Stable Inversion of Piecewise Affine Systems with Application to Feedforward and Iterative Learning Control
Isaac A. Spiegel, Nard Strijbosch, Robin de Rozario, Tom Oomen, Kira Barton
TL;DR
This paper tackles stable inversion of piecewise affine ($PWA$) systems, addressing the challenge that inverses can be non-minimum phase ($NMP$) and hence produce unstable trajectories when naively inverted. It derives closed-form inverse formulas for a broad class of $PWA$ models and provides sufficient conditions for inverse existence and uniqueness, along with stable-inversion methods that decouple stable and unstable modes via a similarity transform. The authors integrate this stable inversion with Invert-Linearize ILC (ILILC) to yield robust feedforward control and validate the approach through simulations on a printhead positioning system. This work enables practical feedforward synthesis from $PWA$ models with $NMP$ components and highlights a pathway to improving tracking when model mismatch and repetitive disturbances are present.
Abstract
Model inversion is a fundamental technique in feedforward control. Unstable inverse models present a challenge in that useful feedforward control trajectories cannot be generated by directly propagating them. Stable inversion is a process for generating useful trajectories from unstable inverses by handling their stable and unstable modes separately. Piecewise affine (PWA) systems are a popular framework for modeling complicated dynamics. The primary contributions of this article are closed-form inverse formulas for a general class of PWA models, and stable inversion methods for these models. Both contributions leverage closed-form model representations to prove sufficient conditions for solution existence and uniqueness, and to develop solution computation methods. The result is implementable feedforward control synthesis from PWA models with either stable or unstable inverses. In practice, feedforward control alone may yield substantial tracking errors due to mismatch between the known system model and the unknowable complete system physics. Iterative learning control (ILC) is a technique for achieving robustness to model error in feedforward control. To demonstrate the primary contributions' validity and utility, this article also integrates PWA stable inversion with ILC in simulations based on a physical printhead positioning system.