The Existence of a Discrete-Event Representation of Linear Continuous-Variable Systems with Quantised State Measurements

Abstract This paper is concerned with the representation of stable linear discrete-time continuous-variable systems with quantised state measurements. It is assumed that the state space IRn is partitioned into disjoint regions Q x and that the number z of the region Q x ( z ) to which the state x belongs at time k is used as the qualitative value of x ( k ): [ x ] = z . For a given z (0) = [ x (0)] the system can generate different qualitative trajectories [ X ] = [ x (0)], [ x (1)], [ x (2)]… The paper presents a necessary and sufficient condition under which the sequence [ X ] is a Markov chain. This condition is rather restrictive and shows that there is, in general, no representation of the qualitative behaviour of linear systems by some stochastic automaton.