The periodical control of discrete-event dynamic system (Q2726009)

The authors consider the discrete-event dynamic system (DEDS) with finite certainty NEWLINE\[NEWLINEX(k)= X(k- 1)\cdot A\oplus U(k)\cdot B,\quad Y(k)= X(k)\cdot C,\tag{1}NEWLINE\]NEWLINE which is derived by \textit{G. Cohen}, \textit{D. Dubois}, \textit{J. P. Quadrat}, and \textit{M. Viot} [IEEE Trans. Autom. Control 30, 210-220 (1985; Zbl 0557.93005)] by using maximal algebra and Petri nets.NEWLINENEWLINENEWLINEAssuming that the matrix \(A\) is \(\lambda\)-periodical and every state of (1) is controllable, the performance of the DEDS (1) is discussed when applying periodical inputs of the form \(U(k+ 1)= \mu U(k)\). The following results are proved in the paper:NEWLINENEWLINENEWLINE(1) If the input is periodical, then so is the state vector of (1),NEWLINENEWLINENEWLINE(2) if the period \(\mu\) of the state vector of (1) is larger than the period \(\lambda\), \(\mu>\lambda\), then the period state vector is still \(\mu\) and the DEDS (1) will have good insensitivity from disturbance to some extent,NEWLINENEWLINENEWLINE(3) if \(\mu= \lambda\) holds, then the utilization is the best,NEWLINENEWLINENEWLINE(4) if \(\mu< \lambda\) holds, then the DEDS (1) is overloaded and will have a congestion.NEWLINENEWLINENEWLINEA simulation for the system \(X(k+1)= X(k)\left(\begin{smallmatrix} \varepsilon & 1\\ 1 & \varepsilon\end{smallmatrix}\right)\oplus U(k)\cdot (0\varepsilon)\) with input \(U(k+1)= 1.5\cdot U(k)\) is also given.