LPV-ARX representations of LPV state-space models with affine dependence

In this paper, we show that an input–output map can be realized by a linear parameter-varying (LPV) state-space representation with an affine and static dependence on the scheduling variables, if and only if this input–output map satisfies certain LPV autoregressive input–output equations. The latter class of equations is linear in the derivatives (for continuous-time) or time-shifts (for discrete-time) of the outputs and control inputs, while the coefficients of this linear equation are polynomials of the shifts of the scheduling variable in discrete-time, or of the high-order derivatives of the scheduling variable in continuous-time. This result is a generalization of the well-known equivalence between linear state-space representations and autoregressive input–output models. Moreover, this result extends the results of Tóth (2010) on LPV state-space representations with a dynamic and meromorphic dependence on the scheduling variables to LPV state-space representations with a static and affine dependence on the scheduling variables.