Integrator Backstepping For Bounded Controls And Control Rates ...

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 2, FEBRUARY 1998 261
By using (21) of , (18) on M, and by imposing
1
sup x
fr(x)g = 1 R (36)
we get the following inequalities:
jy 0 (x)j 1)
jy0(x)j 1 aj([y 0 (x)])j (37)
a
1 jy 0 (x)j )
jy0(x)j(jyj+j 6 (x)j+r(x))
1 1 j([y 0 (x)])j (38)
max
R a ; 2 b
a + b(jyj + j6 (x)j)
2
1j([y 0 (x)])j: (39)
This yields, for all (x; y)
jy 0 (x)j 1 max R
a ; 2 M (y; 6 (x))
b
1j([y 0 (x)])j: (40)
Also, we have
minfR; jy 0 (x)jg R minf1; jy0(x)jg
R j([y 0 (x)])j: (41)
B. Proof of Theorem 1
1) Proof of Global Stability: We propose a Lyapunov function
W (x; y) which belongs to the family of Lyapunov functions described
in [3] and is “flattened” inside the set A 0 , as proposed in
[1]
W (x; y) :=V(x)+ K(y; 6 (x)); when (x; y) 2 A 6
0; when (x; y) 2 A 0 (42)
where K is given by (14). One can verify that W is C 1 , positive
definite, and proper. We next compute W _ in each of the two sets
A 6 and A 0 .
In the set A 0 we obtain, using A2)
_W (x; y) =L f V(x)+L g V(x)y0(x): (43)
Therefore, _W (x; y) is negative definite on A 0 , regardless of the
value of the control variable u.
In the set A 6 we obtain, with u = (x; y) given by (6)
_W (x; y) 0(x)+T(x; y) 0 [y 0 6 (x)]
1 M (y; 6 (x))([y 0 (x)]) (44)
where M is from (17) and
T (x; y) =[y0 6 (x)][L g V (x) +M(y; 6 (x))h(x; y)
0 (a +2bj 6 (x)j)(L f 6 (x) +L g 6 (x)y)]:
It remains to determine the negativeness of _W (x; y) on the set A 6 .
For this we observe that by completing the squares and using (30),
(34), and (35), we get, for all (x; y) 2 A 6
T (x; y) 1 3 (x) +jy06 (x)j
1[(c 2 1(c 0 + a + b) 2 + c 0 c 1 + c 2 2(a + b) 2
+(c 1 +c 2 +c 3)(a + b))jy 0 (x)j
+ c 2 M (y; 6 (x)) minfR; jy 0 (x)jg] (45)
where c 3 is given by the boundedness of L g 6 and 6 . Then, with
(40) and (41), we get more simply
T (x; y) 1 3 (x) +jy06 (x)jM(y; 6 (x)) 1j([y 0 (x)])j
1 1
[c 2 1(c 0 + a + b) 2 + c 0 c 1
+ c 2 2(a + b) 2 +(c 1 +c 2 +c 3 )(a + b)]
1 max R a ; 2 + c 2R : (46)
b
So by imposing that be large enough, we finally arrive at
_W (x; y) 0 2 (x)0 1jy06 (x)j
3 2
1M(y; 6 (x))j([y 0 (x)])j (47)
for all (x; y) 2 A 6 . This proves the negative definiteness of W _ (x; y)
on A 6 .
2) Construction of the Function : With the properties of the
function defined in (32), we can construct a C 1 positive definite
function (x; y) such that
(x; y) 1 j([y 0 (x)])j; when (x; y) 2 A6
4
1
4 (r(x)); when (x; y) 2 (48)
A0
and furthermore is constant outside some compact set. With this
choice for we obtain
where
S(x; y) =
ju 0 (x; y)j (x; y) =) _ W 0S(x; y) (49)
2
3 (x)+ a jy06 (x)j
4
1j([y 0 (x)])j; when (x; y) 2 A 6 (50)
(x); when (x; y) 2 A 0 .
3) Construction of the Functions U and : We define U as
U (x; y) := ln[1 + W (x; y)] (51)
where W is given by (42). We now show that there exists a function
(x; y) such that
• is C 1 and positive definite;
• for all (x; y) 2 IR n 2 IR
S(x; y)
(x; y)
1+W(x; y) ; (52)
• there exist CIR n 2 IR, a compact neighborhood of the origin,
and a constant >0such that for all (x; y) 2C
(x; y) [(x) +(y0(x)) 2 ]: (53)
First, because s 2 s(s) when jsj 1, and (33) holds, there is some
compact neighborhood C of the origin such that for all (x; y) 2C
j([y 0 (x)])j jy 0 (x)j;
1
8 ar(x)2 1 3 (x):
(54)
Let us now bound S(x; y) from below. We begin by observing from
(54) that on A 6 \C
a
4 jy06 (x)jj([y 0 (x)])j
1 4 a[(y 0 (x))2 0 r(x)jy 0 (x)j] (55)
1 4 a[ 1 2 (y 0 (x))2 0 1 2 r(x)2 ] (56)
1 8 a(y 0 (x))2 0 1 3 (x): (57)
Furthermore, because jy 0 (x)j r(x)on A 0 , we have from (54)
that
1
3 (x) 1 8 a(y 0 (x))2 (58)