Multiple Models Approach in Automation: Takagi-Sugeno Fuzzy Systems

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,…, E_k be subspaces of E:

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

[A3.1]

Note that p_i(x) = p(x_i) is a scalar norm defined over subspace E_i; it becomes the vector norm:

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

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

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 ...