# Experimental Validation of a Guaranteed Nonlinear Model Predictive Control
## 3 Guaranteed Controller Design via Validated Simulation
### Validated Nonlinear Model Predictive Control
#### 3.2.2 Interval Cost Function Formulation
Indeed, the dynamic model is proven to be as close as possible to the system's actual behavior (see Section 4.1). Figure 6 shows the measured pendulum angle for the tested scenarios, starting from \(0^{\circ}\) and \(150^{\circ}\). The stabilization is performed in an open-loop by incorporating the simulated input data into the DC motor controller (the computed inputs are drawn in Figures 4c and 5c). Compared to the simulation results, the calculated control inputs stabilize the inverted pendulum in the steady-state regime. The second scenario gives coherent results enabling the pendulum to be stabilized around the target position \(\pi\) thanks to the validated NMPC (see Figure 6 (Right)). However, for the first scenario (starting from \(0^{\circ}\)), we have some synchronization problems between the DC motor sampling period (around 0.028 s) and the sampling horizon of the NMPC method \(T_{c}\). This is due to the lack of good deeds that the system's actuator executes at the right moment and its sensibility to weak signals

Figure 5: Validated NMPC results starting from \([\mathbf{y}_{3_{in}}]=[149^{\circ};\,150^{\circ}]\): (**a**) pendulum angle tubes; (**b**) the corresponding pendulum speed tubes; (**c**) safe and optimal input Intervals \([\mathbf{u}_{1}]\). that destabilize it. Despite these issues, the proposed NMPC enables the system to reach another equilibrium position around \(\pi/2\) (see Figure 6 (Left)), so we will try to handle this issue separately. To sum up, the validated NMPC seems more promising for enabling the system to successfully converge to the reference target with high accuracy and small settling time. The imposed state and control bounds are respected, critical, and necessary for the pendulum's stability and safety. Nevertheless, the computation time of the domain \([\mathbf{u}_{k}]\) is remarkable due to a large number of bisections of the initial input domain and the high computation time required by validated simulations. ## 5 Conclusions and Future Works

The research presented in this paper focuses on developing a reliable and validated nonlinear model predictive control (NMPC), which is more specifically designed using an uncertain mathematical model. The proposed controller is based on validated simulation mixed with interval analysis techniques to handle the system's nonlinearities and uncertainties. Two different procedures are employed: branching procedure to calculate the valid input intervals from an uncertain nonlinear model, allowing the convergence to the set-point interval and the system constraints' insurance. From the generated branches, the optimization procedure is applied to determine the sub-optimal control variable. The efficiency and robustness of the proposed method are validated through numerical and experimental tests using a new nonlinear inverted pendulum. As regards ongoing works, we will focus on studying two issues: firstly, the computational complexity improvement of the validated simulation methods by taking into consideration contraction methods and parallel computing strategies; secondly, the projection of the validated NMPC on a complex mechatronic system (ROV--Remotely Operated underwater Vehicle). It allows us to perform some tasks (trajectory tracking) in the aquatic environment while respecting all the system's constraints. In addition, another relevant aspect enabling completing a system's autonomy is online estimation of the model's parameters via nonlinear observers and interval methods with respect to the environmental conditions, e.g., aerodynamic coefficients. This issue allows for reaching high accuracy and stability of the trajectory tracking task. Conceptualization, M.F. ; software, J.A.d.S. ; validation, M.F. ; writing--review and editing, M.F. and J.A.d.S. ; supervision, J.A.d.S. Both authors have read and agreed to the published version of the manuscript. This research work is funded by the French LABEX Digicosme project \"PREGARI\" and DGA \"CRa 1835 -DGA 2018-2021-Projet F\". Not applicable.