Nonlinear Control Sy.. - Free

5.3. BACKSTEPPING: MORE GENERAL CASES 149
systems. We begin our discussion considering the special case where the system is of order
one (equivalently, k = 1 in the system defined above):
= f(x)+g(x)
= fa(x,0+9a(x,0u.
(5.29)
(5.30)
Assuming that the x subsystem (5.29) satisfies assumptions (i) and (ii) of the backstepping
procedure in Section 5.2, we now endeavor to stabilize (5.29)-(5.30). This system reduces to
the integrator backstepping of Section 5.2 in the special case where f" (x, ) - 0, ga(x, ) - 1.
To avoid trivialities we assume that this is not the case. If ga(x, ) # 0 over the domain of
interest, then we can define
U = O(x, S) def
9.(x, 1 S)
[Ul - Mx' S)].
Substituting (5.31) into (5.30) we obtain the modified system
Cx = f (x) + 9(xX
41 =
t1
(5.31)
(5.32)
(5.33)
which is of the form (5.7)-(5.8). It then follows that, using (5.21), (5.17), and (5.31), the
stabilizing control law and associated Lyapunov function are
9a(x, )
{[f(x)+g(x)] - 19(x) - k1 [ - fi(x)] - fa(x, kl > 0
V2 = V2 (x, S) = Vl (x) + 2 [S -
We now generalize these ideas by moving one step further and considering the system
[[ = f (x) +[ 9(x)1
[ C
S[[1 = f1(x,S[[1)[[+ 91(x,Sl)S2
S2 = f2(x,S1,S2) + 92(x,1,&)6
which can be seen as a special case of (5.29)-(5.30) with
x= l C1
J
, e=e2, l
f
`9 1 ] , 9 = [ 0 ],fa=f2, 9a=92
f, 91
O(x)]2.
(5.34)
(5.35)