Unconstrained Parametrizations of Discrete-Time Linear Input-Output Models
Stability and Dissipativity by Construction
Johan Kon (Eindhoven University of Technology)
Roland Toth (HUN-REN Institute for Computer Science and Control, Eindhoven University of Technology)
Jeroen van de Wijdeven (ASML)
Marcel Heertjes (Eindhoven University of Technology, ASML)
Tom Oomen (TU Delft - Team Jan-Willem van Wingerden, Eindhoven University of Technology)
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Abstract
It is often required that identified models exhibit certain stability and dissipativity properties, e.g., passivity or ℓ2-gain. The aim of this article is to develop an unconstrained parametrization of linear parameter-varying (LPV) input–output (IO) discrete-time (DT) models that guarantees stability/dissipativity by construction, i.e., the model is stable/dissipative for any choice of model parameters. To achieve this, it is shown that any quadratically stable/dissipative DT-LPV-IO model can be generated by a mapping of transformed coefficient functions that are constrained to the unit ball. The unit ball is reparameterized through a Cayley transformation, resulting in a fully unconstrained parameterization. These results immediately apply to linear time-varying IO models. In the linear time-invariant case, an unconstrained parameterization of all stable/dissipative DT transfer functions is obtained. The unconstrained parametrization enables, among others, the use of neural network coefficient functions in LPV system identification while guaranteeing stability and dissipativity.