Realisation problem for weakly positive linear systems (Q1840932)

The author considers a descriptor discrete-time linear system having the representation \[ Ex_{i+1}= Ax_i+ Bu_i,\quad y_i= Cx_i,\quad i\in\mathbb{Z}_+, \] where \(E,A\in \mathbb{R}^{n\times n}\), \(B\in \mathbb{R}^{n\times m}\), \(C\in \mathbb{R}^{p\times n}\), \(\text{det }E= 0\) and \(\text{det }Ez- A\neq 0\) for some \(z\in\mathbb{C}\). Such a system is called weakly positive if the matrices \(E\), \(A\), \(B\), \(C\) have nonnegative entries; in this case it is said to be a positive realization of a given rational matrix \(T(z)\in \mathbb{R}^{p\times m}(z)\) if \(T(z)= C(Ez- A)^{-1}B\). Canonical forms for weakly positive linear systems are introduced and sufficient conditions for the existence of positive realizations in canonical forms are obtained; these conditions are expressed in terms of the signs of the coefficients of the entries of \(T(z)\). Explicit formulae for the positive realizations in canonical forms are obtained.