Unconstrained Parameterization of Stable LPV Input-Output Models
with Application to System Identification
Johan Kon (Eindhoven University of Technology)
Jeroen Van De Wijdeven (ASML)
Dennis Bruijnen (Philips Research)
Roland Tóth (Computer and Automation Research Institute Hungarian Academy of Sciences, Eindhoven University of Technology)
Marcel Heertjes (ASML, Eindhoven University of Technology)
Tom Oomen (TU Delft - Team Jan-Willem van Wingerden, Eindhoven University of Technology)
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Abstract
Ensuring stability of discrete-time (DT) linear parameter-varying (LPV) input-output (IO) models estimated via system identification methods is a challenging problem as known stability constraints can only be numerically verified, e.g., through solving Linear Matrix Inequalities. In this paper, an unconstrained DT-LPV-IO parameterization is developed which gives a stable model for any choice of model parameters. To achieve this, it is shown that all quadratically stable DT-LPV-IO models can be generated by a mapping of transformed coefficient functions that are constrained to the unit ball, i.e., a small-gain condition. The unit ball is then reparameterized through a Cayley transformation, resulting in an unconstrained parameterization of all quadratically stable DT-LPV-IO models. As a special case, an unconstrained parameterization of all stable DT linear time-invariant transfer functions is obtained. Identification using the stable DT-LPV-IO model with neural network coefficient functions is demonstrated on a simulation example of a parameter-varying mass-damper-spring system.