Positive output/state maps and quasi-positive realization of MIMO discrete systems

The positive realization of externally positive systems described by their transfer matrix G(z)G(z) is a complex problem whose analysis has been completed only recently, for the SISO case, on the basis of algebraic and geometric approaches that underline the many constraints that condition its solution. These constraints concern the minimal order of the obtained models, the minimality of their parameterizations and even their existence. This paper considers the new category of quasi-positive   state-space models introduced in Guidorzi (2014) that limit the assumptions on the nonnegativeness of the state-space trajectories to the only trajectories that can be actually generated by the system under nonnegative controls. It is shown that all externally positive systems admit a quasi-positive minimally parameterized state-space realization whose existence is not conditioned by restrictions on the signs of the parameters of the polynomials in G(z)G(z).