Nonlinear Control Sy.. - Free

264 CHAPTER 10. FEEDBACK LINEARIZATION
The following result, known as the Frobenius theorem, will be very important in later sections.
Theorem 10.1 (Frobenius Theorem) Let fl, f2i , fp be a set of linearly independent vector
fields. The set is completely integrable if and only if it is involutive.
Proof: the proof is omitted. See Reference [36].
Example 10.7 [36] Consider the set of partial differential equations
or
with
which can be written as
Oh
2x3
Oh
= 0
C7x1 - 8x2
Oh 8h i9h
-x1-- -2x2a2 +x3ax 3
2x3 -X
8h 8h oh -1 -2X2 I =
Vh[f1 f2]
0 x3
2X3 -xl
fl = -1 , f2 = -2x2
0 x3
= 0
To determine whether the set of partial differential equations is solvable or, equivalently,
whether [fl, f2] is completely integrable, we consider the distribution A defined as follows:
2x3 -xl
= span -1 , -2X2
0 x3
It can be checked that A has dimension 2 everywhere on the set D defined by D = {x E IIF3
xi + x2 # 0}. Computing the Lie bracket [fl, f2], we obtain
(fl,
-4x3
fz] = 2
0
0