Quasi-positive realization of externally positive discrete systems

The positive realization of externally positive systems described by their transfer matrix H(z) is a difficult problem whose analysis has been completed only recently on the basis of algebraic and geometric approaches that underline the many constraints that condition the solution of this problem. These constraints concern the minimal order of the obtained models, the minimality of their parameterizations and even their existence; the positive realization becomes, on the contrary, a standard task under simple sufficiency conditions on H(z) that, however, are seldom fulfilled in practical applications. This paper introduces the new category of quasi-positive state-space models that limit the assumptions on the non negativeness of the state-space trajectories to the only trajectories that can be actually generated by the system under nonnegative controls since all other behaviors are of no interest in control problems. 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 coefficients in H(z).