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Diffeomorphism

18/04/2011

Diffeomorphism is very useful! Also see my question on math stackexchange.

I learned this concept in feedback linearization problems, where it is used for state transformation. For example, x_1,...,x_n are states of a system. Now we have a new set of variables: z_1,...,z_n. The problem is: whether z_1,...,z_n can be a set of states of the system? The answer is: as long as the function z_1(x),...,z_n(x) is diffeomorphism, it can be used to represent the system.

It should be noted that: from z_1(x),...,z_n(x) to x_1,...,x_n, DOFs are preserved. But the value region changes. If x_i \in \Omega _1 then z_i \in \Omega_2. So my question on math stackexchange is not proper. Because x_1,...,x_n may not be an arbitrary vector in \mathbb{R}^n. And of course, z_1(x),...,z_n(x) may not be an arbitrary vector in \mathbb{R}^n. But keep in mind that the DOFs are preserved.

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