Nonlinear Control Systems - Lecture 26
Robust Backstepping
Consider the system
where
!"#$
%'& (*)
%
#$
%
& (
%
%
(+
%
(Note:
and
are bounded model uncertainties)
Let
,
be a stabilizing state feedback control for the upper
system. So we can assume there exists a positive definite
Lyapunov function,
-
such that
.
-
.
0/
,
1 #32"54687:9;
where
9
is positive definite.
As before we rewrite the system to emphasize the variation
about this stabilizing control. The rewritten system is
/
<
,
54<
/
=7
,
54>
?
and we let
@ =7
,
A B7
,
so we can rewrite the above system as
/
,
4@C @*
@ AD @*
Note that this is in the same form as the original system
with the exception that the top system is now asymptotically
stable when
@E2
.
In our original backstepping approach, we let
AF7
.
-
.
<G7H@
and considered the function
-JI
-
LK
M
@
%
to show that
-#N
&
2
and thereby establish asymptotic
stability for the entire system. However, this approach
assumed that
!
and
were both zero. If we keep these
uncertainties then our expression for
-JI
becomes
-I
&
7:9;G7H@
%
.
-
.
O1@>
We don’t know if
-I
will still be negative definite because
these two additional terms. This suggests that our earlier
approach to backstepping may be of limited value since it
relies on exact cancellations.
One way to deal with this is to treat the system as an
interconnection of two systems and then use the small gain
theorem to assure asymptotic stablility. The upper system,
P
/
Q<
,
R4Q<R@S1 #$@*
is already known to be asymptotically stable when
@TU2
with a known Lyapunv function
-
. This
-
is also an ISS-
Lyapunov function so that the upper system is input-to-state
stable. So there must exist class
V
functions
W
)
and class
VSX
function
Y
)
such that
Z
6\[B]_^:`
Y
)
ba
Zc
W
)
@
d
fe
What we’d like to do is choose
A
in the lower system so
that lower system is ISS. This would mean that there exist
VSX
function
Y
%
and
V
function
W
%
such that
@<Z
6[B]_^=`
Y
%
@_a
Z
W
%
d
fe
If we can show
W
)
W
%
&
K
, then we know that the intercon-
nected system is stable through the ISS small gain theorem.
Let’s establish this in a more formal manner. Consider the
original system as before
)
?
%
and as before we let
@gh7
,
M
where
,
stabilizes the
upper system. This means we can rewrite the above system
as
/
Q<
,
54*Q<R@C1
)
@*
@ i7
,
1
%
#$@>
Our choice of control guarantees there are class
V
and
VSX
functions
W
)
and
Y
)
such that
d
6j[B]k^=`
Y
)
!a
$Zc
W
)
@
d
ce
Now let’s require
l
)
#$@>
l
(*)
l
@
l
l
%
#$@>
l
m
l
@
l
%
where
mno
is class
V
.
Now consider the candidate ISS-Lyapunov function
-
@*p K
M
@
%
The directional derivative of
-
is
-
0@
@qr@<sB7
,
1
%
@*$
and we consider the control
tuH<@>pv7@w7xOs@*
where
x
is class
V
and
xO7@*pF7yxOs@*
. This means
@>xOs@*
is always positive if
@Tzg2
.
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