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Multiple Models Approach in Automation: Takagi-Sugeno Fuzzy Systems by Pierre Borne, Mohammed Chadli

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Appendix 3

Stability and Comparison Systems

Study of stability using comparison systems [BOR 74, GRU 76].

A3.1. Vector norms and overvaluing systems

A3.1.1. Definition of a vector norm

Let E = Rn be a vector space and E1, E2,…, Ek be subspaces of E:

Appendix3_image001.jpg

Let x be a vector of Rn for which the projection in subspace Ei is defined by:

[A3.1] Appendix3_image001.jpg

Note that pi(x) = p(xi) is a scalar norm defined over subspace Ei; it becomes the vector norm:

Appendix3_image001.jpg

If x and y are two vectors of space E and ∀i = 1, 2,..,k, then the following relations are verified:

Appendix3_image001.jpg

If k−1 subspaces of Ei are insufficient to define the whole of space E, the norm vector is said to be surjective. Furthermore, if every two subspaces Ei are disjoint, the vector norm is said to be regular:

Appendix3_image001.jpg

A3.1.2. Definition of a system overvalued from a continuous process

We define a process for which the evolution into free state is described by the following equation:

The origin is assumed to be the unique equilibrium point of system [A3.1] and this system ...

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