Nonlinear Control Sy.. - Free

256 CHAPTER 10. FEEDBACK LINEARIZATION
is a function f : D C R" -> 1R1, namely, a mapping that assigns an n-dimensional vector
to every point in the n-dimensional space. Defined in this way, a vector field is an
n-dimensional "column." It is customary to label covector field to the transpose of a vector
field. Throughout this chapter we will assume that all functions encountered are sufficiently
smooth, that is, they have continuous partial derivatives of any required order.
10.1.1 Lie Derivative
When dealing with stability in the sense of Lyapunov we made frequent use of the notion
of "time derivative of a scalar function V along the trajectories of a system d = f (x)." As
we well know, given V : D -+ 1R and f (x), we have
f (x) = VV f (x) = LfV(x).
x
A slightly more abstract definition leads to the concept of Lie derivative.
Definition 10.1 Consider a scalar function h : D C 1R" -* JR and a vector field f : D C
]R" -4 1R" The Lie derivative of h with respect to f, denoted Lfh, is given by
Lfh(x) = ax f (x). (10.1)
Thus, going back to Lyapunov functions, V is merely the Lie derivative of V with respect
to f (x). The Lie derivative notation is usually preferred whenever higher order derivatives
need to be calculated. Notice that, given two vector fields f, g : D C ]R" -* 1R" we have that
and
and in the special case f = g,
Example 10.1 Let
Then, we have
Lfh(x) = axf(x), Lfh(x) = aX-9(x)
L9Lfh(x) = L9[Lfh(x)] = 2(8xh)g(x)
Lf Lf h(x) = L2 fh(x)
_ 8(Lfh)
1
h(x) = 2 (xi + x2)
8x f (x)
_ _ - 2
XIX2
2
A X ) 9(x) _ -x2 + x Jx2
, .