Nonlinear Control Sy.. - Free

10.1. MATHEMATICAL TOOLS 259
10.1.3 Diffeomorphism
Definition 10.3 : (Diffeomorphism) A function f : D C Rn -4 Rn is said to be a diffeomorphism
on D, or a local diffeomorphism, if
(i) it is continuously differentiable on D, and
(ii) its inverse f -1, defined by
exists and is continuously differentiable.
1-1(f(x)) = x dxED
The function f is said to be a global diffeomorphism if in addition
(i) D=R , and
(ii) limx', 11f (x) 11 = 00-
The following lemma is very useful when checking whether a function f (x) is a local
diffeomorphism.
Lemma 10.2 Let f (x) : D E lW -+ 1R" be continuously differentiable on D. If the Jacobian
matrix D f = V f is nonsingular at a point xo E D, then f (x) is a diffeomorphism in a
subset w C D.
Proof: An immediate consequence of the inverse function theorem.
10.1.4 Coordinate Transformations
For a variety of reasons it is often important to perform a coordinate change to transform a
state space realization into another. For example, for a linear time-invariant system of the
form
±=Ax+Bu
we can define a coordinate change z = Tx, where T is a nonsingular constant matrix. Thus,
in the new variable z the state space realization takes the form
z=TAT-1z+TBu = Az+Bu.