Relational Systems

Abstract

The considerations presented in the previous chapter may be extended for static relational systems, i.e. the systems described by relations which are not reduced to functions. Let us consider a static plant with input vector u ∈ U and output vector y ∈ Y, where U and Y are real number vector spaces. The plant is described by a relation

$$ u \rho y \triangleq R(u,y) \subset UxY $$

(3.1)

which may be called a relational knowledge representation of the plant. It is an extension of the traditional functional model y = ф(u) considered in the previous chapter. The description (3.1) given by an expert may have two practical interpretations:

• 1. The plant is deterministic, i.e. at every moment n y n = ф(u _n), but the expert has no full knowledge of the plant and for the given u he can determine only the set of possible outputs: D _y(u) ⊂ Y : {y ∈ Y : (u, y) ∈ R(u, y)}. For example, in one-dimensional case y = cu, the expert knows that c ₁ ≤ c ≤ c ₂; c ₁, c ₂ > 0. Then as the description of the plant he gives a relation presented in the following form

  $$ \left. \begin{gathered} c_1 u \leqslant y \leqslant c_2 u for u \geqslant 0 \hfill \\ c_2 u \leqslant y \leqslant c_1 u for u \leqslant 0 \hfill \\ \end{gathered} \right\}. $$

  (3.2)

  The situation is illustrated in Fig. 3.1, in which the set of points (u _n, y _n) is denoted.

• 2. The plant is not deterministic, which means that at different n we may observe different values y _n for the same values u _n. Then R(u, y) is a set of all possible points (u _n, y _n), denoted for the example (3.2) in Fig. 3.2.