�
2
In
particular,
when
ψ
∞
0,
this
yields
the
definition
of
a
Lyapunov
function.
Finding,
for
a
given
supply
rate,
a
valid
storage
function
(or
at
least
proving
that
one
exists)
is
a
major
challenge
in
constructive
analysis
of
nonlinear
systems.
The
most
com-
mon
approach
is
based
on
considering
a
linearly
parameterized
subset
of
storage
function
candidates
V
defined
by
N
±
x
)
=
φ
q
V
q
(¯
V
=
{
V
(¯
x
)
,
(7.4)
q
=1
where
{
V
q
}
is
a
fixed
set
of
basis
functions,
and
φ
k
are
parameters
to
be
determined.
Here
every
element
of
V
is
considered
as
a
storage
function
candidate
,
and
one
wants
to
set
up
an
eﬃcient
search
for
the
values
of
φ
k
which
yield
a
function
V
satisfying
(7.3).
Example
7.1
Consider
the
finite
state
automata
defined
by
equations
(7.1)
with
value
sets
X
=
{
1
,
2
,
3
}
,
W
=
{
0
,
1
}
,
Y
=
{
0
,
1
}
,
and
with
dynamics
defined
by
f
(1
,
1)
=
2
, f
(2
,
1)
=
3
, f
(3
,
1)
=
1
, f
(1
,
0)
=
1
, f
(2
,
0)
=
2
, f
(3
,
0)
=
2
,
g
(1
,
1)
=
1
, g
(¯
w
)
=
0
�
(¯
w
)
≡
x,
¯
x,
¯
=
(1
,
1)
.
In
order
to
show
that
the
amount
of
1’s
in
the
output
is
never
much
larger
than
one
third
of
the
amount
of
1’s
in
the
input,
one
can
try
to
find
a
storage
function
V
with
supply
rate
ψ
(¯
w
)
=
w
−
3¯
y,
¯
¯
y.
Taking
three
basis
functions
V
1
,
V
2
,
V
3
defined
by
1
,
x
=
k,
¯
V
k
(¯
x
)
=
0
,
x
=
k,
¯
≡
the
conditions
imposed
on
φ
1
,
φ
2
,
φ
3
can
be
written
as
the
set
of
six
aﬃne
inequalities
(7.3),
two
of
which
(with
(¯
w
)
=
(1
,
0)
and
(¯
w
)
=
(2
,
0))
will
be
satisfied
automatically,
while
x,
¯
x,
¯
the
other
four
are
x,
¯
φ
2
−
φ
3
→
1
at
(¯
w
)
=
(3
,
0)
,
x,
¯
φ
2
−
φ
1
→ −
2
at
(¯
w
)
=
(1
,
1)
,
x,
¯
φ
3
−
φ
2
→
1
at
(¯
w
)
=
(2
,
1)
,
x,
¯
φ
1
−
φ
3
→
1
at
(¯
w
)
=
(3
,
1)
.