Nonlinear Control Sy.. - Free

146 CHAPTER 5. FEEDBACK SYSTEMS
G-1
bk
u
Backstepping design for this class of systems can be approached using successive iterations
of the procedure used in the previous section. To simplify our notation, we consider, without
loss of generality, the third order system
= f(x) + 9(x)1
(5.24)
1 = 6 (5.25)
u (5.26)
and proceed to design a stabilizing control law. We first consider the first two "subsystems"
= f(x) + 9(x)1
(5.27)
S1 = 6 (5.28)
and assume that 1 = O(x1) is a stabilizing control law for the system
th = f(x) + g(x)q5(x)
Moreover, we also assume that V1 is the corresponding Lyapunov function for this subsystem.
The second-order system (5.27)-(5.28) can be seen as having the form (5.7)-(5.8) with
6 considered as an independent input. We can asymptotically stabilize this system using
the control law (5.21) and associated Lyapunov function V2:
6 = O(x, bl) = aa(x) [f (x) + 9(x) 1] - ax1 g(x) - k > 0
V2 = V1 + 2 [S1 - O(x)]2
We now iterate this process and view the third-order system given by the first three equations
as a more general version of (5.7)-(5.8) with
x= 1 z, f= f f(x)+09(x) 1 1, 9= L 0 J
Applying the backstepping algorithm once more, we obtain the stabilizing control law:
at(x) . aV2
u = ax xCC- V 9(x[) - k[f2[- O(x)], k > 0
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[a0
- [ a 22, a`1 ] [0 , 11T l - k[S2 - O(x,i)], k > o
ac,