NMPC via qLPV models and Taylor-based Scheduling Parameter Extrapolation: A Cartesian Robot Case Study

Abstract

In this brief paper, we present an overview of recent advances on Model Predictive Control (MPC) synthesis for nonlinear systems using quasi-Linear Parameter Varying (qLPV) embeddings. For such, we consider a highly nonlinear Cartesian robot benchmark as a case study. Specifically, we advocate on the use of recursive Taylor-based extrapolation maps to generate accurate estimates for the future trajectories of the qLPV scheduling parameters, as shown in recent findings. We show how these estimates can be used to enhance and fasten the corresponding MPC algorithms, offering comparable performances to state-of-the-art techniques, while maintaining relieved numerical burden during the implementation. Through realistic simulations of the Cartesian robot, we demonstrate the effectiveness and the real-time capabilities of the discussed method, which is tested against widely acknowledged techniques (the SQP qLPV MPC framework, and the CasADi NMPC solver). I. INTRODUCTIONModel Predictive Control (MPC) is a very wide-spread method for the regulation of constrained processes [1]. Over the last decades, it has had considerable research interest, with extensions developed for a wide variety of systems and settings. The theoretical establishment of MPC corresponds to the formal guarantees of recursive feasibility of the optimisation and closed-loop stability [2].Many works have studied the application of MPC for nonlinear systems (NMPC), e.g. [3], [4]. Nevertheless, such algorithms are usually not trivial and their corresponding online implementation comes at the cost of increases numerical burden due to the inherent nonlinear predictions, which complicates real-time applications. Until the late 10s, even the most efficient NMPC algorithms displayed exponential complexity growth w.r.t. system size. Recent tools have shown how these algorithms can be fastened [5], [6], yet through approximations of the nonlinear optimisation.Anyhow, recent advances have shown how exact NMPC solutions with real-time capabilities can be generated through quasi-Linear Parameter Varying (qLPV) embeddings, see [7], [8]; compare also to the survey [9] and references therein. LPV models are able to describe nonlinear and time-varying behaviours under linear dynamics structures [10], which depend on known, bounded scheduling parameters ρ. *This work has been supported by Campus France (Eiffel Scholarship), by CNRS ("Investissements d'Avenir", ANR-15-IDEX-02), and by CNPq (304032/2019 − 0 researcher grant and PIBIC scientific initiation scholarship). The Authors thank J.P. Jordanou for his helpful insights on CasADi.

A Robust Nonlinear Tracking MPC using qLPV Embedding and Zonotopic Uncertainty Propagation

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