A Direct State-Space Realization of Discrete-Time Linear Parameter-Varying Input-Output Models
Johan Kon, Roland Tóth, Jeroen van de Wijdeven, Marcel Heertjes, Tom Oomen
TL;DR
This work provides a direct nonminimal state-space realization for discrete-time LPV-IO models, avoiding nonlinear dynamic dependencies in the coefficient functions. It establishes reachability under coprimeness and well-posedness conditions, while proving the realization is never observable but completely reconstructible in finite steps, ensuring no unstable hidden dynamics. The approach supports stability/dissipativity analysis and LPV data-driven controller design, with practical validation via LPV and LTI numerical examples. The findings clarify the trade-offs between minimality and direct-IO-structure realizations, offering a usable tool for controller synthesis and data-driven LPV analysis where shifted inputs/outputs can be leveraged directly.
Abstract
A minimal state-space (SS) realization of an identified linear parameter-varying (LPV) input-output (IO) model usually introduces dynamic and nonlinear dependency of the state-space coefficient functions, complicating stability analysis and controller synthesis. The aim of this paper is to introduce and analyze a direct SS realization of this IO model that avoids this nonlinear and dynamic dependency, at the cost of introducing a nonminimal state. It is shown that this direct SS realization 1) is reachable under a coprimeness condition on the coefficient functions of the IO model and a well-posedness condition on the model order, and 2) is never observable but that the unobservable directions converge to zero in a finite amount of steps, i.e., that the realization is reconstructible. The derived results are illustrated through numerical examples in both the LPV and LTI case.