Stability Analysis of Closed Loops of non-linear Systems and LPV Controllers designed using approximated Quasi-LPV Systems

This paper proposes an approach for the stability analysis of feedback configurations consisting of a non-linear system and a linear parameter-varying (LPV) controller. The LPV controller synthesis for non-linear systems is commonly based on a Quasi-LPV representation of a non-linear system. With regard to performance oriented controller design, the approach proposed in this contribution considers the case where the controller is designed using an approximation of a Quasi-LPV representation of the non-linear system. As a consequence, the resulting controller does not necessarily stabilize the non-linear system. Therefore a method to prove the stability of the aforementioned closed loop is introduced. Since the proposed method is based on a Quasi-LPV representation of the closed loop, it results in a feasibility problem for a Quasi-LPV system globally representing the closed loop or otherwise in a simple convex optimization problem. In the latter case, the solution of the convex optimization problem defines an estimate of the Region of Attraction (ROA) of the non-linear closed loop. In both cases the resulting program can be solved using standard tools. Finally the application to a landing vehicle controlled using a 2-DOF LPV-H∞ controller shows the benefits of the proposed method.