Outils et modèles mathématiques pour l'automatique. l'analyse de systemès et le traitement du signal

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

“…We prove Theorem 1 by using known results on the relationship between input-output equations and bilinear state-space representations. This relationship was explored in [24] in DT case, and in [25,30,31] for CT case. Note that bilinear state-space representations can be viewed as a subclass of LPV-SSA without control inputs, if we view scheduling signals as inputs, so the use of bilinear systems theory is quite natural in this context.…”

“…However, despite the existence of a rich literature on bilinear systems, we have not found such algorithms in the literature. The results of [30,31], which use differential algebra, suggest that in the CT case, input-output equations for bilinear state-space representations could, in principle, be obtained by computing formal derivatives of the inputs and outputs and then eliminating the state variable. However, no explicit algorithm has been formulated so far, to the best of our knowledge.…”

“…However, no explicit algorithm has been formulated so far, to the best of our knowledge. The analysis of the proofs of [30,31] does not suggest an obvious way to formulate such algorithms. The problem of designing algorithms for the opposite transformation, i.e., for converting input-output equations to bilinear state-space representations is not obvious either.…”

“…The problem of designing algorithms for the opposite transformation, i.e., for converting input-output equations to bilinear state-space representations is not obvious either. The proofs of [24,25,30,31] do not suggest any easy way to formulate such algorithms.…”

“…This being said, the proof of Theorem 1 is partially constructive. In particu-lar, it is possible to use the proof of Theorem 1 and the proofs from [24,25,30,31] to perform the transformation from an LPV-ARX to an LPV-SSA and back for simple examples. Such an illustrating example is presented in Subsection 4.5 of Section 4.…”