# 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
Since the filtering and branching algorithm begins from the admissible input domain \([-8.05\) N\(\cdot\)m; \(8.05\) N\(\cdot\)m\(]\), it can lead to approximately \(64^{10}\) branches with \(N_{p}=10\). The interval cost function is computed only in the optimization process using these weighting matrices \(\mathbf{Q}=diag[1000,1000,1000,1000]\) and \(\mathbf{R}=1\). Several software tools have been developed to simplify the resolution of the validated NMPC strategy via interval analysis. These tools are globally in the form of libraries to be integrated into a program (e.g., Dynlbex library [21], Numerica [25] or RealPaver [26]). However, it has been shown that these solvers are not much more practical due to their high computation time. In other words, the fast convergence of this kind of solvers is not guaranteed over a finite time (i.e., extreme long response time to get the solution of the problem). That's why the real-time validation of this guaranteed approach is not easy to realize. To do so, it is divided into two steps: a numerically offline calculation of the input variables via validated simulations and then the injection of these inputs into the physical pendulum. Two scenarios have been executed with different initial conditions on the angular pendulum position \([\mathbf{y}_{3_{\text{iii}}}]\) to discuss the capabilities of the proposed controller. Figure 4a-c display the results of the first scenario starting from \([\mathbf{y}_{3_{\text{iii}}}]=[0^{\circ};\,0^{\circ}]\) and Figure 5a-c show those of the second one starting from \([\mathbf{y}_{3_{\text{iii}}}]=[149^{\circ};\,150^{\circ}]\). As can be seen from Figure 4a, the pendulum arm starts from the downward position, and it is stabilized via the validated NMPC in its vertical position interval \([\mathbf{y}_{r}]\) with a small settling-time (around \(t_{7_{\text{yy}}}\approx 0.18\) s). Similarly, the simulation has also been executed when the initial position is close to the desired position \(\pi\). The generated enclosures of the inverted pendulum position are plotted in Figure 5a. As the previous results, the validated NMPC succeeds in stabilizing the pendulum feasibly in its terminal position. On the other hand, the calculated tubes of the pendulum's velocity are depicted in Figures 4b and 5b. As can be noticed in both of these scenarios, the validated velocity respects the imposed constraints and stagnate around zero when the pendulum reaches its

Figure 4: Validated NMPC results starting from \([\mathbf{y}_{3_{\text{iii}}}]=[0^{\circ};\,0^{\circ}]:(\mathbf{a})\) pendulum angle tubes; (\(\mathbf{b}\)) the corresponding pendulum speed tubes; (\(\mathbf{c}\)) safe and optimal input Intervals \([\mathbf{u}_{1}]\). final desired position. In addition, the torques applied by the DC motor for each tested scenario are shown in Figures 4c and 5c. The relationship between the motor's velocity and torque is given by Equation (3). In the first case, a strong input variable with the bang-bang process is applied to enable the pendulum to catch the equilibrium and stabilize in its terminal reference interval (see Figure 4c). Since the pendulum is stabilized, the torque applied by the system's actuator changes the sign to maintain the reference interval. In the second case, the actuator applies maximum inputs for a short time to try to reach the desired interval (see Figure 5c). All the calculated torques fulfill the system's intrinsic constraints. #### 4.2.2 Experimental Validation Using the Inverted Pendulum

To further show the usefulness of the proposed controller, it is tested on the experimental platform (Figure 1). As mentioned before, the real-time implementation of the validated strategy cannot be accomplished due to the high computation time of interval methods. In addition, we are currently working to improve this point by adopting parallel computation and relaxation methods. Consequently, we inject the computed input components into the control part of the pendulum that must somehow behave as the simulated model.