sets are compact connected with nonempty
interiors and obey the condition b of Theorem
their generating function u is well dened and continuous on X
A
the condition t u
MAS
t
of Theorem shows that the
condition of Theorem is fullled for u v
and
the condition implies validity of condition of Theorem
Example Let a second order system of the form be dened by
dX
dt
x
x
x
sin
x
sin
x
x
x
x
sin
x
sin
x
fX
Given
s
X
A
X kXk
e
X
F
X kXk
e
I
What is the largest number
such that the set S fX kXk
e
g is es
timate E
p
fX kXk
e
g fX kXk
eg of the domain D
p
fX
kXk
e
g fX kXk
eg $
Since
fx i fX C
then XtX
C
t
The sets
X
A
X kXk
e
X
F
X kXk
e
S
X kXk
e
are compact connected and with nonempty interiors Let
vX ln kXk
so that
denite
But positive deniteness of the function v is not required in the framework of the
practical stability and we may continue with testing it The results are
X
A
X kXk
e
v
mA
a
X
F
X kXk
e
v
mF
v
mE
F
for E
F
X
F
b
S S
X kXk
e
v
MS
v
MS
c
so that
vX ln kXk
v
mE
F
vY ln kY k
XY
E
F
E
F
Further
D
vX vX
kXk
X
T
x
x
x
sin
x
sin
x
x
x
x
sin
x
sin
x
X
so that
v
MAE
F
We use Theorem and we will determine the set S so that all the conditions
are satised It was shown above that the sets X
A
X
F
and S have the required
properties From and it follows that the function v obeys a and b
of the condition of the theorem The set S fX kXk
e
g should be
determined ie should be calculated so that the conditions and of the
theorem also hold which take the following form due to
v
MS
v
mA
t
t
v
MAS
t
that is that
t t
Hence for t
v
v
t
t
h
v
i
t
t t
For t follows
Now and determine the largest or equivalently the estimate
S E
p
s
X
A
X
F
of the form fX kXk
e
g as
S E
p
fX kXk
e
g fX kXk
eg fX kXk
e
g
Conclusion
The practical stability criteria permit an aggregation function v to be nondenite
with nonsemidenite Dini derivative The function v need not be continuous ev
erywhere The criteria are expressed in a very simple form in terms of extremal
values of v and D
v on compact sets They are suitable for direct calculations
The criteria show qualitative and quantitative tradeo among the sets X
A
X
F
E
the settling and nal time and the form of aggregation function v used The
choice is straightforward in the case of the sets being uniquely bounded where their
generating functions should be taken for the function v
Chapter
Comparison systems and
vector normbased Lyapunov
functions
Introductory comments and denitions
Presentation
This last chapter is devoted to the stability study of large scale systems whose
models are not completely specied either because of an undetermined disturbing
input or because the identication of some varying parameters is not possible This
is represented by the equation
dX
dt
gX d g
n
S
d
n
where X
n
and d
k
S
d
is a vector function corresponding to the modelling
misreading parameter or input disturbance
This book is devoted to nonlinear timeinvariant systems thus the disturbance d
is considered as nonlinear timeinvariant function say d dX however what
follows can be applied to general disturbance dX t without change provided that
a solution X of exists Note that the extension to functional dierential
equations has also been developed
In the following Xt t
X
d denotes as usual the solution of starting
from X
at time t t
and for a given function d S
d
For such large scale systems nding a Lyapunov function that provides conclu
sions independently of disturbance d is a rather di!cult task the introduction of
vector norms VN aims to construct overvaluations of the system behaviours so
that overvaluations no longer depend on disturbances and are described by lower
dX
dt
fX i
related to the practical stability concept belongs to that type of equation
with k In this case d it is an input disturbance and S
d
l
b Equation of Section describing Lurie systems and introduc
ing the absolute stability concept
dX
dt
AX fw
w CX Dfw
is also concerned with d f S
d
N
L In the absolute stability problem
the question is to prove asymptotic stability of x independently of the
actual variation law of the parameter function f in S
d
c In some cases equation can be written as
dX
dt
fX dX
where f is a known function and d S
d
n
is unknown but S
d
is known
The general idea of constructing particular Lyapunov functions on the basis of a
vector norm p of state X is described by the following line of arguments
pX is a vector of size k n each component of which is a scalar norm of
some components of X
pX obeys the overvaluing relation
pXt Zt for any d in S
d
with Zt governed by
dZ
dt
hZ
v vZ is a Lyapunov function for system
then try vpX as a Lyapunov function for system
Moreover studying equilibriums and domains corresponding to is a source
of information concerning the asymptotic and dynamic behaviours of the actual
disturbed large scale system
