Nonlinear Control Sy.. - Free

148 CHAPTER 5. FEEDBACK SYSTEMS
In our case
a¢(xl )
ax,
avl
-(1 + 2axi)
=> O(xl, x2) = -(1 + 2ax1) [axi + x2] - x1 - [X2 + x1 + axi]
where we have chosen k = 1. We now move on to the final step, in which we consider the
third order system as a special case of (5.7)-(5.8) with
x= I
f
x1
x2 J J L 0 J
L f(x1) Og(x1)x2
From the results in the previous section we have that
U = 090(x1, x2) [f (XI) + 9(xl)x2] + aO(xl, x2) X3 _ aV2
axi 09x2 09x2
is a stabilizing control law with associated Lyapunov function
V +21
5.3.2 Strict Feedback Systems
We now consider systems of the form
f (x) +CC 9(x)S1
CC CC
S1 = fl (x, Si) + 91(x, Sl)SCC2
2 = f2(x,e1,6) + 92(x,6,6)e3
9
A,I,
- k[x3 - W(xl, x2)], k > 0
Sk-1 = A-1 (X, 6, S2, Sk-1) + 9k-1(x, SL 61- ' GAG
6 = A(Xi61 2) ...7
G)+A(xi6)6,.,Ck)U
where x E R', Ct E R, and fi, g= are smooth, for all i = 1, , k. Systems of this form
are called strict feedback systems because the nonlinearities f, fzf and gz depend only on
the variables x, C1, that are fed back. Strict feedback systems are also called triangular
0