On the Markov property of quantised state measurement sequences
Considering the linear autonomous discrete-time system \[ {\mathbf x}(k+1)={\mathbf A}{\mathbf x}(k),\quad {\mathbf x}(0)=x_0, \] in which the state vector \({\mathbf x}\in \mathbb{R}^n\) can only be qualitatively measured, Lunze investigates whether the qualitative trajectory \[ [{\mathbf X}({\mathbf x}_0)]=([{\mathbf x}(0)],[{\mathbf x}(1)],\dots) \] is a Markov chain. (Here \([{\mathbf x}]\) is the ``number'' \({\mathbf z} \in \mathbb{Z}^n\) of the set \({\mathcal Q}_x\) to which \(\mathbf x\) belongs.) While a necessary and sufficient condition for this to be the case is given, it is shown that in general no representation of the form \[ [{\mathbf x}(k+1)]={\mathbf f}([{\mathbf x}(k)],{\mathbf v}(k)),\quad [{\mathbf x}(0)]=[{\mathbf x}_0] \] (where \({\mathbf v}(k)\) is a white random noise) of the quantised (or continuous-variable) system exists.
scientific article; zbMATH DE number 1325814