Conditions for the Equivalence of ARMAX and ARX Systems

Abstract

It is shown that an autoregressive moving average with exogenous input (ARMAX) system can be represented as an autoregressive with exogenous input (ARX) model if and only if the transfer function from the noise port to the output port has no transmission zeros. A construction using the matrix fractional description of the system is used to prove this result. This construction shows that, by proper addition of sensor measurements and extending the order of the ARX model, accurate parameter estimates of systems driven by unmeasured disturbances can be obtained.