Constructing and working with such overvaluing systems is the question
we answer in this chapter As a particularly simple case system may be linear
timeinvariant
dZ
dt
MZ q
and then its explicit solution can easily be used for the evaluation of system
behaviours
The notion of comparison system originates in the theory of dierential and in
tegral inequalities which inequalities directly follow from the use of Lyapunovs
methods Since the beginning of the century many works have been devoted to
related topics see for example description and references in Chapter
The comparison concept has also been dened as a formal property of qual
itative concepts that we shall use in the sequel under the following reduced
expression
Denition Let two systems
S
dX
dt
gX d g
n
S
d
n
C
dZ
dt
hZX d h
k
n
S
d
n
and sets X
n
and Z
k
i Notation P CZA QSX B signies If Z has a property or concept
P for system C with regard to the ordered list of arguments A related to
concept P then X has the property or concept Q for system S with regard
to the ordered list of arguments B related to concept Q
ii CZA is a comparison system of SX B with regard to property or con
cept P if
P CZA P SX B
Example The rst Lyapunov method can be stated in term of comparison
systems since the asymptotic stability of Z for
dZ
dt
AZ
implies the asymptotic stability of X for
dX
dt
AX bX
lim
X
b
T
X bX
X
T
X
We can say that f g is a comparison system of f g with regard to the
asymptotic stability property and to the instability property
Example Let the system S be dened by
dX
dt
x
x
X X
Figure Trajectories of system
see Fig and the system C be dened by
dz
dt
z z z
Consider the positive denite function
zx x
x
then it veries equation C
z is an unstable equilibrium of system C and z
e
is an asymptotically
stable one see Fig since the solution of C is
zt t
z
z
z
z
exp t
t
Dfz
e
g
is the asymptotic stability domain of z
e
The convergence of z
to Z fz
e
g for z
implies that state X tends to the unit circle X fX
zX g for any initial condition X
DX
f g
Then asymptotic stability of Z for system C with the asymptotic stability
domain
implies asymptotic stability of X for system S with the asymptot
ic stability domain
f g or asymptotic stability C fg
asymptotic
stability S fX
x
x
g
f g and
C fz
e
g is a comparison system of S fX
x
x
g with
regard to the asymptotic stability property
C fz
e
g
is a comparison system of S fX
x
x
g
f g with regard to the asymptotic stability domain concept
Example We still consider systems and now related to exponential
stability Equation implies that z
e
has the domain D
e
of
exponential stability with regard to see Denition Chapter
Then C fz
e
gD
e
is a comparison system of
S fX
x
x
g fX
x
x
g with regard to
the exponential stability domain concept
Figure Trajectories of system
Dierential inequalities overvaluing systems
Basic results concerning dierential inequalities were provided by Wa-zewski
who gave the necessary and su!cient hypothesis ensuring that the solution of a
system described by
dX
dt
F tX
with initial condition X
at t
with function F tX verifying inequality
F tX GtX
is overvalued by the solution of the so called overvaluing system
dZ
dt
Gt Z
with initial condition Z
X
at t
or in other words conditions on functionGt Z
that guarantee
Zt t
Z
Xt t
X
These conditions and other linked results are the topic of this section
Denition a A function f
k
k
fV f
i
V
i k
is locally quasi
increasing on V
k
if and only if it is continuous wrt V V and for
any V v
i
and W w
i
in V v
i
w
i
and v
j
w
j
j f
i i kg implies f
i
V f
i
W
If V
k
f is globally quasiincreasing
b A function f
n
k
k
fXV f
i
XV
i k
is locally quasi
increasing with regard to the variable V on S V
n
k
if and only if it
is continuous wrt XV S V and for any X in S fX
k
k
is locally quasiincreasing on V
n
k
wrt
ftX V
that are involved in timevarying systems
Quasiincreasing is also called increasing with respect to the diagonal el
ements or uniformly nonsingular monotone or satisfying the
Wa-zewski conditions
Example a Any continuous function f
n
n
fX
%
B
B
B
B
B
B
&
h
x
x
n
j
j j
n
j
j
j h
x
x
x
n
j
n
j
j
n
j j
n
j h
n
x
x
n
'
C
C
C
C
C
C
A
X q
where
ij
and q are constant and h
i
does not depend on x
i
is quasiincreasing
on
n
b Any continuous function f
n
k
k
fXV M X V qX
withM X m
ij
X m
ij
X for i j is quasiincreasing with regard
to V
Denition Let and be nite or innite numbers and T be the
time interval
Let S be a compact connected set S
n
S
i The set S is T S
d
positively invariant for system
dX
dt
gX d
i X
S t
T t t
d S
d
Xt t
X
d S
ii The set S is S
d
positively invariant for if and only if it is
S
d
positively invariant
Denition The system
dOS
dt
hOS d h
n
S
d
n
is a T SS
d
local direct overvaluing system of system i
i the solutions OSt t
OS
d of and Xt t
X
d of exist for
to
OS
X
any t t
and any d S
d
iii S is T S
