# Experimental Validation of a Guaranteed Nonlinear Model Predictive Control
It is based on the online exact linearization of the nonlinear model so as to formulate the optimization problem completely as a Quadratic Program (QP). It can be easily handled by linear solvers to decrease the computational complexity. Nevertheless, they assume that uncertainties, related to the system's modeling and the degree of correctness of measurements, are omitted. Consequently, a control strategy in a guaranteed way is required to ensure robustness toward all the uncertainties occurring in dynamic parameters' estimation. Currently, many researchers apply IA methods to synthesize a guaranteed linear or nonlinear MPC controllers [14; 15]. These guaranteed methods allow the calculation of smooth and safe input intervals over the prediction horizon, i.e., the control vector is expressed here as an interval-vector (or a box that is the Cartesian product of intervals). Appendix A gives a brief presentation outlining different IA tools. For instance, authors in [14] use IA methods to build a new nonlinear MPC law considering the discrete system. It is based on forward-backward contraction using a dynamic model to compute feasible and admissible inputs. This method allows authors to calculate a validated control for an inverse pendulum system. Other techniques include the adjoint method for sensitivity analysis enabling the handling of the rate of change of the system dynamics considering the control variable [6; 16; 17; 18]. Otherwise, this last approach is used to calculate the derivative of dynamics with respect to the input variable that must be limited between some bounds. In addition, a robust and guaranteed MPC of constrained discrete-time nonlinear systems with additive uncertainties is designed in [6; 17] to quantify different uncertainties. Moreover, a branch-and-bound algorithm was applied in [15] to synthesize a NMPC controller. This combinatorial optimization algorithm is promising but needs high computation time to obtain optimal solutions. In addition, a reliable nonlinear MPC via interval arithmetic is developed in [19]. This algorithm uses validated simulation and branching methods to create input interval-vectors (or input boxes) and respect the system's constraints by consecutive bisection of the initial feasible boxes of actuator's inputs. _Contributions_ : This paper focuses on developing a new validated and reliable nonlinear model predictive control (NMPC), which is more specifically designed using an uncertain mathematical model. It encompasses two main research dimensions. (i) This paper is an extension of our previous work introduced in [20] that aims to identify in a guaranteed way the inertial and frictional parameters of our new inverted pendulum (manufactured in our lab and shown in Figure 1). The IA and set-inversion tools have been deployed to express all the needed parameters as intervals instead of constants' values so that the coverage ratio between the physical reality and the mathematical model is significant. (ii) The second contribution lies in the synthesis of a validated NMPC strategy relying on the previous guaranteed identification to compute smooth and safe input intervals. These approaches have been investigated through various simulations with uncertain high-order ordinary differential equations (ODEs) describing the system's behavior. The validated simulations are performed using our Dynl'ke (Dynl'ke Library: [[https://perso.ensta-paris.fr/chapoutot/dynibex/](https://perso.ensta-paris.fr/chapoutot/dynibex/)]([https://perso.ensta-paris.fr/chapoutot/dynibex/](https://perso.ensta-paris.fr/chapoutot/dynibex/)) (accessed on 19 August 2021)) library (a sophisticated solver of ODEs using Runge-Kutta schemes [21]) and several experiments on the actual nonlinear inverted pendulum. The remainder of this paper is organized as follows: firstly, an overview of the direct dynamic model of our inverted pendulum is presented in Section 2. Secondly, Section 3 introduces the guaranteed NMPC approach based on IA.