Parametrization of scalar and multivariable stable linear dynamical systems using orthogonal Hessenberg matrices

Abstract:

If one wants to model a time series by a linear model one can use various possible set-ups, such as an AR (autoregressive), ARMA (autoregressive-moving average) or state space model. Here we argue that using a state-space model has various advantages, one of them being that one has a certain freedom to parameterize the model, due to the fact that one has freedom to choose a basis for the state space. One challenge is to parameterize stable systems, i.e. systems for which the dynamic matrix has all eigenvalues inside the unit circle. Here we show how this can be done using certain orthogonal upper Hessenberg matrices. The advantages of this approach will be mentioned. The method stems from an approach in which lossless systems are modelled using so-called Schur parameters as will be explained.