Qualitative properties of general first order matrix difference systems

Abstract

This paper deals with a discrete - time control system of the form $X(n+1) = AX(n)B$. We present the general solution $X(n)$ in terms of two fundamental matrix solutions of $X(n+1) = AX(n)$ and $X(n+1) = B*X(n)$. We then look at controllability and observability of such systems. The important issue of stability analysis is also taken into consideration. We found out that with the $A$ matrix having eigenvalues outside the unit disk, we can select a $B$ matrix such that the final solution is stable and more importantly asymptotically stable. Proven theorems are presented as well as several examples to illustrate the theoretical results.