d
positively invariant for and
Remarks The mention direct referring to the existence of reverse overvaluing
systems that are used in the instability study shall be omitted in the
following
In the previous denitions # arguments T S or S
d
can be omitted
# if T and S
n
the overvaluing system is S
d
local
# if T the overvaluing system is SS
d
local
# if S
n
the overvaluing system is T S
d
local
# if and do not depend on disturbance d as in
then the overvaluing system is T S local
# if T S
n
and S
d
the overvaluing system is global
The two following theorems are direct adaptations of results by Bitsoris himself
referring to
Theorem Let two systems be with time continuous solutions
dX
dt
gX g
n
n
dOS
dt
hOS h
n
n
and suppose hX gX X
n
Then the system is a global overvaluing system of system i h is
globally quasiincreasing rrr
Theorem Consider the disturbed system and let V X
n
k
be
a continuous function with righthand gradient components of which are positive
denite functions such that along the solutions of the following holds
D
t
V X fX d hXV X d S
d
X
n
t
Suppose that system
dX
dt
gX d a
dV
dt
hXV b
has a unique solution continuous with regard to time for d S
d
Then is
dX
dt
gX d a
D
t
V X D
X
V
T
gX d hXV X b
i h is globally quasiincreasing with regard to the variable V rrr
Proof This result follows from
Denition System is a decoupled S
d
overvaluing system of system
if and only if it is a S
d
overvaluing system of with special form
dX
dt
gX d a
dV
dt
hV X b
which means that comes down to
D
t
V X fX d hV X d S
d
X
n
t
Corollary If system and function V satisfy the hypothesis of Theorem
then system is S
d
decoupled overvaluing system of i hV is quasi
increasing
In this case any motion Zt t
Z
of
dZ
dt
hZ
Z
V X
veries
Zt t
Z
V Xt t
X
d d S
d
This corollary obviously follows from Theorem and Denition It gives a way
to have a suitable stability study of system by comparing it with a simpler
system simpler because is korder k n and is not disturbed
However this result does not concern general S T S
d
local overvaluing
systems that are to be worked out in the next sections leading to stability do
mains estimations Before this we have to study some particular quasiincreasing
Denition and properties
Example shows that any function dened by fX MX q where M is a
constant matrix with nonnegative odiagonal elements and a constant vector is
quasiincreasing
Such matrices M m
ij
ij n
m
ij
i j present strong and useful
properties that are recalled here and that shall be useful in the following sections
Denition A constant real square matrix A is called
i a Zmatrix if and only if all its odiagonal elements are nonpositive which
is also denoted A Z and a Z
matrix if and only if odiagonal elements
are negative this is A Z
ii a Zmatrix i A Z which is also denoted A Z and a Z
matrix
if and only if A Z
A Z
iii a P matrix if and only if all its leading principal minors are positive which
is also denoted A P
P
A a
ij
ij n
a
a
a
n
a
n
a
nn
iv an M matrix Metzlerian or A M if and only if both A Z and A P
v the opposite of an M matrix or an M matrix or A M if and only
if A M
A characteristic of M matrices is to have nonnegative odiagonal elements and
a negative diagonal
Theorem Let M be a constant n n matrix with nonnegative elements
M Z Then the proposition M is a M matrix is equivalent to any of the
following propositions
Any eigenvalue of M has a negative real part
Any real eigenvalue of M is negative
M veries the Koteliansky conditions that is M P
m
m
m
m
m
m
m
m
m
m
m
m
k
m
k
m
kk
n
det M
There exists a constant vector X such that MX
There exists a constant vector X such that MX
There exists a diagonal matrix % with positive diagonal such that N M%
is a matrix with dominant negative principal diagonal ie
n
ii
X
j i
jn
ij
j i n with here jn
ij
j n
ij
M
exists and all its coecients are nonpositive ie M
If N Z and N M then N
exists
For any vector X X there exists an index i such that if Y MX
x
i
y
i
If dA denotes the diagonal of A then for each diagonal matrix R such that
R dA the inverse R
exists and R
A dA where is
the spectral radius ie the maximum of the moduli of all eigenvalues
There exists a permutation matrix P such that PMP
T
T
T
T
lower
triangular matrix T
uppertriangular T
and T
Z and have strictly neg
ative diagonal
rrr
Further properties of M matrices
If M M then there exists a negative real eigenvalue M of M called
the importance value of M such that the real part of any eigenvalue of M is
at most M
If M M there exists a nonnegative eigenvector uM associated to
M and called the importance vector of M
If M M and M is irreducible then uM and in Theorem one
can choose X uM in proposition and % diaguM in proposi
tion
We consider here the linear system dened as in by
dX
dt
MX q M n n is a M matrix q and X
n
q
Its solution is
Xt t
X
exp
M t t
!
