Qualitative modelling of continuous-variable systems by means of non-deterministic automata

Abstract

This paper considers the problem of qualitative modelling of discrete-time continuous-variable dynamical systems, for which only a quantised measurement [x (k)] of the state x(k) is available. The qualitative model has to describe the qualitative trajectory x(1), x(2), … for given qualitative initial state x(0) and qualitative input sequence. First, it is shown that the qualitative trajectory of the system is ambiguous. Hence, the qualitative model has to be non-deterministic. Secondly, it is shown that non-deterministic automata provide reasonable qualitative models of the continuous-variable system. The relation between the automaton and the given system shows what knowledge about the system has to be available if the qualitative model is to be set up. Thirdly, we propose to use stochastic automata, which provide a means for weighting each state concerning its appearance on the qualitative trajectory of the continuous-variable system. On this basis, the set of spurious solutions, which exist for any qualitative model, can be reduced. The suitability of the model becomes obvious by designing a qualitative controller. The results are illustrated by the problem of stabilising an ‘inverted pendulum’.