Dynamical Systems

Abstract

The relational knowledge representation for the dynamical plant may have the form analogous to that for the static plant presented in Sect. 3.1. The deterministic dynamical plant is described by the equations

$$ \left. \begin{gathered} s_{n + 1} = f(s_n ,u_n ), \hfill \\ y_n = \eta (s_n ) \hfill \\ \end{gathered} \right\} $$

(1)

where n denotes the discrete time and s _n ∈ S, u _n ∈ U, y _n ∈ Y are the state, the input and the output vectors, respectively. In the relational dynamical plants the functions f and ν are replaced by relations

$$ \left. \begin{gathered} R_I (u_n ,s_n ,s_{n + 1} ) \subseteq U x S x S, \hfill \\ R_{II} (s_n ,y_n ) \subseteq S x Y. \hfill \\ \end{gathered} \right\} $$

(1)

The relations R _I and R _II form a relational knowledge representation of the dynamical plant. For a nonstationary plant the relations R _I and R _II depend on n . The relations R _I and R _II may have the form of equalities and/or inequalities concerning the components of the respective vectors. In particular the relations are described by inequalities

$$ \begin{gathered} f_1 (u_n ,s_n ) \leqslant s_{n + 1} \leqslant f_2 (s_n ,u_n ), \hfill \\ \eta _1 (s_n ) \leqslant y_n \leqslant \eta _2 (s_n ), \hfill \\ \end{gathered} $$

i.e. by a set of inequalities for the respective components of the vectors. The formulations of the analysis and decision problems may be similar to those in Sect. 3.2. Let us assume that $s_0 \in D_{s0} \subset S.$