Unconstrained Parametrizations of Discrete-Time Linear Input–Output Models: Stability and Dissipativity by Construction

It is often required that identified models exhibit certain stability and dissipativity properties, e.g., passivity or <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _{2}$</tex-math></inline-formula>-gain. The aim of this paper 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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">any</i> 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.